Galerkin Approximations Of Elliptic Problems Research Articles

Two-Level Schwarz Methods for a Discontinuous Galerkin Approximation of Elliptic Problems with Jump Coefficients

We present two-level nonoverlapping and overlapping Schwarz preconditioners for the linear algebraic system arising from the weighted symmetric interior penalty Galerkin approximation of elliptic problems with highly heterogeneous coefficients. The coarse space is constructed by the local Dirichlet-to-Neumann maps the theoretical results show that the condition number of the preconditioned system is independent of the discontinuous coefficient, the number of subdomains and the mesh size for the nonoverlapping case. For the overlapping case adding an extra assumption of coefficient distribution, the similar conclusion is also obtained. Numerical experiments validate the theoretical results and illustrate the performance and robustness of the proposed two-level methods.

A BDDC algorithm with adaptive primal constraints for staggered discontinuous Galerkin approximation of elliptic problems with highly oscillating coefficients

A BDDC (Balancing Domain Decomposition by Constraints) algorithm for a staggered discontinuous Galerkin approximation is considered. After applying domain decomposition method, a global linear system on the subdomain interface unknowns is obtained and solved by the conjugate gradient method combined with a preconditioner. To construct a preconditioner that is robust to the coefficient variations, a generalized eigenvalue problem on each subdomain interface is solved and primal unknowns are selected from the eigenvectors using a predetermined tolerance. By the construction of the staggered discontinuous Galerkin methods, the degrees of freedom on subdomain interfaces are shared by only two subdomains, and hence the construction of primal unknowns are simplified. The resulting BDDC algorithm is shown to have the condition number bounded by the predetermined tolerance. A modified algorithm for parameter dependent problems is also introduced, where the primal unknowns are only computed in an offline stage. Numerical results are included to show the performance of the proposed method and to verify the theoretical estimate.

A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

In this paper we design and analyze a uniform preconditioner for a class of high-order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high-order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by numerical tests.

Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients

We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods.

Algebraic multilevel iteration method for lowest order Raviart–Thomas space and applications

AbstractAn optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems, that have their weak formulation in the space H(div), is presented. The algorithm is developed for the discrete problem obtained by using the lowest‐order Raviart–Thomas space. The method is theoretically analyzed and supporting numerical examples are presented. Furthermore, as a particular application, the algorithm is used for the solution of the discrete minimization problem which arises in the functional‐type a posteriori error estimates for the discontinuous Galerkin approximation of elliptic problems. Copyright © 2011 John Wiley & Sons, Ltd.

Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems

AbstractIn this article, we develop functional a posteriori error estimates for discontinuous Galerkin (DG) approximations of elliptic boundary‐value problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximations developed by S. Repin (see e.g., Math Comp 69 (2000) 481–500). On these grounds, we derive two‐sided guaranteed and computable bounds for the errors in “broken” energy norms. A series of numerical examples presented confirm the efficiency of the estimates. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems

We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.

Multiplicative Schwarz Methods for Discontinuous Galerkin Approximations of Elliptic Problems

In this paper we introduce and analyze some non-overlapping multiplicative Schwarz meth- ods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods. For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion on the issue of preconditioning non-symmetric DG approximations of elliptic problems is also included. Extensive numerical experiments to confirm the theoretical results and to assess the robustness and the efficiency of the proposed preconditioners are provided.

An accurate [formula omitted] flux reconstruction for discontinuous Galerkin approximations of elliptic problems

We introduce a new H ( div ) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. The reconstructed flux is computed elementwise and its divergence equals the L 2 -orthogonal projection of the source term onto the discrete space. Moreover, the energy-norm of the error in the flux is bounded by the discrete energy-norm of the error in the primal variable, independently of diffusion heterogeneities. To cite this article: A. Ern et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h) , where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.

Energy norm a posteriori error estimation for discontinuous Galerkin methods

In this paper we present a residual-based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous Galerkin approximations of elliptic problems. The theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed. We also present some illustrating numerical examples.

Discontinuous Galerkin approximations for elliptic problems

In this article, we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability and error estimates in various norms are proven. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 365–378, 2000

Superconvergence of the gradient of Galerkin approximations for elliptic problems

On considere la superconvergence pour un gradient moyen de la methode de Galerkin appliquee aux problemes de Dirichlet sur un carre. On utilise les produits tensoriels d'espaces de fonctions continues polynomiales par morceaux. On illustre les resultats theoriques par des exemples numeriques

Superconvergence of the gradient of Galerkin approximations for elliptic problems

On considere la superconvergence pour un gradient moyen de la methode de Galerkin appliquee aux problemes de Dirichlet sur un carre. On utilise les produits tensoriels d'espaces de fonctions continues polynomiales par morceaux. On illustre les resultats theoriques par des exemples numeriques