# <span class=strong>Reception and Poster Session</span><br><br/><br/><b>Poster submissions welcome from all participants</b><br><br/><br/><a<br/><br/>href=/visitor-folder/contents/workshop.html#poster><b>Instructions</b></a>

Monday, November 29, 2010 - 4:00pm - 5:30pm

Lind 400

**Scalable electromagnetic simulations with the**

Auxiliary-space Maxwell Solver (AMS)

Tzanio Kolev (Lawrence Livermore National Laboratory)

Second-order definite Maxwell problems arise in many practical applications, such as the modeling of electromagnetic diffusion in ALE-MHD simulations. Typically, these problems are discretized with Nedelec finite elements resulting in a large sparse linear system which is challenging for linear solvers due to the large nullspace of curl-operator. In this poster we describe our work on the Auxiliary-space Maxwell Solver (AMS) which is a provably efficient scalable code for solving definite Maxwell problems based on the recent Hiptmair-Xu (HX) decomposition of the lowest-order Nedelec space. We demonstrate the scalability of the method and its robustness with respect to jumps in material coefficients. We also report some results from recent work on the algebraic extension of the AMS algorithm and theory to linear systems obtained by explicit element reduction.**Robin-Robin domain decomposition methods for**

Stokes-Darcy model

Xiaoming He (Missouri University of Science and Technology)

Two parallel domain decomposition methods for solving the coupled Stokes-Darcy model are proposed and analyzed. One is for a steady state Stokes-Darcy model with BJ condition and the other one is for a time-dependent Stokes-Darcy model with BJS condition. Robin boundary conditions are utilized to decouple the Stokes and Darcy parts of the system. Two numerical examples are used to illustrate the features of the two methods and confirm the theoretical results respectively.**Convergence and optimality of adaptive finite element methods**

Christian Kreuzer (Universität Duisburg-Essen)

We present convergence and optimality results for a standard AFEM

SOLVE -> ESTIMATE -> MARK -> REFINE.

The results range from plain convergence for inf-sup problems to

contraction properties and quasi-optimal rates of an AFEM with Dorfler

marking for elliptic problems.

Beyond that we show how these results can be used to design convergent

adaptive methods for a linear parabolic pde and a nonlinear stationary

Stokes problem.**H-LU factorization of stabilized saddle point problems**

Sabine Le Borne (Tennessee Technological University)

The (mixed finite element) discretization of the linearized Navier-Stokes

equations leads to a linear system of equations of saddle point type.

The iterative solution of this linear system requires the construction

of suitable preconditioners, especially in the case of high Reynolds

numbers. In the past, a stabilizing approach has been suggested which

does not change the exact solution but influences the accuracy of

the discrete solution as well as the effectiveness of iterative solvers.

This stabilization technique can be performed on the continuous side before

the discretization,

where it is known as grad-div stabilization, as well as on the discrete

side where it is known as an augmented Lagrangian technique

(and does not change the discrete solution).

We study the applicability of H-LU factorizations to solve the

arising subproblems in the different variants of stabilized saddle point

systems.**A comparison of two domain decomposition methods for a**

linearized contact problem

Jungho Lee (Argonne National Laboratory)

We compare two domain decomposition methods for a linearized contact

problem. The first method we consider has been used in an engineering

community; we provide theoretical and numerical evidence that this

method is not scalable with respect to the number of subdomains

(processors). We propose a scalable alternative and analyze its

properties, both theoretically and numerically. We also solve a model

problem using a combination of a primal-dual active set method, viewed

as a semismooth Newton method, and the scalable alternative.**3D boundary integral analysis by a precorrected fast Fourier transform algorithm**

Sylvain Nintcheu Fata (Oak Ridge National Laboratory)

An acceleration of a Galerkin boundary integral equation (BIE)

method for solving the three-dimensional Laplace equation is

investigated in the context of the precorrected fast Fourier

transform (PFFT) scheme. The PFFT technique is an

algorithm for rapid computation of the

dense matrix-vector products arising in an iterative solution

of discretized integral equations. In the PFFT method, the

problem domain is overlaid with a

regular Cartesian grid that serves as an auxiliary platform for

computation. With the aid of the fast Fourier transform (FFT)

procedure, the necessary influence matrices of the discretized

problem are rapidly evaluated on the Fourier grid in a

sparse manner resulting in a significant reduction in execution

time and computer memory requirements.

This research was supported by the Office of Advanced

Scientific Computing Research,

U.S. Department of Energy, under contract DE-AC05-00OR22725

with UT-Battelle, LLC.**Finite element analysis and a fast solver approach to a nonlocal dielectric continuum model**

Dexuan Xie (University of Wisconsin)

The nonlocal continuum dielectric model is an important extension of the classical Poisson dielectric model. This poster will report some recent results we made on the finite element analysis and fast solver development for one commonly-used nonlocal continuum dielectric model. We first prove that the finite element equation of this model has the unique solution but leads to a dense linear system, which is very expansive to be solved. Surprisingly, we then discover and prove that such a dense linear system can be converted to a system of two sparse finite element equations in a form similar to the standard mixed finite element equation. In this way, fast numerical solvers can be developed to solve the nonlocal continuum dielectric model in an optimal order. Some numerical results in free energy calculation will also be presented to demonstrate the great promise of nonlocal dielectric modeling in improving the accuracy of the classic Poisson dielectric model in computing electrostatic potential energies. This project is a joined work with Prof. Ridgeway Scott, Peter Brune, (both from University of Chicago), and Yi Jiang under the support of NSF grant #DMS-0921004.**An efficient rearrangement algorithm for shape**

optimization on eigenvalue problems

Chiu-Yen Kao (The Ohio State University)

In this poster, an efficient rearrangement algorithm is proposed to find the optimal shape and topology for eigenvalue problems in an inhomogeneous media. The method is based on Rayleigh quotient formulation of eigenvalue and a monotone iteration process to achieve the optimality. The common numerical approach for these problems is to start with an initial guess for the shape and then gradually evolve it, until it morphs into the optimal shape. One of the difficulties is that the topology of the optimal shape is unknown. Developing numerical techniques which can automatically handle topology changes becomes essential for shape and topology optimization problems. The level set approach based on both shape derivatives and topological derivatives has been well known for its ability to handle topology changes. However, CFL constrain significantly slows down the algorithm when the mesh is further refined. Due to the efficient rearrangement, the new method not only has the ability of topological changes but also is exempt from CFL condition. We provides numerous numerical examples to demonstrate the robustness and efficiency of our approach.**Two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method**

Eun-Hee Park (Louisiana State University)

The weakly over-penalized symmetric interior penalty (WOPSIP) method was introduced for second order elliptic problems by Brenner et al. in 2008. It belongs to the family of discontinuous Galerkin methods. We will discuss two-level additive Schwarz preconditioners for the WOPSIP method. The key ingredient of the two-level additive Schwarz preconditioner is the construction of the subdomain solvers and the coarse solver. In our approach, we consider different choices of coarse spaces and intergrid transfer operators. It is shown that the condition number estimates previously obtained for classical finite element methods also hold for the WOPSIP method. Numerical results will be presented, which illustrate the parallel performance of these preconditioners.

This is joint work with Andrew T. Barker, Susanne C. Brenner, and Li-yeng Sung.**Adaptivity for the Hodge decomposition of Maxwell's**

equations

Recently a new numerical method for the two-dimensional

Maxwell's equation based on the Hodge decomposition for

divergence-free vector fields has been introduced by Brenner,

Cui, Nan and Sung. The advantage of this new approach is that

an approximation of the vector field is obtained by solving

several standard second order scalar elliptic boundary value

problems instead of using more complicated methods. For the

linear Courant finite elements standard energy residual type a

posteriori error techniques can be applied to obtain guaranteed

upper bounds for the L^{2}-error. For smooth solutions a duality

argument shows reliability of an L^{2}residual type a posteriori

error estimator for the H(curl)-error. A dual weighted residual

error estimator is derived for singular solutions. Numerical

experiments verify reliability and show empirically efficiency

of the proposed error estimators. It is shown that adaptive

mesh-refinement numerically leads to optimal convergence rates

for general domains that are non-convex and may include holes.**Domain decomposition preconditioning for the hp-version**

of the discontinuous Galerkin method

Paola Antonietti (Politecnico di Milano)

We address the problem of efficiently solving the algebraic

linear systems of equations arising from

the discretization of a symmetric, elliptic boundary value

problem using hp discontinuous Galerkin finite element methods.

We introduce a class of domain decomposition preconditioners based

on the Schwarz framework, and prove bounds on the condition

number of the resulting iteration operators. Numerical results

confirming the theoretical estimates are also presented.

Joint work with

Paul Houston, University of Nottingham, UK.**Fast adaptive collocation by radial basis functions**

Tobin Driscoll (University of Delaware)

Radial basis

functions provide flexible, meshfree approximations to functions and

solutions of differential equations. Naive algorithms suffer from

dense linear algebra and severe ill conditioning. Simple multiscale

adaptive techniques for the nodes and shape parameters have previously

proven very effective in controlling ill conditioning for small node

sets. We present a new fast summation method suitable for adaptively

generated basis functions with varying shape parameters. When coupled

with an easily parallelized restricted additive Schwarz

preconditioner, the method can find RBF coefficients in near O(N log

N) time for N nodes.**Energy minimization algebraic multigrid: Robustness and**

flexibility in multilevel software

Ray Tuminaro (Sandia National Laboratories)

Energy minimization provides a general framework for developing a family of multigrid algorithms.

The proposed strategy is applicable to Hermitian, non-Hermitian, definite, and indefinite problems.

