# Poster Session and Reception (4th Floor Lind)

Wednesday, October 22, 2014 - 5:15pm - 6:45pm

Lind 400

**Time-Domain FEM for Wave Propagation in Metamaterials**

In this poster, we present our recent works on the mathematical study and time-domain finite element simulation of wave propagation in metamaterials.

Metamaterial is a new type nano materials successfully constructed first in 2000, and its many exotic properties have attracted intensive investigations by researchers from many areas. Here we present two interesting phenomena: backward wave propagation in metamaterials, and cloaking with metamaterials.**Speed-up of Finite Element Method in Reservoir Simulation**

Guosheng Fu (University of Minnesota, Twin Cities)

We use the mixed FEM to solve the pressure equation in reservoir simulation. And we speed up the method using the hybridization technique.

(The material in the poster is part of the work I've done at ExxonMobil as a summer intern this summer.)**A Posteriori Error Estimates of Discontinuous Galerkin Methods for the Signorini Problem**

Kamana Porwal (Louisiana State University)

A reliable and efficient a posteriori error estimator is derived for a class of

discontinuous Galerkin (DG) methods for the Signorini problem. A common property shared by many DG methods leads to a unified error analysis with the help of a constraint preserving enriching map. The error estimator of DG methods is comparable with the error estimator of the conforming methods. Numerical experiments illustrate the performance of the error estimator.**Mixed FEM For The Stefan Problem With Surface Tension**

Shawn Walker (Louisiana State University)

A mixed formulation is proposed for the Stefan problem with surface

tension (Gibbs-Thomson law). The method uses a mixed form of the heat

equation in the solid and liquid domains, and imposes the interface motion

law (on the solid-liquid interface) as a constraint. Well-posedness of

the time semi-discrete and fully discrete (finite element) formulations is

proved in 3-D, as well as an a priori bound, conservation law, and error

estimates. Simulations are presented in 2-D.**The Periodic Table of the Finite Elements**

Douglas Arnold (University of Minnesota, Twin Cities)

This poster displays a sort of periodic table of the

primary finite element spaces used to discretize the

fundamental operators of vector calculus. The

systematic tabulation is based on the principles of

finite element exterior calculus. Specifically,

elements are organized according to four families, two

each for simplices and cubes, dimension, differential form degree, and polynomial degree. In this way 108 elements

(all cases with dimension at most 3 and polynomial degree at

most 3) are presented graphically in the poster

and at an accompanying website femtable.org.**Optimal Symmetric Mixed Finite Elements for Linear Elasticity on Triangular and on Tetrahedral Grids**

Shangyou Zhang (University of Delaware)

It was pointed out, by D. N. Arnold in his plenary address

to the 2002 International Congress of Mathematicians,

that four decades of

searching for mixed finite elements for elasticity did not yield any

stable elements with polynomial shape functions.

In the same year, Arnold and Winther constructed first ever

family of

mixed finite elements of polynomial degree 3 or higher,

on triangular grids.

In 2008, Awanou, Arnold and Winther contructed a family

of mixed finite elements of degree 4 or higher on tetrahedral grids.

However, all these finite elements are not of optimal order.

But they are the only symmetric elements ever found.

We construct a new family of finite elements in 2D, and

another family of symmetric mixed

finite elements in 3D.

The new elements are of optimal order, one order (in 2D) and

two orders (in 3D) higher

the above mentioned, all existing conforming elements.**Immersed Finite Element Methods with Enhanced Stability**

Xu Zhang (Purdue University)

In this poster, we present new immersed finite element (IFE) methods for the second-order elliptic interface problems. Comparing with classic IFE schemes using Galerkin formulation, our new IFE methods contain additional stability terms. A priori error estimates show that these new methods converge optimally in corresponding energy norms. Numerical experiments indicate that these stabilized IFE methods outperform classic IFEs at vicinity of interfaces.**An Angular Momentum Method for the Wave Map to the Sphere**

Franziska Weber (University of Oslo)

We present a convergent finite difference method for approximating wave maps into the sphere. The method is based on a reformulation of the second order equation as a first order system by using the angular momentum as an auxilary variable. This way, the method is able to conserve the length constraint as well as a discrete version of the energy associated to the equation.**A Structure-Preserving Discretization and Robust Preconditioner for Incompressible MHD Equations**

Yicong Ma (The Pennsylvania State University)

It is important to preserve the divergence-free condition of magnetic field when discretizing MHD equations. A stable finite element discretization will be presented, so is a uniform solver for this discretization. Numerical results are provided to verify the structure-preserving property and demonstrate the effectiveness of the solver.**Finite Element Methods for the Evolution Problem in General Relativity**

Vincent Quenneville-Bélair (University of Minnesota)

Gravitational waves can be understood as small ripples in the fabric of the Universe, caused by moving masses. To detect such weak waves, several new gravitational wave observatories are being built. Computer simulations are essential for determining expected signals and interpreting the data. This project consists of the design and implementation of a new mixed finite element method for the propagation of gravitational waves, by adapting the recently developed Finite Element Exterior Calculus framework.**A Saddle Point Least Squares Method**

Constantin Bacuta (University of Delaware)

We introduce a saddle point least-squares method for solving first order systems of PDEs.

The discretization algorithm is based on the availability of nested finite element approximation spaces for the test space. The way the discrete test spaces are constructed is dual to the DPG method. A discrete trial spaces is obtained as the image of the discrete test space via a related differential operator. On each fixed level an efficient solver such as the the conjugate gradient algorithm for inverting a Schur complement is implemented. As a main application of our approach we define the saddle point least-squares method for solving first order systems of PDEs. We apply our saddle point least-squares method to discretizing the time harmonic Maxwell equations. The variables of interest for the Maxwell equations, the magnetic and electric vector fields, become the constrained variable in the saddle point least-squares formulation, and bases or stiffness matrices associated with these variables are not needed. The saddle point least-squares iterative discretization we build does not involve edge elements or spaces of bubble functions, and is suitable for preconditioning.**Higher-Order FEM To A Class of Interface Problems**

Manuel Sanchez Uribe (Brown University)

We present a higher-order finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side of the natural method. We prove optimal error estimates of the method on general quasi-uniform meshes in the maximum norms. In addition, we apply the method to a Stokes interface problem obtaining optimal result.**Numerical Weather Prediction with Topography**

We aim to study a finite volume scheme to solve the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions. In that respect, a version of a projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. The resulting scheme allows for a significant reduction of the errors near the topography when compared to more standard finite volume schemes. We then report on numerical experiments using realistic parameters.

Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated. The numerical results show that such a forcing is responsible for recurrent large-scale patterns to emerge in the temperature and velocity fields.**Equation Free Method for Computing Self-Semilar Solutions of Smoluchowski's Dynamics**

Xingjie Li (Brown University)

We illustrate the computation of coarsely self-

similar solutions of Smoluchowski 's dynamics with additive kernel

K(x,y)=x+y using the recently developed equation-free approach.

Usually, traditional long-time numerical integrator fails to get self-

similar solutions due to accumulated errors, and the number of particles in the particle simulation decreases below the minimal resolution requirement because the dynamic system is about the collision of sticky particles.

However, we show that dynamic renormalization and fixed-point algorithms successfully overcome these difficulties. This approach is promising for numerically finding one-parameter families of self-similar solutions and their corresponding exponents for certain types of initial conditions. It works for

exponential decayed self-similar solutions and heavy-tailed solutions, which are maximally-skewed alpha-stable L'{e}vy distributions.

Moreover, this method captures asymptotical behaviors of exact self- similar solutions and approximate the first moments of these solutions

with satisfactory accuracy.**A Generalized Multiscale Finite Element Method for the Brinkman Equation**

Guanglian Li (Texas A & M University)Ke Shi (Texas A & M University)

In this paper we consider the numerical upscaling of the Brinkman equation in the presence

of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized

Multiscale Finite Element Method (GMsFEM) for the Brinkman model. In the fine grid, we use

mixed finite element method with the velocity and pressure being continuous piecewise quadratic

and piecewise constant finite element spaces, respectively. Using the GMsFEM framework we

construct suitable coarse-scale spaces for the velocity and pressure that yield a robust mixed

GMsFEM. We develop a novel approach to construct a coarse approximation for the velocity

snapshot space and a robust small offline space for the velocity space. The stability of the mixed

GMsFEM and a priori error estimates are derived. A variety of two-dimensional numerical

examples are presented to illustrate the effectiveness of the algorithm.**Certified Reduced Basis Method and Reduced Collocation Method**

Yanlai Chen (University of Massachusetts Dartmouth)

Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized partial differential equation are desired in a fast/real-time fashion. These include simulation-based design, parameter optimization, optimal control, multi-model/scale analysis, uncertainty quantification etc. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RBM solution is maintained through a rigorous a posteriori error estimator whose efficient development is critical.

