Preconditioning for Heterogeneous Problems
Part of the
Lecture Notes in Computational Science and Engineering
book series (LNCSE, volume 40)
The main focus of this paper is to suggest a domain decomposition method for mixed finite element approximations of elliptic problems with anisotropic coefficients in domains. The theorems on traces of functions from Sobolev spaces play an important role in studying boundary value problems of partial differential equations. These theorems are commonly used for a priori estimates of the stability with respect to boundary conditions, and also play very important role in constructing and studying effective domain decomposition methods. The trace theorem for anisotropic rectangles with anisotropic grids is the main tool in this paper to construct domain decomposition preconditioners.
KeywordsDomain Decomposition Conjugate Gradient Method Domain Decomposition Method Saddle Point Problem Lanczos Method
D. N. Arnold and F. Brezzi. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Math. Model. Numer. Anal.
, 19:7–32, 1985.MathSciNetMATHGoogle Scholar
L. C. Cowsar, J. Mandel, and M. F. Wheeler. Balancing domain decomposition for mixed finite elements. Math. Comp.
, 64(211):989–1015, July 1995.MathSciNetCrossRefMATHGoogle Scholar
Y. A. Kuznetsov and M. F. Wheeler. Optimal order substructuring preconditioners for mixed finite element methods on nonmatching grids. East-West J. Numer. Math.
, 3(2):127–143, 1995.MathSciNetMATHGoogle Scholar
D. Kwak, S. Nepomnyaschikh, and H. Pyo. Domain decomposition for model heterogeneous anisotropic problem. Numer. Linear Algebra
, 10:129–157, 2003.MathSciNetCrossRefMATHGoogle Scholar
A. M. Matsokin and S. V. Nepomnyaschikh. On using the bordering method for solving systems of mesh equations. Sov. J. Numer. Anal. Math. Modeling
, 4:487–492, 1989.MathSciNetCrossRefGoogle Scholar
T. Rusten and R. Winther. A preconditioned iterative method for saddle point problems. SIAM J. Matrix Anal.
, 13:887–904, 1992.MathSciNetCrossRefMATHGoogle Scholar
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