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A sharp interface finite volume method for elliptic equations on Cartesian grids

Please always quote using this URN: urn:nbn:de:0297-zib-10900
  • We present a second order sharp interface finite volume method for the solution of the three-dimensional poisson equation with variable coefficients on Cartesian grids. In particular, we focus on interface problems with discontinuities in the coefficient, the source term, the solution, and the fluxes across the interface. The method uses standard piecewiese trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27-point stencil. Singularities associated with vanishing partial volumes of intersected grid cells are removed by a two-term asymptotic approach. In contrast to the 2D method presented by two of the authors in [M.~Oevermann, R.~Klein: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J.~Comp.~Phys.~219 (2006)] we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz. This simplifies the treatment of the different cut-cell types and avoids additional special operations for degenerated interface topologies. The resulting set of linear equations has been solved with a BiCGSTAB solver preconditioned with an algebraic multigrid. In various testcases -- including large coefficient ratios and non-smooth interfaces -- the method achieves second order of accuracy in the L_inf and L_2 norm.

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Metadaten
Author:Michael Oevermann, Carsten Scharfenberg, Rupert Klein
Document Type:ZIB-Report
Tag:Finite Volumen; diskontinuierliche Lösung; eingebettete Ränder; elliptische Gleichungen; variable und diskontinuierliche Koeffizienten
Elliptic equations; discontinuous solution; embedded interface; finite volume methods; variable and discontinuous coefficients
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
Contributing Corporation:Deutsche Forschungsgemeinschaft
Date of first Publication:2008/09/24
Series (Serial Number):ZIB-Report (08-38)
ISSN:1438-0064
ZIB-Reportnumber:08-38
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