In this poster, I will give a brief introduction of the RBM and discuss recent and ongoing efforts to develop RCM, and the accompanying parametric analytical preconditioning techniques which are capable of improving the quality of the error estimation uniformly on the parameter domain, and speeding up the convergence of the reduced solution to the truth approximation significantly.**Adaptivity for Two Dimensional Time Harmonic Maxwell Equations**

Joscha Gedicke (Louisiana State University)

We present an adaptive finite element method for time harmonic Maxwell equations with impedance boundary condition on the outer boundary and perfectly conducting boundary conditions on the inner part of the boundary. The analysis is presented for inhomogeneous and anisotropic permittivity and possibly sign changing permeability. This approach is based on the Hodge decomposition in two dimensions and simplifies Maxwell equations to solving standard second order elliptic boundary value problems. An a posteriori error estimator is developed which is proven to be reliable. Numerical experiments show efficient approximations of a cloaking simulation and the simulation of a flat lens.**Stable and Efficient Schemes for Parabolic Problems using the Method of Lines Transpose**

Hana Cho (Michigan State University)

As followed up to [1], we present a novel numerical scheme suitable for solving parabolic differential equation model using the Method of Lines Transpose (MOLT) combined with the successive convolution operators.

The primary advantage is that the operator can be computed quickly in O(N) work, to high precision;

and a multi dimensional solution is formed by dimensional sweeps. We demonstrate our solver on the Allen-Cahn and Cahn-Hilliard equation.**Free Boundary Problems with Surface Tension Effects**

Harbir Antil (George Mason University)

We are interested in PDE constrained optimization problems governed by free boundary problems. The state system is based on coupling of bulk equations

(Laplace or Stokes) with a Young-Laplace equation on the free boundary to account for surface tension.**Multigrid Methods for Constrained Minimization Problems and Application to Saddle Point Problems**

Long Chen (University of California)

The first order condition of the constrained minimization problem leads to a saddle point problem. A multigrid method using a multiplicative Schwarz smoother for saddle point problems can thus be interpreted as a successive subspace optimization method based on a multilevel decomposition of the constraint space. Convergence theory is developed for successive subspace optimization methods based on two assumptions on the space decomposition: stable decomposition and strengthened Cauchy-Schwarz inequality, and successfully applied to the saddle point systems arising from mixed finite element methods for Poisson and Stokes equations. Uniform convergence is obtained without the full regularity assumption of the underlying partial differential equations. As a byproduct, a V-cycle multigrid method for non-conforming finite elements is developed and proved to be uniform convergent with even one smoothing step.**A New Covariant form of the Equations of Geophysical Fluid Dynamics and Their Structure-Preserving Discretization**

Werner Bauer (Universität Hamburg)

I introduce n-dimensional equations of geophysical fluid dynamics (GFD). Using straight and twisted differential forms, I split these analytical equations into a metric-free part, consisting of a topological momentum and continuity

equation, and into a metric-dependent part, consisting of closure equations that map between the topological equations by using the Hodge star operator.

This splitting of the analytical equations of GFD enables their systematic, structure-preserving, discretization using methods of Discrete Exterior Calculus and Finite Element Exterior Calculus. The approximation of the topological manifold by discrete meshes and of the differential forms by suitable finite elements provides metric-free algebraic momentum and continuity equations. Such equations exist on independent primal and dual meshes. A discrete Hodge star (e.g. a diagonal matrix) connects these meshes,

carries metric information, and closes the algebraic set of equations.

This discretization technique, once applied to the split equations of GFD, enables to preserve in the discrete case, too, the separation into topological and metric-dependent terms. In comparison to other formulations, this translates into a clear advantage in terms of easiness of discretization

and efficiency of implementation.**Well-posedness and Robust Preconditioners for Discretized Fluid-Structure Interaction Systems**

Kai Yang (The Pennsylvania State University)

In our work we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models. After the time discretization, we formulate the fluid-structure interaction equations as saddle point problems and prove the uniform well-posedness. Then we discretize the space dimension by finite element methods and prove their uniform well-posedness by two different approaches under appropriate assumptions. The uniform well-posedness makes it possible to design robust preconditioners for the discretized fluid-structure interaction systems.**Non-Conforming Approaches to Discrete de Rham Complexes**

Martin Licht (University of Oslo)

Distributional finite element spaces are ubiquitous in computational partial differential equations, most prominently non-conforming methods (such as DG-FEM) and a posteriori error estimation. However, discrete de Rham complexes have only found scattered usage, for example in the construction of equilibrated residual error estimators for Maxwell's equations. We present complexes of discrete distributional differential forms within the framework of finite element exterior calculus.

On the one hand, their harmonic forms realize the de Rham cohomology on domains of any topology, and we also include general partial boundary conditions in the theory. This yields a new isomorphism between the harmonic forms of the finite element complex and the simplicial homology of the triangulation. On the other hand, uniform bounds for Poincaré-Friedrichs-type inequalities for mesh-dependent norms hold. Their derivation involves a generalized flux reconstruction that is of independent interest.

