==> acta-info.txt <== Title: Finite element exterior calculus, homological techniques, and applications Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: Acta Numerica 15 (2006), pp. 1-155 Status: Published Abstract: Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Laplacian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners. Keywords: finite element, exterior calculus, mixed method, Hodge, de Rham Subj. class.: 65N30, 58A10, 58A12, 58A14, 65N12, 65N15, 65N25, 65N55, 74S05, 78M20 URL: http://ima.umn.edu/~arnold/papers/acta.pdf ==> bdlayer2-info.txt <== Title: Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model Authors: Douglas N. Arnold and Richard S. Falk Source: SIAM J. Math. Anal. 27 (1996), pp. 486-514 Status: Published Abstract: We investigate the structure of the solution of the Reissner-Mindlin plate equations in its dependence on the plate thickness for various boundary conditions, developing asymptotic expansions in powers of the plate thickness for the main physical quantities. These expansions are uniform up to the boundary for the transverse displacement, but for other variables there is a boundary layer, whose strength depends on the boundary conditions. We give rigorous error bounds for the errors in the expansions in Sobolev norms and make various applications. Keywords: Reissner, Mindlin, plate, boundary layer Subj. class.: 73K10, 73K25 URL: http://ima.umn.edu/~arnold/papers/bdlayer2.pdf ==> bdlayer-info.txt <== Title: The boundary layer for the Reissner-Mindlin plate model Authors: Douglas N. Arnold and Richard S. Falk Source: SIAM J. Math. Anal. 21 (1990), pp. 281-312 Status: Published Abstract: The structure of the solution of the Reissner-Mindlin model of a clamped plate is investigated, emphasizing its dependence on the plate thickness. Asymptotic expansions in powers of the plate thickness are developed for the main physical quantities and the boundary layer is studied. Rigorous error bounds are given for the errors in the expansions in Sobolev norms. As applications, new regularity results for the solutions and new estimates for the difference between the Reissner-Mindlin solution and the solution to the biharmonic equation are derived. Boundary conditions for a clamped edge are considered for most of the paper, and the very similar case of a hard simply-supported plate is discussed briefly at the end. Keywords: Reissner, Mindlin, plate, boundary layer Subj. class.: 73K10, 73K25 URL: http://ima.umn.edu/~arnold/papers/bdlayer.pdf ==> bistetima-info.txt <== Title: Tetrahedral bisection and adaptive finite elements Authors: Douglas N. Arnold and Arup Mukherjee Source: in: "Grid Generation and Adaptive Algorithms" (M. Bern, J. E. Flaherty, and M. Luskin, eds.), IMA Volumes in Mathematics and its Applications 113, Springer-Verlag, New York-Heidelberg-Berlin, 1999, pp. 29-42 Status: Published Abstract: An adaptive finite element algorithm for elliptic boundary value problems in $\R^3$ is presented. The algorithm uses linear finite elements, a-posteriori error estimators, a mesh refinement scheme based on bisection of tetrahedra, and a multi-grid solver. We show that the repeated bisection of an arbitrary tetrahedron leads to only a finite number of dissimilar tetrahedra, and that the recursive algorithm ensuring conformity of the meshes produced terminates in a finite number of steps. A procedure for assigning numbers to tetrahedra in a mesh based on a-posteriori error estimates, indicating the degree of refinement of the tetrahedron, is also presented. Numerical examples illustrating the effectiveness of the algorithm are given. Keywords: finite elements, adaptive mesh refinement, error estimators, bisection of tetrahedra Subj. class.: 65N50 URL: http://ima.umn.edu/~arnold/papers/bistetima.pdf ==> bistet-info.txt <== Title: Locally adapted tetrahedral meshes using bisection Authors: Douglas N. Arnold, Arup Mukherjee, and Luc Pouly Source: SIAM Journal on Scientific Computing 22 (2000), pp. 431-448 Status: Published Abstract: We present an algorithm for the construction of locally adapted conformal tetrahedral meshes. The algorithm is based on bisection of tetrahedra. A new data structure is introduced, which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected. We prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity, and bound the amount of additional refinement that is needed to achieve conformity. Numerical examples of the effectiveness of the algorithm are presented. Keywords: bisection, tetrahedral meshes, adaptive refinement, similarity classes, finite elements Subj. class.: 65N50 URL: http://ima.umn.edu/~arnold/papers/bistet.pdf ==> cai-info.txt <== Title: Computer-aided instruction Authors: Douglas N. Arnold Source: Microsoft Encarta Online Encyclopedia 1997- Status: Published Abstract: Enclyclopedia article on computer-aided instruction URL: http://ima.umn.edu/~arnold/papers/cai.pdf ==> collocation-info.txt <== Title: On the asymptotic convergence of collocation methods Authors: Douglas N. Arnold, Wolfgang L. Wendland Source: Math. Comp. 41 (1981), pp. 349-381 Status: Published Abstract: We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence. Keywords: collocation, spline, integral equation Subj. class.: 65R20, 65R99, 65L10, 65N99, 45J05, 45L10, 45F15, 35S99, 30C30, 73K30, 31A30 URL: http://ima.umn.edu/~arnold/papers/collocation.pdf ==> colpde-info.txt <== Title: On the asymptotic convergence of spline collocation methods for partial differential equations Authors: Douglas N. Arnold and Jukka Saranen Source: SIAM J. Numer. Anal. 21 (1984), pp. 459-472 Status: Published Abstract: We examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations. Specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions. We prove optimal rates of convergence in L2 for partial derivatives of the approximate solution which are of order at least two in one variable, while the solution itself and its gradient converge in L2 at rates less than the optimal approximation theoretic results. URL: http://ima.umn.edu/~arnold/papers/colpde.pdf ==> colvsgal-info.txt <== Title: Collocation versus Galerkin procedures for boundary integral methods Authors: Douglas N. Arnold and Wolfgang L. Wendland Source: in: "Boundary Element Methods in Engineering" (C. A. Brebbia, ed.), Springer-Verlag, New York-Heidelberg-Berlin, 1983, pp. 18-33 Status: Published Abstract: We compare the efficiency of the solution of two-dimensional elliptic boundary value problems via boundary integral methods using two different discretization procedures with comparable convergence rates: Galerkin procedures with numerical integration and collocation. Keywords: boundary integral methods, boundary element methods, Galerkin method, collocation URL: http://ima.umn.edu/~arnold/papers/colvsgal.pdf ==> complejos-info.txt <== Title: Complejos diferenciales y estabilidad numérica Note: Spanish translation of "Differential complexes and numerical analysis," Proceedings of the International Congress of Mathematicians, Beijing 2002, Volume I: Plenary Lectures Authors: Douglas N. Arnold Source: La Gaceta de la Real Sociedad Matemática Española 8.2 (2005), pp. 335-360. Status: Published Abstract: Complejos diferenciales como el de de Rham desempeñan un papel cada vez más importante en el diseño y en el análisis de métodos numéricos para ecuaciones en derivadas parciales. El diseño de discretizaciones estables de sistemas de ecuaciones en derivadas parciales a menudo se basa en capturar sutiles aspectos de la estructura del sistema a discretizar. Con frecuencia la estructura geométrico-diferencial capturada por un complejo diferencial ha resultado ser un elemento clave, y un complejo diferencial discreto relacionado con el complejo anterior ha sido esencial. Este nuevo enfoque geométrico está suponiendo una unificación en la forma de entender una variedad de métodos numéricos innovadores que se han ido desarrollando en las ´ ultimas décadas y ha abierto el camino para obtener discretizaciones estables de problemas para los que no disponíamos de ellas; parece probable que en el futuro este enfoque nos permita abordar algunos problemas, hoy por hoy intratables, en ecuaciones en derivadas parciales numéricas. Keywords: