# 2-dimensional primal domain decomposition theory in detail

• Volume: 60, Issue: 3, page 265-283
• ISSN: 0862-7940

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## Abstract

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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 $O\left({\left(1+log\left(H/h\right)\right)}^{2}\right)$, 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.

## How to cite

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Lukáš, Dalibor, et al. "2-dimensional primal domain decomposition theory in detail." Applications of Mathematics 60.3 (2015): 265-283. <http://eudml.org/doc/270114>.

@article{Lukáš2015,
abstract = {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.},
author = {Lukáš, Dalibor, Bouchala, Jiří, Vodstrčil, Petr, Malý, Lukáš},
journal = {Applications of Mathematics},
keywords = {domain decomposition method; finite element method; preconditioning; domain decomposition method; finite element method; preconditioning},
language = {eng},
number = {3},
pages = {265-283},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {2-dimensional primal domain decomposition theory in detail},
url = {http://eudml.org/doc/270114},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Lukáš, Dalibor
AU - Bouchala, Jiří
AU - Vodstrčil, Petr
AU - Malý, Lukáš
TI - 2-dimensional primal domain decomposition theory in detail
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 265
EP - 283
AB - 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.
LA - eng
KW - domain decomposition method; finite element method; preconditioning; domain decomposition method; finite element method; preconditioning
UR - http://eudml.org/doc/270114
ER -

## References

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1. Bramble, J. H., Pasciak, J. E., Schatz, A. H., 10.1090/S0025-5718-1986-0842125-3, Math. Comput. 47 (1986), 103-134. (1986) Zbl0615.65112MR0842125DOI10.1090/S0025-5718-1986-0842125-3
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3. Dryja, M., Widlund, O. B., Some domain decomposition algorithms for elliptic problems, Iterative Methods for Large Linear Systems Austin, TX, 1988. Academic Press, Boston 273-291 (1990). (1990) MR1038100
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5. George, A., 10.1137/0710032, SIAM J. Numer. Anal. 10 (1973), 345-363. (1973) Zbl0259.65087MR0388756DOI10.1137/0710032
6. Mandel, J., Brezina, M., 10.1090/S0025-5718-96-00757-0, Math. Comput. 65 (1996), 1387-1401. (1996) Zbl0853.65129MR1351204DOI10.1090/S0025-5718-96-00757-0
7. Mandel, J., Tezaur, R., 10.1007/s002110050201, Numer. Math. 73 (1996), 473-487. (1996) Zbl0880.65087MR1393176DOI10.1007/s002110050201
8. Payne, L. E., Weinberger, H. F., 10.1007/BF00252910, Arch. Ration. Mech. Anal. 5 (1960), 286-292. (1960) Zbl0099.08402MR0117419DOI10.1007/BF00252910
9. Toselli, A., Widlund, O., Domain Decomposition Methods---Algorithms and Theory, Springer Series in Computational Mathematics 34 Springer, Berlin (2005). (2005) Zbl1069.65138MR2104179

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