Maximum principle preserving schemes for interface problems with discontinuous coefficients.

*(English)*Zbl 1001.65115The authors consider the elliptic problem

$${\left(\beta {u}_{x}\right)}_{x}+{\left(\beta {u}_{y}\right)}_{y}-\kappa (x,y)u=f(x,y)$$

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.

Reviewer: Rolf Dieter Grigorieff (Berlin)

##### MSC:

65N06 | Finite difference methods (BVP of PDE) |

35J25 | Second order elliptic equations, boundary value problems |

35R05 | PDEs with discontinuous coefficients or data |

65N12 | Stability and convergence of numerical methods (BVP of PDE) |

65N50 | Mesh generation and refinement (BVP of PDE) |