elemento finito, estabilidad numérica, complejo diferencial Subj. class.: 65N12 URL: http://ima.umn.edu/~arnold/papers/complejos.pdf ==> constrained-info.txt <== Title: Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials Authors: Douglas N. Arnold and Richard S. Falk Source: Arch. Rational Mech. Anal. 98 (1987), pp. 143-165 Status: Published Abstract: We consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained, and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well-posed. For materials fulfilling the condition, we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system. As an application we determine the orthotropic materials for which the fundamental problems are well-posed in terms of their Young's moduli, shear moduli, and Poisson ratios. Finally, we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements. For a two-dimensional constrained material a further reduction to a single scalar equation is outlined. Keywords: elasticity, anisotropic, constraint, well-posed URL: http://ima.umn.edu/~arnold/papers/constrained.pdf ==> corners-info.txt <== Title: Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon Authors: Douglas N. Arnold, L. Ridgway Scott, and Michael Vogelius Source: Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 15 (1988), pp. 169-192 Status: Published Abstract: We consider the existence of regular solutions to the boundary value problem div U = f on a plane polygonal domain with the Dirichlet boundary condition U=g. We formulate simultaneously necessary and sufficient conditions on f and g in order that a solution U exist in the Sobolev space W^{s+1}_p. In addition to the obvious regularity and integral conditions these consist of at most one compatibility condition at each vertex of the polygon. In the special case of homogeneous boundary data, it is necessary and sufficient that f belong to W^s_p, have mean value zero, and vanish at each vertex. (The latter condition only applies if s is large enough that the point values make sense.) We construct a solution operator which is independent of s and p. As intermediate results we obtain various new trace theorems for Sobolev spaces on polygons. Keywords: divergence, trace, Sobolev space Subj. class.: 35B65, 46E35 URL: http://ima.umn.edu/~arnold/papers/corners.pdf ==> delta-info.txt <== Title: The delta-trigonometric method using the single-layer potential representation Authors: Douglas N. Arnold and Raymond S. S. Cheng Source: J. Integral Equations 1 (1988), pp. 517-547 Status: Published Abstract: The Dirichlet problem for Laplace's equation is often solved by means of the single layer potential representation, leading to a Fredholm integral equation of the first kind with logarithmic kernel. We propose to solve this integral equation using a Petrov-Galerkin method with trigonometric polynomials as test functions and, as trial functions, a span of delta distributions centered at boundary points. The approximate solution to the boundary value problem thus computed converges exponentially away from the boundary and algebraically up to the boundary. We show that these convergence results hold even when the discretization matrices are computed via numerical quadratures. Finally, we discuss our implementation of this method using the fast Fourier transform to compute the discretization matrices, and present numerical experiments in order to confirm our theory and to examine the behavior of the method in cases where the theory doesn't apply due to lack of smoothness. Subj. class.: 65R20, 65N30, 65E05, 45L10 URL: http://ima.umn.edu/~arnold/papers/delta.pdf ==> dgerr-info.txt <== Title: Unified analysis of discontinuous Galerkin methods for elliptic problems Authors: Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini Source: SIAM Journal on Numerical Analysis 39 (2002), pp. 1749-1779 Status: Published Abstract: We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems by diverse communities over three decades. Keywords: elliptic problems, discontinuous Galerkin, interior penalty Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/dgerr.pdf ==> dg-info.txt <== Title: Discontinuous Galerkin methods for elliptic problems Authors: Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and Donatella Marini Source: in: "Discontinuous Galerkin Methods: Theory, Computation and Applications" (B. Cockburn, G. Karniadakis, and C. W. Shu, eds.), Lecture Notes in Computational Science and Engineering 11, Springer-Verlag, New York, 2000, pp. 89-101. Status: Published Abstract: We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cockburn and Shu, and the method of Baumann and Oden. It also includes the so-called interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others. URL: http://ima.umn.edu/~arnold/papers/dg.pdf ==> dgrm2-info.txt <== Title: Locking-free Reissner-Mindlin elements without reduced integration Authors: Douglas N. Arnold, Franco Brezzi, Richard S. Falk, and L. Donatella Marini Source: Computer Methods in Applied Mechanics and Engineering 196 (2007), pp. 3660-3671 Status: Published Abstract: In a recent paper of Arnold, Brezzi, and Marini, the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner--Mindlin plate problem. By following their basic approach, but making different choices of finite element spaces, we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator, which has been a central feature of many locking-free methods. For k>1, all the methods use piecewise polynomials of degree k to approximate the transverse displacement and (possibly subsets) of piecewise polynomials of degree k-1 to approximate both the rotation and shear stress vectors. The approximation spaces for the rotation and the shear stress are always identical. The methods vary in the amount of interelement continuity required. In terms of smallest number of degrees of freedom, the simplest method approximates the transverse displacement with continuous, piecewise quadratics and both the rotation and shear stress with rotated linear Brezzi-Douglas-Marini elements. Keywords: discontinuous Galerkin, Reissner-Mindlin plate, locking Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/dgrm2.pdf ==> dgrm-info.txt <== Title: A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate Authors: Douglas N. Arnold, Franco Brezzi, and Donatella Marini Source: Journal of Scientific Computing 22 (2005), pp. 25-41 Status: Published Abstract: We develop a family of locking-free elements for the Reissner-Mindlin plate using Discontinuous Galerkin techniques, one for each odd degree, and prove optimal error estimates. A second family uses conforming elements for the rotations and nonconforming elements for the transverse displacement, generalizing the element of Arnold and Falk to higher degree. Keywords: discontinuous Galerkin, Reissner-Mindlin plate, locking Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/dgrm.pdf ==> dundee-info.txt <== Title: Adaptive finite elements and colliding black holes Authors: Douglas N. Arnold, Arup Mukherjee, and Luc Pouly Source: in: "Numerical Analysis 1997" (D. F. Griffiths, D. J. Higham, and G. A. Watson, eds.), Addison Wesley Longman, Essex, 1998, pp. 1-15 Status: Published Abstract: According to the theory of general relativity, the relative acceleration of masses generates gravitational radiation. Although gravitational radiation has not yet been detected, it is believed that extremely violent cosmic events, such as the collision of black holes, should generate gravity waves of sufficient amplitude to detect on earth. The massive Laser Interferometer Gravitational-Wave Observatory, or LIGO, is now being constructed to detect gravity waves. Consequently there is great interest in the computer simulation of black hole collisions and similar events, based on the numerical solution of the Einstein field equations. In this note we introduce the scientific, mathematical, and computational problems and discuss the development of a computer code to solve the initial data problem for colliding black holes, a nonlinear elliptic boundary value problem posed in an unbounded three dimensional domain which is a key step in solving the full field equations. The code is based on finite elements, adaptive meshes, and a multigrid solution process. Here we will particularly emphasize the mathematical and algorithmic issues arising in the generation of adaptive tetrahedral meshes. Keywords: adaptivity, finite elements, black holes, Einstein equations Subj. class.: 65N50, 65N30, 83C05, 83C35, 83C57 URL: http://ima.umn.edu/~arnold/papers/dundee.pdf ==> ecbc-info.txt <== Title: Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations Authors: Douglas N. Arnold and Nicolae Tarfulea Source: Electronic Journal on Differential Equations, Conf. 15 (2007), pp. 11-27 Status: Published Abstract: Specifying boundary conditions continues to be a challenge in numerical relativity in order to obtain a long time convergent numerical simulation of Einstein's equations in domains with artificial boundaries. In this paper, we address this problem for the Einstein--Christoffel (EC) symmetric hyperbolic formulation of Einstein's equations linearized around flat spacetime. First, we prescribe simple boundary conditions that make the problem well posed and preserve the constraints. Next, we indicate boundary conditions for a system that extends the linearized EC system by including the momentum constraints and whose solution solves Einstein's equations in a bounded domain. Finally, we extend our results to the case of inhomogeneous boundary conditions. Keywords: general relativity, Einstein equations, boundary condition Subj. class.: 35Q75, 35L50, 83C99 URL: http://ima.umn.edu/~arnold/papers/ecbc.pdf ==> einsteinhyper-info.txt <== Title: New first-order formulation for the Einstein equations Authors: Alexander M. Alekseenko and Douglas N. Arnold Source: gr-qc/0210071, Phys. Rev. D 68 (2003) Status: Published Abstract: We derive a new first-order formulation for Einstein's equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the $3+1$ decomposition with arbitrary lapse and shift. In the reduction to first order form only 8 particular combinations of the 18 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of symmetric hyperbolic system in 14 unknowns, namely the components of the extrinsic curvature perturbation and the 8 new variables, from whose solution the metric perturbation can be computed by integration. Keywords: relativity, Einstein equations, symmetric hyperbolic Subj. class.: PACS 04.20.Ex, 04.25.Dm URL: http://ima.umn.edu/~arnold/papers/hyper.pdf ==> elas3dfamily-info.txt <== Title: Finite elements for symmetric tensors in three dimensions Authors: Douglas N. Arnold, Gerard Awanou, and Ragnar Winther Source: Mathematics of Computation Status: Submitted 2007 Abstract: We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed formulation of the elasticty equations, when standard discontinous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther. Keywords: finite element, elasticity, mixed method Subj. class.: 65N30, 74S05 URL: http://ima.umn.edu/~arnold/papers/elas3dfamily.pdf ==> elascomplexes2d-info.txt <== Title: Differential complexes and stability of finite element methods. II. The elasticity complex Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: in: "Compatible Spatial Discretizations for PDE", IMA Volumes in Mathematics and its Applications 142, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides, and M. Shashkov, eds., Springer, 2006, pp. 23-46 Status: Published Abstract: A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field. We also discuss how the strongly symmetric methods proposed in [8] can be derived in the present framework. The method of construction works in both two and three space dimensions, but for simplicity the discussion here is limited to the two dimensional case. Keywords: mixed finite element method, Hellinger-Reissner principle, elasticity Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/elascomplexes2d.pdf ==> elas-info.txt <== Title: A new mixed formulation for elasticity Authors: Douglas N. Arnold and Richard S. Falk Source: Numer. Math. 53 (1988), pp. 13-30 Status: Published Abstract: We propose a new mixed variational formulation for the equations of linear elasticity. It does not require symmetric tensors and consequently is easy to discretize by adapting mixed finite elements developed for scalar second order elliptic equations. Keywords: mixed finite element method, elasticity Subj. class.: 65N30, 73C35, 73K25 URL: http://ima.umn.edu/~arnold/papers/elas.pdf ==> enumath-info.txt <== Title: Numerical problems in general relativity Authors: Douglas N. Arnold Source: in: "Numerical Mathematics and Advanced Applications" (P> Neittaanmäki, T. Tiihonen, and P. Tarvainen, eds.), World Scientific, 2000, pp. 3-15 Status: Published Abstract: The construction of gravitational wave observatories is one of the greatest scientific efforts of our time. As a result, there is presently a strong need to numerically simulate the emission of gravitation radiation from massive astronomical events such as black hole collisions. This entails the numerical solution of the Einstein field equations. We briefly describe the field equations in their natural setting, namely as statements about the geometry of space time. Next we describe the complicated system that arises when the field equations are recast as partial differential equations, and discuss procedures for deriving from them a more tractable system consisting of constraint equations to be satisfied by initial data and together with evolution equations. We present some applications of modern finite element technology to the solution of the constraint equations in order to find initial data relevant to black hole collisions. We conclude by enumerating some of the many computational challenges that remain. Keywords: general relativity, gravitational radiation, black holes, Einstein equations Subj. class.: 65N30, 83C05, 83C35, 83C57 URL: http://ima.umn.edu/~arnold/papers/enumath.pdf ==> fedf-info.txt <== Title: Finite element differential forms Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: Proceedings in Applied Mathematics and Mechanics Status: to appear Abstract: A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree $1$ and of degree $n-1$ (where $n$ is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades. Keywords: mixed method, finite element, differential form, exterior calculus Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/fedf.pdf ==> hdivcurl-info.txt <== Title: Multigrid in H(div) and H(curl) Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: Numerische Mathematik 85 (2000), pp. 197-218 Status: Published Abstract: We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl) in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products. Keywords: multigrid, preconditioner, mixed method, finite element Subj. class.: 65N55, 65N30 URL: http://ima.umn.edu/~arnold/papers/hdivcurl.pdf ==> hdivdd9-info.txt <== Title: Preconditioning in H(div) and applications Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: in: "Ninth International Conference on Domain Decomposition Methods (Bergen, 1996)" (P. Bjorstad, M. Espedal, and D. Keyes, eds.), DDM.org, 1998, pp. 12-19 Status: Published Abstract: Summarizing the work of [1], we show how to construct preconditioners using domain decomposition and multigrid techniques for the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I - grad div. These preconditioners are shown to be spectrally equivalent to the inverse of the operator and thus may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems. Keywords: preconditioner, mixed method, least squares, finite element, multigrid, domain decomposition Subj. class.: 65N55, 65N30 URL: http://ima.umn.edu/~arnold/papers/hdivdd9.pdf ==> hdiv-info.txt <== Title: Preconditioning in H(div) and applications Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: Math. Comp. 66 (1997), pp. 957-984 Status: Published Abstract: We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I - grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems. Keywords: preconditioner, mixed method, least squares, finite element, multigrid, domain decomposition Subj. class.: 65N55, 65N30 URL: http://ima.umn.edu/~arnold/papers/hdiv.pdf ==> hdivn-info.txt <== Title: Multigrid preconditioning in H(div) on non-convex