Partial evaluation of the discrete solution of elliptic boundary value problems

Wolfgang Hackbusch and Florian Drechsler

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Submission date: 19. Sep. 2012
published in: Computing and visualization in science, 15 (2012) 5, p. 227-245 
DOI number (of the published article): 10.1007/s00791-013-0211-6
Bibtex
MSC-Numbers: 65F05, 65N22, 65N99
Keywords and phrases: elliptic boundary value problem, finite element discretisation, hierarchical matrices, homogenisation
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Abstract:
The technique of hierarchical matrices is used to construct a solution operator for a discrete elliptic boundary value problem. The solution operator can be determined once for all from a recursive domain decomposition structure. Then, given boundary values and a source term, the solution can be evaluated by applying the solution operator. The complete procedure yields all components of the solution vector. The data size and computational cost is O(nlog ^*n), where n is the number of unknowns.

Once the data of the solution operator are constructed, components related to small subdomains can be truncated. This reduces the storage amount and still enables a partial evaluation of the solution (restricted to the skeletons of the remaining subdomains). The latter approach is in particular suited for problems with oscillatory coefficients, where one is not interested in all details of the solution.