A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces
Please always quote using this URN: urn:nbn:de:0297-zib-8996
- We present a finite volume method for the solution of the two-dimensional Poisson equation $ \nabla\cdot( \beta( {\mbox{\boldmath $x$}}) \nabla u({\mbox{\boldmath $x$}})) = f(\mbox{\boldmath $x$}) $ with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution $u({\mbox{\boldmath $x$})$ resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the $L^\infty$ and $L^2$ norm.
Author: | Michael Oevermann, Rupert Klein |
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Document Type: | ZIB-Report |
Tag: | Poisson equation; ariable and discontinuous coefficients; discontinuous solution; embedded interface; finite volume methods |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J25 Boundary value problems for second-order elliptic equations |
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N12 Stability and convergence of numerical methods | |
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods | |
Date of first Publication: | 2006/02/13 |
Series (Serial Number): | ZIB-Report (06-05) |
ZIB-Reportnumber: | 06-05 |
Published in: | Appeared in: J. Comput. Phys., 219, 749--769, (2006) |