polygons Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: Computational and Applied Mathematics 17 (1998), pp. 307-319 Status: Published Abstract: In an earlier paper we constructed and analyzed a multigrid preconditioner for the system of linear algebraic equations arising from the finite element discretization of boundary value problems associated to the differential operator I - grad div. In this paper we analyze the procedure without assuming that the underlying domain is convex and show that, also in this case, the preconditioner is spectrally equivalent to the inverse of the discrete operator. Keywords: preconditioner, finite element, multigrid, nonconvex domain Subj. class.: 65N55, 65N30 URL: http://ima.umn.edu/~arnold/papers/hdivn.pdf ==> heat-info.txt <== Title: Collocation versus Galerkin procedures for boundary integral methods Authors: Douglas N. Arnold and Wolfgang L. Wendland Source: in: "Boundary Element Methods in Engineering" (C. A. Brebbia, ed.), Springer-Verlag, New York-Heidelberg-Berlin, 1983, pp. 18-33 Status: Published Abstract: We compare the efficiency of the solution of two-dimensional elliptic boundary value problems via boundary integral methods using two different discretization procedures with comparable convergence rates: Galerkin procedures with numerical integration and collocation. Keywords: boundary integral methods, boundary element methods, Galerkin method, collocation URL: http://ima.umn.edu/~arnold/papers/colvsgal.pdf ==> heatnote-info.txt <== Title: Coercivity of the single layer heat potential Authors: Douglas N. Arnold and Patrick J. Noon Source: J. Comput. Math. 7 (1989), pp. 100-104 Status: Published Abstract: The single layer heat potential operator, K, arises in the solution of initial-boundary value problems for the heat equation using boundary integral methods. In this note we show that K maps a certain anisotropic Sobolev space isomorphically onto its dual, and, moreover, satisfies the coercivity inequality \ge c|q|^2. We thereby establish the well-posedness of the operator equation K q=f and provide a basis for the analysis of the discretizations. Keywords: coercivity, heat potential, anisotropic Sobolev space URL: http://ima.umn.edu/~arnold/papers/heatnote.pdf ==> hierarchical-info.txt <== Title: Asymptotic estimates of hierarchical modeling Authors: Douglas N. Arnold and Alexandre L. Madureira Source: Mathematical Models and Methods in Applied Sciences (M3AS) 13 (2003) Status: Published Abstract: In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the influence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers). Keywords: hierarchical modeling, dimension reduction, asymptotic estimates Subj. class.: 35C20 URL: http://ima.umn.edu/~arnold/papers/hierarchical.pdf ==> hopes-info.txt <== Title: A family of higher order mixed finite element methods for plane elasticity Authors: Douglas N. Arnold, Jim Douglas, Jr., and Chaitan P. Gupta Source: Numer. Math. 45 (1984), pp. 1-22 Status: Published Abstract: The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor in L^2 and optimal order negative norm estimates are obtained in (H^s)' for a range of s depending on the index of the finite element space. An optimal order estimate inL in L_infinity for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail. Keywords: finite element methods, plane elasticity Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/hopes.pdf ==> icm2002-info.txt <== Title: Differential complexes and numerical stability Authors: Douglas N. Arnold Source: Proceedings of the International Congress of Mathematicians, Beijing 2002, Volume I: Plenary Lectures Status: Published Abstract: Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial differential equations often hinges on capturing subtle aspects of the structure of the system in the discretization. In many cases the differential geometric structure captured by a differential complex has proven to be a key element, and a discrete differential complex which is appropriately related to the original complex is essential. This new geometric viewpoint has provided a unifying understanding of a variety of innovative numerical methods developed over recent decades and pointed the way to stable discretizations of problems for which none were previously known, and it appears likely to play an important role in attacking some currently intractable problems in numerical PDE. Keywords: finite element, numerical stability, differential complex Subj. class.: 65N12 URL: http://ima.umn.edu/~arnold/papers/icm2002.pdf ==> icmsc-info.txt <== Title: Mixed finite elements for elasticity in the stress-displacement formulation Authors: Douglas N. Arnold and Ragnar Winther Source: in: "Current Trends in Scientific Computing" (Z. Chen, R. Glowinski, and K. Li, eds), Contemporary Mathematics, AMS, 2003 Status: Published Abstract: We present a family of pairs of finite element spaces for the unaltered Hellinger--Reissner variational principle using polynomial shape functions on a single triangular mesh for stress and displacement. There is a member of the family for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and each is stable and affords optimal order approximation. The simplest element pair involves 24 local degrees of freedom for the stress and 6 for the displacement. We also construct a lower order element involving 21 stress degrees of freedom and 3 displacement degrees of freedom which is, we believe, likely to be the simplest possible conforming stable element pair with polynomial shape functions. For all these conforming elements the approximate stress not only belongs to H(div), but is also continuous at element vertices, which is more continuity than may be desired. We show that for conforming finite elements with polynomial shape functions, this additional continuity is unavoidable. To overcome this obstruction, we construct as well some non-conforming stable mixed finite elements, which we show converge with optimal order as well. The simplest of these involves only 12 stress and 6 displacement degrees of freedom on each triangle. Keywords: mixed method, finite element, elasticity Subj. class.: 65N30, 74S05 URL: http://ima.umn.edu/~arnold/papers/icmsc.pdf ==> imacs92-info.txt <== Title: Quadratic velocity/linear pressure Stokes elements Authors: Douglas N. Arnold and Jinshui Qin Source: in: "Advances in Computer Methods for Partial Differential Equations-VII" (R. Vichnevetsky, D. Knight, G. Richter, eds.), IMACS, 1992, pp. 28-34 Status: Published Abstract: We study the finite element approximation of the stationary Stokes equations in the velocity-pressure formulation using continuous piecewise quadratic functions for velocity and discontinuous piecewise linear functions for pressure. For some meshes this method is unstable, even after spurious pressure modes are removed. For other meshes there are spurious local pressure modes, but once they are removed the method is stable, and in particular, the velocity converges with optimal order. On yet other meshes there are no spurious pressure modes and the method is stable and optimally convergent for both pressure and velocity. Keywords: finite element, Stokes equations URL: http://ima.umn.edu/~arnold/papers/imacs92.pdf ==> intpen-info.txt <== Title: An Interior Penalty Finite Element Method with Discontinuous Elements Authors: Douglas N. Arnold Source: SIAM J. Numer. Anal. 19 (1982), pp. 742-760 Status: Published Abstract: A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and L2-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general. URL: http://ima.umn.edu/~arnold/papers/intpen.pdf ==> kdv-info.txt <== Title: A superconvergent finite element method for the Korteweg-deVries equation Authors: Douglas N. Arnold, Ragnar Winther Source: Math. Comp. 38 (1982), pp. 23-36 Status: Published Abstract: An unconditionally stable fully discrete finite element method for the Korteweg-de Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small time-independent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values. Keywords: superconvergence, finite element, Korteweg-de Vries equation Subj. class.: 65N30, 65N15, 76B15 URL: http://ima.umn.edu/~arnold/papers/kdv.pdf ==> minc-info.txt <== Title: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates Authors: Douglas N. Arnold and Franco Brezzi Source: Mathematical Modelling and Numerical Analysis 19 (1985), pp. 7-32 Status: Published Abstract: We discuss a technique of implementing certain mixed finite elements based on the use of Lagrange multipliers to impose interelement continuity. The matrices arising from this implementation are positive definite. Considering some well-known mixed methods, namely the Raviart-Thomas methods for second order elliptic problems and the Hellan-Hermann-Johnson method for biharmonic problems, we show that the computed Lagrange multipliers may be exploited in a simple postprocess to produce better approximation of the original variables. We further extablish an equivalence between the mixed methods and certain modified versions of well-known nonconforming methods, notably the Morley method in the case of the biharmonic problem. The equivalence is exploited to provide error estimates for both the mixed and nonconforming methods. Keywords: mixed finite element, Lagrange multiplier Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/minc.pdf ==> mindlin-info.txt <== Title: A uniformly accurate finite element method for the Reissner-Mindlin plate Authors: Douglas N. Arnold and Richard S. Falk Source: SIAM J. Numer. Anal. 26 (1989), pp. 1276-1290 Status: Published Abstract: We present and analyze a simple finite element method for the Mindlin-Reissner plate model in the primitive variables. Our method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging. We prove that the method converges with optimal order uniformly with respect to thickness. Keywords: Reissner-Mindlin plate, finite element, nonconforming Subj. class.: 65N30,73K10,73K25 URL: http://ima.umn.edu/~arnold/papers/mindlin.pdf ==> mini-info.txt <== Title: A stable finite element for the Stokes equations Authors: Douglas N. Arnold, Franco Brezzi, and Michel Fortin Source: Calcolo 21 (1984), pp. 338-344 Status: Published Abstract: We present in this paper a new velocity-pressure finite element for the computation of Stokes flow. We discretize the velocity field iwth continuous piecewise linear functions enriched by bubble functions, and the pressure by piecewise linear functions. We show that this element satisfies the usual inf-sup condition and converges with first order for both velocities and pressures. Finally we relate this element to families of higher order elements and to the popular Taylor-Hood element. Keywords: finite element, Stokes equations Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/mini.pdf ==> mixedelas3d-info.txt <== Title: Mixed finite element methods for linear elasticity with weakly imposed symmetry Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: Mathematics of Computation 76 (2007), pp. 1699-1723 Status: Published Abstract: In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger--Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de~Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field. Keywords: mixed method, finite element, elasticity Subj. class.: 65N30, 74S05 URL: http://ima.umn.edu/~arnold/papers/mixedelas3d.pdf ==> mixedelas-info.txt <== Title: Mixed finite elements for elasticity Authors: Douglas N. Arnold and Ragnar Winther Source: Numerische Mathematik 42 (2002), pp. 401-419 Status: Published Abstract: There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available. Keywords: mixed method, finite element, elasticity Subj. class.: 65N30, 74S05 URL: http://ima.umn.edu/~arnold/papers/mixedelas.pdf ==> mixedelasrect-info.txt <== Title: Rectangular mixed finite elements for elasticity Authors: Douglas N. Arnold and Gerard Awanou Source: Math. Models and Methods in Appl. Sci. 15 (2005), pp. 1417-1429 Status: Published Abstract: We present a family of stable rectangular mixed finite elements for plane elasticity. Each member of the family consists of a space of piecewise polynomials discretizing the space of symmetric tensors in which the stress field is sought, and another to discretize the space of vector fields in which the displacement is sought. These may be viewed as analogues in the case of rectangular meshes of mixed finite elements recently proposed for triangular meshes. As for the triangular case the elements are closely related to a discrete version of the elasticity differential complex. Keywords: mixed method, finite element, elasticity, rectangular Subj. class.: 65N30, 74S05 URL: http://ima.umn.edu/~arnold/papers/mixedelasrect.pdf ==> mixed-info.txt <== Title: Mixed finite element methods for elliptic problems Authors: Douglas N. Arnold Source: Comput. Methods Appl. Mech. Engrg. 82 (1990), pp. 281-300 Status: Published Abstract: This paper treats the basic ideas of mixed finite element methods at an introductory level. Although the viewpoint presented is that of a mathematician, the paper is aimed at practitioners and the mathematical prerequisites are kept to a minimum. A classification of variational principles and of the corresponding weak formulations and Galerkin methods--displacement, equilibrium, and mixed--is given and illustrated through four significant examples. The advantages and disadvantages of mixed methods are discussed. The concepts of convergence, approximability, and stability and their interrelations are developed, and a resume is given of the stability theory which governs the performance of mixed methods. The paper concludes with a survey of techniques that have been developed for the construction of stable mixed methods and numerous examples of such methods. Keywords: mixed method, finite element, variational principle Subj. class.: 65N30, 73C35, 73K25 URL: http://ima.umn.edu/~arnold/papers/mixed.pdf ==> nacomplexes-info.txt <== Title: Differential complexes and stability of finite element methods. I. The de Rham complex Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: in: "Compatible Spatial Discretizations for PDE", IMA Volumes in Mathematics and its Applications 142, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides, and M. Shashkov, eds., Springer, 2006, pp. 47-68 Status: Published Abstract: In this paper we explain the relation between certain piecewise polynomial subcomplexes of the de Rham complex and the stability of mixed finite element methods for elliptic problems. Keywords: mixed finite element method, de Rham complex, stability Subj. class.: 65N12 URL: http://ima.umn.edu/~arnold/papers/nacomplexes.pdf ==> ncelas-info.txt <== Title: Nonconforming mixed elements for elasticity Authors: Douglas N. Arnold and Ragnar Winther Source: Mathematical Models and Methods in Applied Sciences (M3AS) 13 (2003), pp. 295-307 Status: Published Abstract: We construct first order, stable, nonconforming mixed finite elements for plane elasticity and analyze their convergence. The mixed method is based on the Hellinger-Reissner variational formulation in which the stress and displacement fields are the primary unknowns. The stress elements use polynomial shape functions but do not involve vertex degrees of freedom. Keywords: mixed method, finite element, nonconforming, elasticity Subj. class.: 65N30, 74S05 URL: http://ima.umn.edu/~arnold/papers/ncelas.pdf ==> ncstokes-info.txt <== Title: On nonconforming linear-constant elements for some variants of the Stokes equations Authors: Douglas N. Arnold Source: Istit. Lombardo Accad. Sci. Lett. Rend. A 127 (1993), pp. 83-93 Status: Published Abstract: Nonconforming piecewise linear finite elements for the velocity field and piecewise constant elements for the pressure field give a simple stable, optimal order approximation to the Stokes equations, but are not stable for the equations of incompressible elasticty, which differ from the Stokes equations only in that the vector Laplace operator is replaced by the Lame operator. However, we show that if we replace the divergence by the rotation, then the nonconforming linear-constant element is stable both for the system involving the Laplacian and for that involving the Lame operator. Finally we discuss an application to the Reissner-Mindlin plate. Keywords: Stokes equations, mixed finite element method,nonconforming finite elements, Reissner-Mindlin plate Subj. class.: 