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

On Some Versions of the Element Agglomeration AMGe Method

The present paper deals with element-based AMG methods that target linear systems of equations coming from finite element discretizations of elliptic PDEs. The individual element information (element matrices and element topology) is the main input to construct the AMG hierarchy. We study a number of variants of the spectral agglomerate element based AMG method. The core of the algorithms relies on element agglomeration utilizing the element topology (built recursively from fine to coarse levels). The actual selection of the coarse degrees of freedom (dofs) is based on solving large number of local eigenvalue problems. Additionally, we investigate strategies for adaptive AMG as well as multigrid cycles that are more expensive than the V-cycle utilizing simple interpolation matrices and nested conjugate gradient (CG) based recursive calls between the levels. The presented algorithms are illustrated with an extensive set of experiments based on a matlab implementation of the methods

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

AMG by element agglomeration and constrained energy minimization interpolation

This paper studies AMG (algebraic multigrid) methods that utilize energy minimization construction of the interpolation matrices locally, in the setting of element agglomeration AMG. The coarsening in element agglomeration AMG is done by agglomerating fine-grid elements, with coarse element matrices defined by a local Galerkin procedure applied to the matrix assembled from the individual fine-grid element matrices. This local Galerkin procedure involves only the coarse basis restricted to the agglomerated element. To construct the coarse basis, one exploits previously proposed constraint energy minimization procedures now applied to the local matrix. The constraints are that a given set of vectors should be interpolated exactly, not only globally, but also locally on every agglomerated element. The paper provides algorithmic details, as well as a convergence result based on a ''local-to-global'' energy bound of the resulting multiple-vector fitting AMG interpolation mappings. A particular implementation of the method is illustrated with a set of numerical experiments