Tensor Structured Iterative Solution of Elliptic Problems with Jumping Coefficients

Sergey Dolgov, Boris N. Khoromskij, Ivan V. Oseledets, and Eugene E. Tyrtyshnikov

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Submission date: 28. Sep. 2010
Pages: 28
published in: Linear algebra and its applications, 436 (2012) 9, p. 2980-3007 
DOI number (of the published article): 10.1016/j.laa.2011.09.010
Bibtex
with the following different title: A reciprocal preconditioner for structured matrices arising from elliptic problems with jumping coefficients
MSC-Numbers: 65F30, 65F50, 65N35
Keywords and phrases: structured matrices, elliptic operators, preconditioners, multi-dimensional matrices, tensors, finite elements, numerical methods
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Abstract:

We study separability properties of solutions of elliptic equations with piecewise constant coefficients in ℝ^d, d ≥ 2. Besides that, we develop efficient tensor-structured preconditioner for the diffusion equation with variable coefficients. It is based only on rank structured decomposition of the tensor of reciprocal coefficient and on the decomposition of the inverse of the Laplacian operator. It can be applied to full vector with linear-logarithmic complexity in the number of unknowns N. It also allows low-rank tensor representation, which has linear complexity in dimension d, hence, it gets rid of the “curse of dimensionality” and can be used for large values of d. Extensive numerical tests are presented.