65N30,65N12,76M10,76D07,73K10 URL: http://ima.umn.edu/~arnold/papers/ncstokes.pdf ==> newmixed-info.txt <== Title: A new mixed formulation for the numerical solution of elasticity problems Authors: Douglas N. Arnold Source: in: "Advances in Computer Methods for Partial Differential Equations--V" (R. Vichnevetsky and R. S. Stepleman, eds.), IMACS 1984, pp. 353-356 Status: Published Abstract: A mixed formulation for boundary value problems in linear elastostatics is presented. This formulation differs slightly from the classical Hellinger-Reissner formulation. The unknown fields are the displacement and a tensor related but not equal to the stress. The tensors appearing in the formulation need not be symmetric, and consequently mixed finite elements developed for scalar second order elliptic problems may be applied directly. Keywords: elasticity, mixed finite element Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/newmixed.pdf ==> orthotropic-info.txt <== Title: Continuous dependence on the elastic coefficients for a class of anisotropic materials Authors: Douglas N. Arnold and Richard S. Falk Source: IMA preprint 165, Institute for Mathematics and its Applications, 36 pages, 1985 Status: Published Abstract: We prove apriori estimates and continuous dependence on the elastic moduli for the equations of homogeneous orthotropic elasticity. These results are uniform with respect to the three Poisson rations, Young's moduli, and shear moduli of the material for certain ranges of these constants. These ranges include the possibility that the compliance tensor is singular such as occurs for incompressible materials. Keywords: orthotropic elasticity, incompressible, constrained material Subj. class.: 73C30; 73C35 URL: http://ima.umn.edu/~arnold/papers/orthotropic.pdf ==> ozf-info.txt <== Title: Analysis of a linear-linear finite element for the Reissner-Mindlin plate model Authors: Douglas N. Arnold and Richard S. Falk Source: Math. Models and Methods in Appl. Sci. 7 (1997), pp. 217-238 Status: Published Abstract: An analysis is presented for a recently proposed finite element method for the Reissner-Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual "locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t=O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations confirm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t fixed, the method does not converge as the mesh size h tends to zero. Keywords: Reissner, Mindlin, plate, finite element, nonconforming Subj. class.: 65N30, 73K10, 73K25 URL: http://ima.umn.edu/~arnold/papers/ozf.pdf ==> paramdep-info.txt <== Title: Discretization by finite elements of a model parameter dependent problem Authors: Douglas N. Arnold Source: Numer. Math. 37 (1981), pp. 405-421 Status: Published Abstract: The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures. Keywords: mixed finite element method, reduced integration, Timoshenko beam, parameter Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/paramdep.pdf ==> peers-info.txt <== Title: PEERS: A new mixed finite element for elasticity Authors: Douglas N. Arnold, Franco Brezzi, and Jim Douglas, Jr. Source: Japan Journal of Applied Mathematics 1 (1984), pp. 347-367 Status: Published Abstract: A mixed fintie element procedure for plane elasticity is introduced and analyzed. The symmetry of the stress tensor is enforced through the introduction of a Lagrange multiplier. An additional Lagrange multiplier is instroduced to simplify the algebraic system. Applications are made to incompressible elastic problems and to plasticity problems. Keywords: finite element methods, plane elasticity Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/peers.pdf ==> platederiv-info.txt <== Title: Derivation and justification of plate models by variational methods Authors: Stephen M. Alessandrini, Douglas N. Arnold, Richard S. Falk, and Alexandre L. Madureira Source: in: "Plates and Shells (Quebec 1996)", M. Fortin, editor, CRM Proceeding and Lecture Notes, vol. 21, American Mathematical Society, Providence, RI, 1999, pp. 1-20 Status: Published Abstract: We consider the derivation of two-dimensional models for the bending and stretching of a thin three-dimensional linearly elastic plate using variational methods. Specifically we consider restriction of the trial space in two different forms of the Hellinger-Reissner variational principle for 3-D elasticity to functions with a specified polynomial dependence in the transverse direction. Using this approach many different plate models are possible and we classify and investigate the most important. We study in detail a method which leads naturally not only to familiar plate models, but also to error bounds between the plate solution and the full 3-D solution. Keywords: plate, dimensional reduction, Reissner-Mindlin Subj. class.: 73K10, 73C02 URL: http://ima.umn.edu/~arnold/papers/platederiv.pdf ==> plates-info.txt <== Title: Innovative finite element methods for plates Authors: Douglas N. Arnold Source: Mat. Apl. Comput. 10 (1991), pp. 77-88. Status: Published Abstract: Finite element methods for the Reissner-Mindlin plate theory are discussed. Methods in which both the tranverse displacement and the rotation are approximated by finite elements of low degree mostly suffer from locking. However a number of related methods have been devised recently which avoid locking effects. Although the finite element spaces for both the rotation and transverse displacement contain little more than piecewise linear functions, optimal order convergence holds uniformly in the thickness. The main ideas leading to such methods are reviewed and the relationships between various methods are clarified. Keywords: Reissner, Mindlin, plate, finite element Subj. class.: 65N30, 73K10 URL: http://ima.umn.edu/~arnold/papers/plates.pdf ==> prermdd9-info.txt <== Title: Preconditioning discrete approximations of the Reissner-Mindlin plate model Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: in: "Ninth International Conference on Domain Decomposition Methods (Bergen, 1996)" (P. Bjorstad, M. Espedal, and D. Keyes, eds.), DDM.org, 1998, pp. 215-221 Status: Published Abstract: We consider iterative methods for the solution of linear systems of equations arising from mixed finite element discretization of the Reissner-Mindlin plate model. We show how to construct symmetric positive definite block diagonal preconditioners for these indefinite systems such that the resulting systems have spectral condition numbers independent of both the mesh size h and the plate thickness t. Keywords: preconditioner, Reissner, Mindlin, plate, finite element Subj. class.: 65N30, 65N22, 65F10, 73V05 URL: http://ima.umn.edu/~arnold/papers/prermdd9.pdf ==> prerm-info.txt <== Title: Preconditioning discrete approximations of the Reissner-Mindlin plate model Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther Source: Math. Modelling Numer. Anal. 31 (1997), pp. 517-557 Status: Published Abstract: We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner-Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear system has spectral condition number independent of both the mesh size h and the plate thickness t. We further discuss how this preconditioner may be implemented and then apply it to efficiently solve this indefinite linear system. Although the mixed formulation of the Reissner-Mindlin problem has a saddle-point structure common to other mixed variational problems, the presence of the small parameter t and the fact that the matrix in the upper left corner of the partition is only positive semidefinite introduces new complications. Keywords: preconditioner, Reissner, Mindlin, plate, finite element Subj. class.: 65N30, 65N22, 65F10, 73V05 URL: http://ima.umn.edu/~arnold/papers/prerm.pdf ==> principles-info.txt <== Title: Finite element methods: principles for their selection Authors: Douglas N. Arnold, Ivo Babuska, and John E. Osborn Source: Computer Methods in Applied Mechanics and Engineering 45 (1984), pp. 57-96 Status: Published Abstract: Principles for the selection of a finite element method for a particular problem are discusses. These principles are stated in terms of the notion of approximability, optimality, and stability. Several examples are discussed in details as illustrations. Conclusions regarding the selection of finite element methods are summarized in the final section of the paper. Keywords: finite elements, aproximability, stability, optimality Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/principles.pdf ==> psri-info.txt <== Title: The partial selective reduced integration method and applications to shell problems Authors: Douglas N. Arnold and Franco Brezzi Source: Comput. & Stuctures 64 (1997), pp. 879-880 Status: Published Abstract: We briefly present the main idea of partial selective reduced integration as developed in other works of the authors. The idea is quite general and can be applied to a number of different situations, but we concentrate on the case of the Naghdi shell model. Keywords: partial selective reduced integration, Naghdi shell Subj. class.: 65N30, 73K15, 73V05 URL: http://ima.umn.edu/~arnold/papers/psri.pdf ==> quadapprox-info.txt <== Title: Approximation by quadrilateral finite elements Authors: Douglas N. Arnold, Daniele Boffi, and Richard S. Falk Source: Mathematics of Computation 71 (2002), pp. 909-922 Status: Published Abstract: We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r+1 in L2 and order r in H1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements. Keywords: quadrilateral, finite element, approximation, serendipity, mixed finite element Subj. class.: 65N30, 41A10, 41A25, 41A27, 41A63 URL: http://ima.umn.edu/~arnold/papers/quadapprox.pdf ==> quadmeshes-info.txt <== Title: Finite element approximation on quadrilateral meshes Authors: Douglas N. Arnold, Daniele Boffi, Richard S. Falk, and Lucia Gastaldi Source: Communications in Numerical Methods in Engineering 17 (2001), pp. 805-812 Status: Published Abstract: Quadrilateral finite elements are generally constructed by starting from a given finite dimensional space of polynomials V^ on the unit reference square K^. The elements of V^ are then transformed by using the bilinear isomorphisms F_K which map K^ to each convex quadrilateral element K. It has been recently proven that a necessary and sufficient condition for approximation of order r+1 in L^2 and r in H^1 is that V^ contains the space Q_r of all polynomial functions of degree r separately in each variable. In this paper several numerical experiments are presented which confirm the theory. The tests are taken from various examples of applications: the Laplace operator, the Stokes problem and an eigenvalue problem arising in fluid-structure interaction modeling. Keywords: quadrilateral, finite element, approximation, serendipity, mixed finite element Subj. class.: 65N30, 41A10, 41A25, 41A27, 41A63 URL: http://ima.umn.edu/~arnold/papers/quadmeshes.pdf ==> rmelts-info.txt <== Title: Some new elements for the Reissner-Mindlin plate model Authors: Douglas N. Arnold and Franco Brezzi Source: in: "Boundary Value Problems for Partial Differential Equations" (J.-L. Lions and C. Baiocchi, eds.), Masson, Paris, pp. 287-292 Status: Published Abstract: We report on a new approach to obtaining stable locking free finite element discretizations for the Reissner-Mindlin plate. Keywords: Reissner-Mindlin plate, finite element Subj. class.: 65N30,73K10,73K25 URL: http://ima.umn.edu/~arnold/papers/rmelts.pdf ==> rmexamples-info.txt <== Title: Edge effects in the Reissner-Mindlin plate theory Authors: Douglas N. Arnold and Richard S. Falk Source: in: "Analytic and Computational Models of Shells" (A. K. Noor, T. Belytschko, and J. C. Simo, eds.), A.S.M.E., New York, 1989, pp. 71-90 Status: Published Abstract: We investigate the structure of the solution of the Reissner-Mindlin plate equations in its dependence on the plate thickness for various boundary conditions, developing asymptotic expansions in powers of the plate thickness for the main physical quantities. These expansions are uniform up to the boundary for the transverse displacement, but for other variables there is a boundary layer, whose strength depends on the boundary conditions. We give rigorous error bounds for the errors in the expansions in Sobolev norms and make various applications. Keywords: Reissner, Mindlin, plate, boundary layer, edge effect Subj. class.: 73K10, 73K25 URL: http://ima.umn.edu/~arnold/papers/rmexamples.pdf ==> rmint-info.txt <== Title: Interior estimates for a low order finite element method for the Reissner-Mindlin plate Authors: Douglas N. Arnold and Xiaobo Liu Source: Advances in Computational Mathematics 7 (1997), pp. 337-360 Status: Published Abstract: Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner-Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold-Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion. Keywords: Reissner-Mindlin plate, boundary layer, mixed finite element, interior error estimate Subj. class.: 65N30,73K10 URL: http://ima.umn.edu/~arnold/papers/rmint.pdf ==> rmkl-info.txt <== Title: On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models Authors: Douglas N. Arnold, Alexandre Madureira, and Sheng Zhang Source: Journal of Elasticity Status: Published Abstract: We show that the Reissner-Mindlin plate bending model has a wider range of applicability than the Kirchhoff-Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner-Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff-Love model fails. Keywords: plate, Reissner-Mindlin, Kirchhoff-Love Subj. class.: 73K10, 73C02 URL: http://ima.umn.edu/~arnold/papers/rmkl.pdf ==> rmquad-info.txt <== Title: Remarks on quadrilateral Reissner-Mindlin plate elements Authors: Douglas N. Arnold, Daniele Boffi, and Richard S. Falk Source: in: "WCCM V - Fifth World Congress on Computational Mechanics" (H. A. Mang, F. G. Rammerstorfer, and J. Eberhardsteiner, eds.), 2002 Status: Published Abstract: Over the last two decades, there has been an extensive effort to devise and analyze finite elements schemes for the approximation of the Reissner­Mindlin plate equations which avoid locking, numerical overstiffness resulting in a loss of accuracy when the plate is thin. There are now many triangular and rectangular finite elements, for which a mathematical analysis exists to certify them as free of locking. Generally speaking, the analysis for rectangular elements extends to the case of parallograms, which are defined by affine mappings of rectangles. However, for more general convex quadrilaterals, defined by bilinear mappings of rectangles, the analysis is more complicated. Recent results by the authors on the approximation properties of quadrilateral finite elements shed some light on the problems encountered. In particular, they show that for some finite element methods for the approximation of the Reissner-Mindlin plate, the obvious generalization of rectangular elements to general quadrilateral meshes produce methods which lose accuracy. In this paper, we present an overview of this situation. Keywords: Reissner-Mindlin plate, finite element, locking-free, isoparametric Subj. class.: 65N30,74K20 URL: http://ima.umn.edu/~arnold/papers/rmquad.pdf ==> robustness-info.txt <== Title: Robustness of finite element methods for a model parameter dependent problem Authors: Douglas N. Arnold Source: in: "Advances in Computer Methods for Partial Differential Equations--IV" (R. Vichnevetsky and R. S. Stepleman, eds.), IMACS 1981, pp. 18-22 Status: Published Abstract: A convergence analysis is presented for standard and mixed finite element discretizations of a model system of equations for a transversely loaded beam. The equations depend parametrically on the beam thickness and the emphasis of the analysis is on the robustness of the methods with respect to this parameter. The mixed methods are shown to be far more robust than the standard methods employing elements of the same degree. Moveover they entail no additional computational expense. Computational results are included to illustrate the main results. Keywords: beam, mixed finite element, parameter-dependence Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/robustness.pdf ==> santiago-info.txt <== Title: Dimensional reduction for plates based on mixed variational principles Authors: Stephen M. Alessandrini, Douglas N. Arnold, Richard S. Falk, and