Double complexes emerge as an important concept in finite element differential forms, and jump-term-like operators are identified as differential operators on their own.**Mixed Finite Element Method for a Pressure Poisson Equations Reformulation of the Incompressible Navier-Stokes Equations**

Dong Zhou (Temple University)

Popular schemes for the incompressible Navier-Stokes equations (NSE), such as projection methods, are efficient but may introduce numerical boundary layers or have limited temporal accuracy due to their fractional step nature. Pressure Poisson equation (PPE) reformulations represent a class of methods that replace the incompressibility constraint by a Poisson equation for the pressure, with a suitable choice of the boundary condition so that the incompressibility is maintained. PPE reformulations of the NSE have important advantages: the pressure is no longer implicitly coupled to the velocity, thus can be directly recovered by solving a Poisson equation, and no numerical boundary layers are generated. We focus on numerical approaches of the Shirokoff-Rosales PPE reformulation. Interestingly, the electric boundary conditions provided for the velocity render classical nodal finite elements non-convergent. We thus present an alternative methodology, mixed finite element methods, and demonstrate that these approaches allow for arbitrary order of accuracy both in space and in time.**Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System**

Amanda Diegel (University of Tennessee)

We devise and analyze a mixed finite element method for a Cahn-Hilliard

equation coupled with a non-steady Darcy-Stokes flow that models phase

separation and coupled fluid flow in immiscible binary fluids and

diblock

copolymer melts. The time discretization is based on a convex splitting

of

the energy of the equation. We prove that our scheme is unconditionally

energy stable with respect to a spatially discrete analogue of the

continuous free energy of the system and unconditionally uniquely

solvable. We prove that the phase variable is bounded in L∞ (0, T

;

L∞) and the chemical potential is bounded in L∞ (0, T ; L2),

for any time and space step sizes, in two and three dimensions, and for

any finite final time T . We subsequently prove that these variables

converge with optimal rates in the appropriate energy norms in both two

and three dimensions.**Finite Element Differential Forms on Curvilinear Cubic Meshes and Their Approximation Properties**

Francesca Bonizzoni (University of Vienna )

We study the approximation properties of a wide class of finite element differential forms

on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element

is the image of a cubical reference element under a diffeomorphism, and finite element

spaces in which the shape functions and degrees of freedom are obtained from the reference

element by pullback of differential forms. In the case where the diffeomorphisms from

the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard

that the rate of convergence in L2 exceeds by one the degree of the largest full polynomial

space contained in the reference space of shape functions. When the diffeomorphism is

multilinear, the rate of convergence for the same space of reference shape function

may degrade severely, the more so when the form degree is larger. The main

gives a sufficient condition on the reference shape functions to obtain a given

rate of convergence.**M-Adaptation for the Electric Vector Wave Equation**

Vrushali Bokil (Oregon State University)

We present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time-domain electric vector wave equation on square meshes based on the mimetic finite difference (MFD) method. We compare this strategy, called M-adaptation, to the lowest order N ed elec edge element discretization. Both discrete methods use the same edge-based degrees of freedom, while the temporal discretization is performed using the standard explicit Leapfrog scheme. To obtain efficient and explicit time stepping methods, both schemes are further mass lumped. We perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms of their stability regions and phase error. Our results indicate that the N ed elec method is second order accurate for numerical dispersion. The result of M-adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy. Numerical simulations are provided that illustrate the theoretical results.**B-Splines Approximation for a Singularly Perturbed Reaction-Diffusion Equation Using Graded Meshes**

Mariana Prieto (University of Buenos Aires)

It is known that the solution of singularly perturbed reaction-diffusion

problem presents boundary layers and special methods are needed to obtain

good numerical approximations. For bilineal finite element, it was proved

that a good strategy to recover the order of convergence is to consider a

family of graded meshes. These meshes are more refined near the boundary

layers. It has been proved optimal order of convergence respect to the

degrees of freedom and quasi-uniform respect to the perturbation

parameter. These results are based on a precise knowledge of the boundary

layers. In particular, some a priori estimates of the solution in weighted

Sobolev spaces are needed, where the weights are powers of the distance to

the boundary layers.

In this work, we generalize these results to the approximation by

B-splines of arbitrary order. In this way, we can obtain optimal

approximation of high order and uniform in the perturbation parameter. The

graduation parameter is chosen according to the order of the B-spline

used.**EYE DEC A Software Environment to Model Blood Flow in the Eye**

Andrea Dziubek (State University of New York Institute of Technology)Edmond Rusjan (State University of New York Institute of Technology)

Together with external collaborators we developed a model based on first principles to to study the relation between perfusion pressure and ocular blood flow in the retina. We use a combination of Discrete Exterior Calculus (DEC), Finite Element Exterior Calculus (FFEC) and traditional Finite Elements to discretize the model. The application of this model to the blood flow in the retina supports the hypothesis that changes of the blood flow in the retina play a role in retinal pathologies, including open eye glaucoma and myopia.