Results 1  10
of
45
Finite Element Methods and Their Convergence for Elliptic and Parabolic Interface Problems
 Numer. Math
, 1996
"... In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of ar ..."
Abstract

Cited by 51 (8 self)
 Add to MetaCart
(Show Context)
In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Mathematics Subject Classification (1991): 65N30, 65F10. A running title: Finite element methods for interface problems. Correspondence to: Dr. Jun Zou Email: zou@math.cuhk.edu.hk Fax: (852) 2603 5154 1 Institute of Mathematics, Academia Sinica, Beijing 100080, P.R. China. Email: zmchen@math03.math.ac.cn. The work of this author was partially supported by China National Natural Science Foundation. 2 Department of Mathematics, the Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: zou@math.cuhk....
Some Nonoverlapping Domain Decomposition Methods
, 1998
"... . The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the Neumann ..."
Abstract

Cited by 47 (8 self)
 Add to MetaCart
(Show Context)
. The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the NeumannNeumanntype methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include localglobal and globallocal techniques. The analyses for both two and threedimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived. Key words. nonoverlapping domain decomposition, Schur complement, localglobal and globallocal techniques, jumps in coe#cients, substructuring, NeumannNeumann, balancing methods AMS subject classifications. 65N30, 65N55, 65F10 PII. S0036144596306800 1. Introduction. T...
The Auxiliary Space Method And Optimal Multigrid Preconditioning Techniques For Unstructured Grids
 Computing
, 1996
"... . An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxi ..."
Abstract

Cited by 35 (3 self)
 Add to MetaCart
(Show Context)
. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...
Global Convergence of Subspace Correction Methods for Convex Optimization Problems
, 1998
"... A general space decomposition technique is used to solve nonlinear convex minimization problems. The dierential of the minimization functional is required to satisfy some growth conditions that are weaker than Lipschitz continuity and strong monotonicity. Optimal rate of convergence is proved. If th ..."
Abstract

Cited by 25 (6 self)
 Add to MetaCart
A general space decomposition technique is used to solve nonlinear convex minimization problems. The dierential of the minimization functional is required to satisfy some growth conditions that are weaker than Lipschitz continuity and strong monotonicity. Optimal rate of convergence is proved. If the dierential is Lipschitz continuous and strongly monotone, then the algorithms have uniform rate of convergence. The algorithms can be used for domain decomposition and multigrid type of techniques. Applications to linear elliptic and some nonlinear degenerated partial dierential equation are considered. 1 Introduction Domain decomposition and multigrid methods have been intensively studied for linear partial dierential equations. Recent research, see for example [32], reveals that domain decomposition and multigrid methods can be analysed using a same framework, see also [3], [13], [23], [17]. The present work uses this framework to analyse the convergence of two algorithms for convex...
Uniform hp convergence results for the mortar finite element method
 Math. Comp
, 1997
"... Abstract. The mortar finite element is an example of a nonconforming method which can be used to decompose and recompose a domain into subdomains without requiring compatibility between the meshes on the separate components. We obtain stability and convergence results for this method that are unif ..."
Abstract

Cited by 21 (4 self)
 Add to MetaCart
(Show Context)
Abstract. The mortar finite element is an example of a nonconforming method which can be used to decompose and recompose a domain into subdomains without requiring compatibility between the meshes on the separate components. We obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used, without assuming quasiuniformity for the meshes. Our results establish that the method is optimal when nonquasiuniform h or hp methods are used. Such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain. We also give an error estimate for when the p version is used. Numerical results for h, p and hp mortar FEMs show that these methods behave as well as conforming FEMs. An hp extension theorem is also proved. 1.
Multiscale Domain Decomposition Methods for Elliptic Problems with High Aspect Ratios
"... Abstract In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
Abstract In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiplescale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiplescale coefficients, as well problems with continuous scales.
A Convergence Theory of Multilevel Additive Schwarz Methods on Unstructured Meshes
 Numer. Algorithms
, 1996
"... . We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, they need not be related to one ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
. We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, they need not be related to one another, and the domains formed by the coarse and fine grids may not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for nonselfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusion...
Parallel Adaptive Subspace Correction Schemes with Applications to Elasticity
 Comput. Methods Appl. Mech. Engrg
, 1999
"... : In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main featur ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
: In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main features of each of the three distinct topics and treat the historical background and modern developments. Furthermore, we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic PDEs and especially of linear elasticity problems. We report on numerical experiments for the adaptive parallel multilevel solution of some test problems, namely the Poisson equation and Lam'e's equation. Here, we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute, which is achieved through hash storage techniques and spacefilling curves. Keywords: subspace correction, iter...
Convergence analysis of domain decomposition algorithms with full overlapping for the advectiondi usion problems
"... The aim of this paper is to study the convergence properties of a Time Marching Algorithm solving AdvectionDi usion problems on two domains using incompatible discretizations. The basic algorithm is rst presented, and theoretical or numerical results illustrate its convergence properties. This work ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
The aim of this paper is to study the convergence properties of a Time Marching Algorithm solving AdvectionDi usion problems on two domains using incompatible discretizations. The basic algorithm is rst presented, and theoretical or numerical results illustrate its convergence properties. This work has been supported by the Hermes Research program under grant number
Schwarz Methods: To Symmetrize Or Not To Symmetrize
 SIAM J. Numer. Anal
, 1997
"... A preconditioning theory is presented which establishes su#cient conditions for multiplicative and additive Schwarz algorithms to yield selfadjoint positive definite preconditioners. It allows for the analysis and use of nonvariational and nonconvergent linear methods as preconditioners for conju ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
A preconditioning theory is presented which establishes su#cient conditions for multiplicative and additive Schwarz algorithms to yield selfadjoint positive definite preconditioners. It allows for the analysis and use of nonvariational and nonconvergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. It is illustrated why symmetrizing may be a bad idea for linear methods. It is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration. Also, it is shown that absence of symmetry in the linear preconditioner is advantageous when the linear method is accelerated by using the BiCGstab method. Numerical examples are presented for two test problems which illustrate the theory and conjectures.