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A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. (English) Zbl 1143.35022
The authors propose a second-order accurate method for the solution of elliptic equations with variable coefficients and discontinuities across an embedded interface. The interface is represented by a level set approach. In contrast to existing methods in the literature the authors exploit a finite volume approach on Cartesian grids using ideas from finite element methods in reconstructing the solution within grid cells. The authors present a piecewise bilinear finite element for irregular cut cells taking into account known jump conditions of the solution and the normal gradient across the interface. They resolve singularities arising from the bilinear ansatz itself and the position of the interface relative to the grid by a two-step asymptotic approach. The method achieves second-order of accuracy in $L^\infty$ and $L^2$ norms.

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