Exact de Rham Sequences of Spaces Defined on Macro-elements in Two and Three Spatial Dimensions

This paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure. The main requirement is that the agglomerates share faces that have closed boundaries composed of 1-d edges. The spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of H{sup 1}-conforming, H(curl) conforming, H(div) conforming and piecewise constant spaces associated with an unstructured 'fine' mesh. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an illustration we generate coarser spaces from the sequence corresponding to the lowest order Nedelec spaces, lowest order Raviart-Thomas spaces, and for piecewise linear H{sup 1}-conforming spaces, all in three-dimensions. The resulting V-cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case

Exact de Rham sequences of spaces defined on macro-elements in two and three spatial dimensions

Abstract. This paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure. The main requirement is that the agglomerates share faces that have closed boundaries composed of 1-d edges. The spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of H 1 -conforming, H(curl) conforming, H(div) conforming and piecewise constant spaces associated with an unstructured "fine" mesh. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an illustration we generate coarser spaces from the sequence corresponding to the lowest order Nédélec spaces, lowest order Raviart-Thomas spaces, and for piecewise linear H 1 -conforming spaces, all in three-dimensions. The resulting V -cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case

Parallel auxiliary space AMG for definite Maxwell problems

Motivated by the needs of large multi-physics simulation codes, we are interested in algebraic solvers for the linear systems arising in time-domain electromagnetic simulations. Our focus is on finite element discretization, and we are developing scalable parallel preconditioners which employ only fine-grid information, similar to algebraic multigrid (AMG) for diffusion problems. In the last few years, the search for efficient algebraic preconditioners for H(curl) bilinear forms has intensified. The attempts to directly construct AMG methods had some success, see [12, 1, 7]. Exploiting available multilevel methods on auxiliary mesh for the same bilinear form led to efficient auxiliary mesh preconditioners to unstructured problems as shown in [4, 8]. A computationally more attractive approach was recently proposed by Hiptmair and Xu [5]. In contrast to the auxiliary mesh idea, the method in [5] uses a nodal H{sup 1}-conforming auxiliary space on the same mesh. This significantly simplifies the computation of the corresponding interpolation operator. In the present talk, we consider several options for constructing unstructured mesh AMG preconditioners for H(curl) problems and report a summary of computational results from [10, 9]. Our approach is slightly different than the one from [5], since we apply AMG directly to variationally constructed coarse-grid operators, and therefore no additional Poisson matrices are needed on input. We also consider variable coefficient problems, including some that lead to a singular matrix. Both type of problems are of great practical importance and are not covered by the theory of [5]

Parallel H1-based auxiliary space AMG solver for H(curl) problems

This report describes a parallel implementation of the auxiliary space methods for definite Maxwell problems proposed in [4]. The solver, named AMS, extends our previous study [7]. AMS uses ParCSR sparse matrix storage and the parallel AMG (algebraic multigrid) solver BoomerAMG [1] from the hypre library. It is designed for general unstructured finite element discretizations of (semi)definite H(curl) problems discretized by Nedelec elements. We document the usage of AMS and illustrate its parallel scalability and overall performance

Overlapping Schwarz for Nonlinear Problems. An Element Agglomeration Nonlinear Additive Schwarz Preconditioned Newton Method for Unstructured Finite Element Problems

This paper extends previous results on nonlinear Schwarz preconditioning ([4]) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The non-local finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in [8]. Then, the algebraic construction from [9] of the corresponding non-linear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. Numerical illustration is also provided

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property

A combined first and second order variational approach for image reconstruction

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure