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A sharp interface finite volume method for elliptic equations on Cartesian grids. (English) Zbl 1169.65343
Summary: We present a second order sharp interface finite volume method for the solution of the three-dimensional elliptic equation $\nabla \cdot (\beta (\overrightarrow x)\nabla u(\overrightarrow x)) = f(\overrightarrow x)$ 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 piecewise 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 [{\it M. Oevermann} and {\it R. Klein}, J. Comput. Phys. 219, No. 2, 749--769 (2006; Zbl 1143.35022)], 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 $\beta $-ratios and non-smooth interfaces -- the method achieves second order of accuracy in the $L_{\infty }$ and $L_{2}$ norm.

65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
65N12Stability and convergence of numerical methods (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
Full Text: DOI
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