Alexandre L. Madureira Source: in: "Shells (Santiago de Compostela 1997)", Cursos Congr. Univ. Santiago de Compostela, vol. 105, Univ. Santiago de Compostela, 1997, pp. 25-28 Status: Published Abstract: We consider the derivation and rigorous justification of models for thin linearly elastic plates using mixed variational principles. Keywords: plate, dimensional reduction, mixed variational principle Subj. class.: 73K10, 73C02 URL: http://ima.umn.edu/~arnold/papers/santiago.pdf ==> selection-info.txt <== Title: Selection of finite element methods Authors: Douglas N. Arnold, Ivo Babuska, and John E. Osborn Source: in: "Hybrid and Mixed Finite Element Methods" (S. N. Atluri R. H. Gallagher, and O. C. Zienkiewicz, eds.), John Wiley,1983, pp. 433-451 Status: Published Abstract: The goal of engineering computations is to obtain quantitative information about engineering problems. This goal is usually achieved by the approximation solution of a mathematically formulated problem. Although a relevant mathematical formulation of the problem and its approximation solution are closely related, here we shall suppose that a mathematical formulation has already been determined and is amenable to an approximate treatment. We shall discuss a broad class of approaches based on variational methods of discretization which allow one to find the approximation solution within a desired range of accuracy. We discuss properties of these methods which enable us to distinguish among them and which aid in the selection or design of a method which is effective in achieving the given goals of the computation. Keywords: finite elements, variational methods Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/selection.pdf ==> shellelt-info.txt <== Title: Locking free finite element methods for shells Authors: Douglas N. Arnold and Franco Brezzi Source: Math. Comp. 66 (1997), pp. 1-14 Status: Published Abstract: We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer k. The methods are based on a nonstandard mixed formulation, and the kth method employs triangular Lagrange finite elements of degree k+2 augmented by bubble functions of degree k+3 for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree k for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone, as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the H1 norm of the displacement and rotation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order k+1 uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants. Keywords: shell, locking, finite element Subj. class.: 65N30,73K15,73V05 URL: http://ima.umn.edu/~arnold/papers/shellelt.pdf ==> singint-info.txt <== Title: The effect of the test functions on the convergence of spline projection Authors: Douglas N. Arnold Source: in: "Numerical Solution of Singular Integral Equations" (A. Gerasoulis and R. Vichnevetsky, eds.), 1984, pp. 1-4 Status: Published Abstract: We investigate the asymptotic convergence properties of a variety of methods for the numerical solution of the system of singular integral equations arising from the traction problem of plane elasticity. Various sorts of Galerkin methods and collocation methods are considered, all of which determine a spline approximation via paring with certain test functions; the test functions may be splines of the same degree as the trial functions (ordinary Galerkin methods), splines of different degree (Petrov-Galerkin methods), delta functions (collocation), or trigonometric polynomials (spline-trig methods). The choice of test functions is shown to have a significant influence on the convergence properties. Keywords: integral equations, Galerkin methods, collocation Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/singint.pdf ==> soboleveqn-info.txt <== Title: Superconvergence of the finite element appoximation of the solution of a Sobolev partial differential equation in a single space variable Authors: Douglas N. Arnold, Jim Douglas, Jr., and Vidar Thomée Source: Math. Comp. 36 (1981), pp. 53-63 Status: Published Abstract: A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit single-step methods is discussed. Keywords: superconvergence, finite element, Sobolev equation Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/soboleveqn.pdf ==> splinecol-info.txt <== Title: The convergence of spline collocation for strongly elliptic equations on curves Authors: Douglas N. Arnold and Wolfgang L. Wendland Source: Numer. Math. 47 (1985), pp. 317-341 Status: Published Abstract: Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations. Subj. class.: 65R20, 65N99, 65N30, 65E05, 30C30, 73K30, 65N35 URL: http://ima.umn.edu/~arnold/papers/splinecol.pdf ==> splinetrig-info.txt <== Title: A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method Authors: Douglas N. Arnold Source: Math. Comp. 41 (1981), pp. 383-397 Status: Published Abstract: We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions. We analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the Dirichlet problem in the interior or exterior of an analytic plane curve. In contrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hilbert spaces including all the Sobolev spaces of negative order. As a consequence we prove that the approximate solution to the Dirichlet problem and all its derivatives converge pointwise with exponential rate. Keywords: spline, spline-trigonometric, Galerkin method, boundary integral Subj. class.: 65R20, 65N30, 65E05, 45L10, 45B05, 41A15 URL: http://ima.umn.edu/~arnold/papers/splinetrig.pdf ==> stokes-info.txt <== Title: Local error estimates for finite element discretizations of the Stokes equations Authors: Douglas N. Arnold and Xiaobo Liu Source: Math. Modelling Numer. Anal. 29 (1995), pp. 367-389 Status: Published Abstract: Local error estimates are derived which apply to most stable finite mixed finite element discretizations of the stationary Stokes equations. Keywords: Stokes equations, mixed finite element method, local error estimates, interior error estimates Subj. class.: 65N30, 65N15, 76M10, 76D07 URL: http://ima.umn.edu/~arnold/papers/stokes.pdf ==> supercon-info.txt <== Title: Superconvergence of the Galerkin approximation of a quasilinear parabolic equation in a single space variable Authors: Douglas N. Arnold and Jim Douglas, Jr. Source: Calcolo 16 (1979), pp. 346-369 Status: Published Abstract: The asympotic expansion of the Galerkin solution of a parabolic equation by means of a sequence of elliptic projections that was introduced by Douglas, Dupont, and Wheeler is carried out for a quasilinear equation. This quasi-projection can be applied to establish knot superconvergence in the case of a single space variable. In addition, an optimal order error estimate in L-infinity(L-infinity) is derived for a single space variable. Keywords: finite element, parabolic equation, superconvergence Subj. class.: 65N30 URL: http://ima.umn.edu/~arnold/papers/supercon.pdf ==> vecquad-info.txt <== Title: Quadrilateral H(div) finite elements Authors: Douglas N. Arnold, Daniele Boffi, and Richard S. Falk Source: SIAM J. Numer. Anal. 42 (2005), pp. 2429-2451 Status: Published Abstract: We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to a space of vector fields on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism of the square onto the element. For affine isomorphisms, a necessary and sufficientcondition for approximation of order r+1 in L2 is that each component ofthe given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms,the situation is more complicated and we give a precise characterization of what is needed for optimal order L2-approximation of the function and of its divergence. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of H(div). We also derive new estimates for approximation by quadrilateral Raviart-Thomas elements (requiring less regularity) and propose a new quadrilateral finite element space which provides optimal order approximation in H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares finite element aproximations of the solution of Poisson's equation. Keywords: quadrilateral, finite element, approximation, mixed finite element Subj. class.: 65N30, 41A10, 41A25, 41A27, 41A63 URL: http://ima.umn.edu/~arnold/papers/vecquad.pdf