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