124 research outputs found
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods
An efficient -multigrid scheme is presented for local discontinuous
Galerkin (LDG) discretizations of elliptic problems, formulated around the idea
of separately coarsening the underlying discrete gradient and divergence
operators. We show that traditional multigrid coarsening of the primal
formulation leads to poor and suboptimal multigrid performance, whereas
coarsening of the flux formulation leads to optimal convergence and is
equivalent to a purely geometric multigrid method. The resulting
operator-coarsening schemes do not require the entire mesh hierarchy to be
explicitly built, thereby obviating the need to compute quadrature rules,
lifting operators, and other mesh-related quantities on coarse meshes. We show
that good multigrid convergence rates are achieved in a variety of numerical
tests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curved
domains using implicitly defined meshes and for multi-phase elliptic interface
problems with complex geometry. Extension to non-LDG discretizations is briefly
discussed
Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems
Algebraic multilevel preconditioners for algebraic problems arising from the discretization of a class of systems of coupled elliptic partial differential equations (PDEs) are presented. These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods. A convergence theory for the twolevel case is presented
Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media
In this paper we propose and analyze a preconditioner for a system arising
from a finite element approximation of second order elliptic problems
describing processes in highly het- erogeneous media. Our approach uses the
technique of multilevel methods and the recently proposed preconditioner based
on additive Schur complement approximation by J. Kraus (see [8]). The main
results are the design and a theoretical and numerical justification of an
iterative method for such problems that is robust with respect to the contrast
of the media, defined as the ratio between the maximum and minimum values of
the coefficient (related to the permeability/conductivity).Comment: 28 page
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
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