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Maximum principle preserving schemes for interface problems with discontinuous coefficients. (English) Zbl 1001.65115
The authors consider the elliptic problem $$ (\beta u_x)_{x}+(\beta u_y)_{y}- \kappa(x,y)u=f(x,y) $$ in a domain $\Omega$ that contains a smooth curve $\Gamma$ across which $\beta$ and $f$ may have jump discontinuities. The standard 5-point finite difference scheme is set up but modified near $\Gamma$ in such a way that a discrete maximum principle is valid while the resulting scheme exhibits overall first order acuracy. Based on these properties convergence is proved. The authors study also a second order scheme starting from the 9-point finite difference stencil. The needed properties are in this case verified numerically. Numerical examples illustrate the theoretical results.

65N06Finite difference methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
35R05PDEs with discontinuous coefficients or data
65N12Stability and convergence of numerical methods (BVP of PDE)
65N50Mesh generation and refinement (BVP of PDE)
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