Two-Level Additive Schwarz Methods for a Discontinuous Galerkin Approximation of Second Order Elliptic Problems

A second order elliptic problem with discontinuous coefficient in 2-D or 3-D is considered. The problem is discretized by a symmetric weighted interior penalty discontinuous Galerkin finite element method with nonmatching simplicial elements and piecewise linear functions. The resulting discrete problem is solved by a two-level additive Schwarz method with a relatively coarse grid and with local solves restricted to subdomains which can be as small as single element. In this way the method has a potential for a very high level of fine grained parallelism. Condition number estimate depending on the relative sizes of the underlying grids is provided. The rate of convergence of the method is independent of the jumps of the coefficient if its variation is moderate inside coarse grid substructures or on local solvers’ subdomain boundaries. Numerical experiments are reported which confirm theoretical results.

Two-level Schwarz methods for Wilson element approximation of elasticity problem

Scaling function and wavelet preconditioners for second order elliptic problems

The original additive average Schwarz method and its variants were introduced for solving second order elliptic boundary value problems. In this paper, we extend the original idea to fourth order problems by designing a variant of the additive average Schwarz method for the plate bending problem using the nonconforming Morley finite element for the discretization. Like the original average method this method uses only nonoverlapping subdomains, and its coarse space does not need any explicit coarse mesh. An analysis of the method, and some numerical experiments to validate the theory are presented.

A non-symmetric interface Schur complement arises from non- selfcon-jugate second order elliptic problems with domain decomposition methods. The usual numerical methods for solving it are GMRES, OR-THOMIN, and BICGSTAB, but they take a large amount of computer time and memory. The authors find in this paper that the nonsymmetric Schur complement can in fact be changed into a symmetric one by scaling. Then an efficient preconditioner can be provided by which the preconditioned system can be solved iteratively by a modified PCG method. When the problem is imposed on a rectangular region, the condition number is estimated and is nearly one. Numerical experiments are also presented. Non-selfconjugate problems arise in mathematical modeling and numerical simulation of fluid flows and transport in porous media.

Local ultraconvergence of linear and bilinear finite element method for second order elliptic problems

An economical cascadic multigrid method for the weak Galerkin finite element approximation of second order elliptic problems

A new approximate method for second order elliptic problems with rapidly oscillating coefficients based on the method of multiscale asymptotic expansions

We consider the multiplicative and additive Schwarz methods for solving linear systems of equations and we compare their asymptotic rate of convergence. Moreover, we compare the multiplicative Schwarz method with the weighted restricted additive Schwarz method. We prove that the multiplicative Schwarz method is the fastest method among these three. Our comparisons can be done in the case of exact and inexact subspaces solves. In addition, we analyse two ways of adding a coarse grid correction --- multiplicatively or additively.

Quadarant diagonal partitioning (QDP) method is an iterative method for solving linear system proposed by D. J. Evans et al. (see [1],[2],[3]) appropriate for paralell implementation. Here we show that in case of second order elliptic problem, the classical, color and multicolor ordering technique [4] are applicable for further paralellization of QDP iterative methods. Moreover we give a direct algorithm with arithmetical complexity O(n) for solving a special quadrant tridiagonal matrix and using these results we propose a quadrant tridiagonal preconditioned (QTP) conjugate gradient method for solving second order elliptic problems. The numerical experiments presented here shows that this method in several cases are more effective than other PCG algorithm.

We address the question of the rates of convergence of the p-version interior penalty discontinuous Galerkin method (p-IPDG) for second order elliptic problems with non-homogeneous Dirichlet boundary conditions. It is known that the p-IPDG method admits slightly suboptimal a-priori bounds with respect to the polynomial degree (in the Hilbertian Sobolev space setting). An example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated in practice through numerical experiments is presented. Moreover, the performance of p-IPDG on the related problem of p-approximation of corner singularities is assessed both theoretically and numerically, witnessing an almost doubling of the convergence rate of the p-IPDG method.

We will discuss the spectral equivalence of hierarchical matrix approximations for second order elliptic problems. Our theory will show that a modified variant of the hierarchical matrix Cholesky decomposition which preserves test vectors while truncating blocks to lower rank will lead to a spectrally equivalent approximation when using an adapted truncation threshold. Our theory also covers the usual hierarchical Cholesky decomposition which does not preserve test vectors but expects a significantly more restrictive threshold adaption to obtain a spectrally equivalent approximation. Numerical experiments indicate that the adaption of the truncation parameter seems to be necessary for the traditional hierarchical Cholesky preconditioner to obtain mesh-independent convergence while the variant which preserves test vectors works in practice quite well even with a fixed parameter.

Using the weighted residual formulation we derive a-posteriori estimates for Discontinuous Galerkin approximations of second order elliptic problems in mixed form. We show that our approach allows to include in a unified way all the methods presented so far in the literature.

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.MSC:35J30, 35J40.

This is a further development of [9] regarding multilevel preconditioning for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. We assume that the mesh on the finest level is a results of a geometrically refined fixed coarse mesh. The preconditioner is a multilevel method that uses a sequence of finite element spaces of either continuous or piecewise constant functions. The spaces are nested, but due to the penalty term in the DG method the corresponding forms are not inherited. For the continuous finite element spaces we show that the variable V-cycle provides an optimal preconditioner for the DG system. The piece-wise constant functions do not have approximation property so in order to control the energy growth of the inter-level transfer operator we apply W–cycle MG. Finally, we present a number of numerical experiments that support the theoretical findings.