2-Dimensional primal domain decomposition theory in detail

We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into t...

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Bibliographic Details
Published in:Applications of mathematics (Prague) 2015-06, Vol.60 (3), p.265-283
Main Authors: Lukáš, Dalibor, Bouchala, Jiří, Vodstrčil, Petr, Malý, Lukáš
Format: Article
Language:English
Subjects:
Analysis
Applications of Mathematics
Applied mathematics
Boundary conditions
Calculus
Classical and Continuum Physics
Decomposition
Dirichlet problem
Discretization
Domain decomposition
Finite element analysis
Jumping
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Optimization
Studies
Theoretical
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Summary:We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into triangular subdomains that align with the coefficients jumps. We prove that the condition number of the vertex-based DD preconditioner is O ((1 + log( H / h )) 2 ), independently of the coefficient jumps, where H and h denote the discretization parameters of the coarse and fine triangulations, respectively. Although this preconditioner and its analysis date back to the pioneering work J.H.Bramble, J. E.Pasciak, A.H. Schatz (1986), and it was revisited and extended by many authors including M.Dryja, O.B.Widlund (1990) and A.Toselli, O.B.Widlund (2005), the theory is hard to understand and some details, to our best knowledge, have never been published. In this paper we present all the proofs in detail by means of fundamental calculus.
ISSN:0862-7940
1572-9109
DOI:10.1007/s10492-015-0095-5