Results 1  10
of
20
Domain decomposition for multiscale PDEs
 Numer. Math
"... We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises of ..."
Abstract

Cited by 50 (19 self)
 Add to MetaCart
(Show Context)
We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains,
Robust Domain Decomposition Algorithms for Multiscale PDEs, submitted
, 2006
"... In this paper we describe a new class of domain deomposition preconditioners suitable for solving elliptic PDEs in highly fractured or heterogeneous media, such as arise in groundwater flow or oil recovery applications. Our methods employ novel coarsening operators which are adapted to the heterogen ..."
Abstract

Cited by 25 (13 self)
 Add to MetaCart
(Show Context)
In this paper we describe a new class of domain deomposition preconditioners suitable for solving elliptic PDEs in highly fractured or heterogeneous media, such as arise in groundwater flow or oil recovery applications. Our methods employ novel coarsening operators which are adapted to the heterogeneity of the media. In contrast to standard methods (based on piecewise polynomial coarsening), the new methods can achieve robustness with respect to coefficient discontinuities even when these are not resolved by a coarse mesh. This situation arises often in practical flow computation, in both the deterministic and (MonteCarlo simulated) stochastic cases. An example of a suitable coarsener is provided by multiscale finite elements. In this paper we explore the linear algebraic aspects of the multiscale algorithm, showing that it involves a blend of both classical overlapping Schwarz methods and nonoverlapping Schur methods. We also extend the algorithm and the theory from its additive variant to obtain new hybrid and deflation variants. Finally we give extensive numerical experiments on a range of heterogeneous media problems illustrating the properties of the methods. c ○ 2006 John Wiley & Sons, Inc. I.
R.: Towards a rigorously justified algebraic preconditioner for highcontrast diffusion problems
 Comput. Vis. Sci
, 2008
"... Abstract. In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with highcontrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions ..."
Abstract

Cited by 17 (11 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with highcontrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the freedoms into “high ” and “low ” permeability regions which may be of arbitrary geometry. This partition yields a corresponding blocking of the stiffness matrix and hence a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. The structure of the required subblock inverses in the high contrast case is revealed by a singular perturbation analysis (with the contrast playing the role of a large parameter). This shows that for high enough contrast each of the subblock inverses can be approximated well by solving only systems with constant coefficients. The second phase of the algorithm involves the approximation of these constant coefficient systems using multigrid methods. The result is a general method of algebraic character which (under suitable hypotheses) can be proved to be robust with respect to both the contrast and the mesh size. While a similar performance is also achieved in practice by algebraic multigrid (AMG) methods, this performance is still without theoretical justification. Since the first phase of our
Coefficientexplicit condition number bounds for overlapping additive Schwarz, in Domain Decomposition Methods in Science and Engineering XVII (Langer, Discacciati, et al., Eds
 Lecture Notes in Computational Science and Engineering
, 2008
"... In this paper we discuss new domain decomposition preconditioners for piecewise linear finite element discretisations of boundaryvalue problems for the model elliptic problem − ∇ · (A∇u) = f, (1) ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
(Show Context)
In this paper we discuss new domain decomposition preconditioners for piecewise linear finite element discretisations of boundaryvalue problems for the model elliptic problem − ∇ · (A∇u) = f, (1)
Domain Decomposition for Heterogeneous Media
"... Summary. Elliptic problems with multiscale coefficients have been studied to a great extent recently. Preconditioners based on standard domain decomposition methods often perform poorly when the variation of the coefficients inside the subdomains is large. In this paper we study the behaviour of dom ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Summary. Elliptic problems with multiscale coefficients have been studied to a great extent recently. Preconditioners based on standard domain decomposition methods often perform poorly when the variation of the coefficients inside the subdomains is large. In this paper we study the behaviour of domain decomposition methods based on linear coarsening for such problems and we also propose improved methods which use the notion of multiscale finite elements to define coarsening operators. 1 Problem Description Typical examples of elliptic multiscale problems occur among others in fluid flow in strongly heterogeneous media or heat conduction in composite media. Let us therefore consider the second order partial differential equation of Poisson type −∇.a(x)∇u(x) = f(x) for x ∈ Ω, (1) with Ω ⊂ R d, where a(x) is the conductivity, which for simplicity is assumed to be scalar valued, symmetric and positive, but which is allowed to vary very
OPTIMIZED SCHWARZ AND 2LAGRANGE MULTIPLIER METHODS FOR MULTISCALE PDES ∗
"... Abstract. In this article, we formulate and analyze a twolevel preconditioner for Optimized Schwarz and 2Lagrange Multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of lowfrequency mo ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In this article, we formulate and analyze a twolevel preconditioner for Optimized Schwarz and 2Lagrange Multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of lowfrequency modes of the subdomain DirichlettoNeumann maps. Under a suitable change of basis, the preconditioner is a 2 × 2 block upper triangular matrix with the identity matrix in the upperleft block. We show that the spectrum of the preconditioned system is included in the disk having center z = 1/2 and radius r = 1/2 − , where 0 < < 1/2 is a parameter that we can choose. We further show that the GMRES algorithm applied to our heterogeneous system converges in O(1/) iterations (neglecting certain polylogarithmic terms). The number can be made arbitrarily large by automatically enriching the coarse space. Our theoretical results are confirmed by numerical experiments. Key words. Domain decomposition, coefficient dependent coarse space, adaptive coarse space enrichment, Dirichlet to Neumann generalized eigenproblem, multiscale PDEs, heterogeneous media AMS subject classifications. 65N55, 65F10, 65N30, 65N22
ROBUST MULTISCALE ITERATIVE SOLVERS FOR NONLINEAR FLOWS IN HIGHLY HETEROGENEOUS MEDIA
"... Abstract. In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steadystate Richards ’ equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steadystate Richards ’ equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. Theproposediterativesolversconsistoftwokindsofiterations, outerandinneriterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a largescale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods (see [15, 17]), we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results. 1.