Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. contrast such involving one smooth non-intersecting interface, been extensively studied, there are very few papers addressing interface intersecting interfaces of coefficient jumps. It is well known if the values permeability in four subregions around a point intersection two all different, solution has singularity significantly affects accuracy approximation vicinity point. present propose fourth-order 9-point scheme on uniform Cartesian meshes an problem whose constant rectangular subdomains overall two-dimensional domain. Moreover, special case when lines grid point, compact scheme, relatively simple formulas computation stencil coefficients, can even reach sixth order accuracy. Furthermore, show resulting linear system M-matrix, and prove theoretical convergence rate using discrete maximum principle. Our numerical experiments demonstrate (for case) at least fourth general orders proposed schemes. case, derive third-order also yielding M-matrix. addition, principle, third cross-interface problem.