Each column of the grid transfer operator P is minimized in an energy-based norm while enforcing two types of constraints: a defined sparsity pattern and preservation of specified modes in the range of P. A Krylov-based

strategy is used to minimize energy, which is equivalent to solving A P = 0 with the constraints ensuring a

nontrivial solution. For the Hermitian positive definite case, a conjugate gradient-based (CG)

method is utilized to construct grid transfers, while methods based on generalized minimum residual

(GMRES) and CG on the normal equations (CGNR) are explored for the general case.

One of the main advantages of the approach is that it is flexible, allowing for arbitrary coarsenings,

unrestricted sparsity patterns, straightforward long distance interpolation, and general use of constraints,

either user-defined or auto-generated. We illustrate how this flexibility can be used to adapt an

algebraic multigrid scheme to an extended finite element discretization suitable for modeling fracture.

Computational results are presented illustrate that this particular energy minimization scheme gives rise

to mesh independent convergence rates and is relatively insensitive to the number and location of cracks

being modeled.**Multigrid methods for two-dimensional Maxwell's equations on graded meshes**

Jintao Cui (University of Minnesota, Twin Cities)

In this work we investigate the numerical solution for

two-dimensional Maxwell's equations on graded meshes.

The approach is based on the Hodge decomposition. The solution*u*of Maxwell's equations is approximated by

solving standard second order elliptic problems. Quasi-optimal

error estimates for both*u*and

curl of*u*in the*L*norm are obtained on graded meshes. We_{2}

prove the uniform convergence of the W-cycle and

full multigrid algorithms for the resulting discrete problem.**Two-level additive Schwarz preconditioners for the local discontinuous**

Galerkin method

Andrew Barker (Louisiana State University)

We propose and analyze two-level overlapping additive Schwarz

preconditioners for the local discontinuous Galerkin discretization. We

prove a condition number estimate and show numerically that the method is

scalable in terms of linear iterations. We also present numerical

evidence that a parallel implementation of the method shows good

scalability and speedup.**Adaptive solution of parametric eigenvalue problems for partial differential equations**

Eigenvalue problems for partial differential equations (PDEs) arise in a large number of current technological applications, e.g., in the computation of the acoustic field inside vehicles (such as cars, trains or airplanes). Another current key application is the noise compensation in highly efficient motors and turbines. For the analysis of standard adaptive finite element methods an exact solution of the discretized algebraic eigenvalue problem is required, and the error and complexity of the algebraic eigenvalue problems are ignored. In the context of eigenvalue problems these costs often dominate the overall costs and because of that, the error estimates for the solution of the algebraic eigenvalue problem with an iterative method have to be included in the adaptation process. The goal of our work is to derive adaptive methods of optimal complexity for the solution of PDE-eigenvalue problems including problems with parameter variations in the context of homotopy methods. In order to obtain low (or even optimal) complexity methods, we derive and analyse methods that adapt with respect to the computational grid, the accuracy of the iterative solver for the algebraic eigenvalue problem, and also with respect to the parameter variation. Such adaptive methods require the investigation of a priori and a posteriori error estimates in all three directions of adaptation. As a model problem we study eigenvalue problems that arise in convection-diffusion problems. We developed robust a posteriori error estimators for the discretization as well as for the iterative solver errors, first for self-adjoint second order eigenvalue problems (undamped problem, diffusion problem), and then bring in the non-selfadjoint part (damping, convection) via a homotopy, where the step-size control for the homotopy is included in the adaptation process.**A jumping multigrid method via finite element extrapolation**

Shangyou Zhang (University of Delaware)

The multigrid method solves the finite element equations in optimal

order, i.e., solving a linear system of $O(N)$ equations in

$O(N)$ arithmetic operations.

Based on low level solutions, we can use finite element extrapolation

to obtain the high-level finite element solution on some

coarse-level element boundary, at an higher accuracy $O(h_i4)$.

Thus, we can solve higher level $(h_j, j\underset{\sim}{ finite element problems locally on each such

coarse-level element.

That is, we can skip the finite element problem

on middle levels, $h_{i+1},h_{i+2},\dots, h_{j-1}$.

Roughly speaking,

such a jumping multigrid method solves an order $O(N)=O(2^{2di})$

linear system of equations by

a memory of $O(\sqrt N)=O(2^{di})$, and by a parallel computation

of $O(\sqrt N)$, where $d$ is the space dimension.**Hypre: A scalable linear solver library**

Ulrike Yang (Lawrence Livermore National Laboratory)

Hypre is a software library for solving large, sparse linear systems of equations on massively parallel computers. The library was created with the primary goal of providing users with advanced parallel preconditioners. The library features parallel multigrid solvers for both structured and unstructured grid problems. For ease of use, these solvers are accessed from the application code via hypre’s conceptual linear system interfaces, which allow a variety of natural problem descriptions. The motivation for the design of hypre, an overview of its interfaces and some of its performance highlights are presented.