论文信息 - Balancing domain decomposition

Balancing domain decomposition

The Neumann–Neumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite-element discretizations of difficult problems with strongly discontinuous coefficients (De Roeck and Le Tallec, 1991). However, this algorithm suffers from the need to solve in each iteration an inconsistent singular problem for every subdomain, and its convergence deteriorates with increasing number of subdomains due to the lack of a coarse problem to propagate the error globally. We show that the equilibrium conditions for the singular problems on subdomains lead to a simple and natural construction of a coarse problem. The construction is purely algebraic and applies also to systems such as those that arise in elasticity. A convergence bound independent of the number of subdomains is proved and results of computational tests are reported.

J. Mandel

[1] D. O’Leary,et al. Generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. [Solution of sparse, symmetric, positive-definite systems of linear equations arising from discretization of boundary-value problems for elliptic partial differential equations] , 1976 .

[2] I. Gustafsson,et al. Preconditioning and two-level multigrid methods of arbitrary degree of approximation , 1983 .

[3] J. Pasciak,et al. The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[4] Jan Mandel,et al. A multigrid method for three-dimensional elasticity and algebraic convergence estimates , 1987 .

[5] R. Bank,et al. 5. Variational Multigrid Theory , 1987 .

[6] R. Glowinski,et al. Variational formulation and algorithm for trace operation in domain decomposition calculations , 1988 .

[7] Jan Mandel,et al. Two-level domain decomposition preconditioning for the p-version finite element method in three dimensions , 1990 .

[8] Barry F. Smith. A domain decomposition algorithm for elliptic problems in three dimensions , 1990 .

[9] P. Tallec,et al. Domain decomposition methods for large linearly elliptic three-dimensional problems , 1991 .

[10] Charbel Farhat,et al. An Unconventional Domain Decomposition Method for an Efficient Parallel Solution of Large-Scale Finite Element Systems , 1992, SIAM J. Sci. Comput..