High-order finite difference schemes for elliptic problems with intersecting interfaces

In this work we derive high-order (second and fourth of convergence, but with first and third-order local truncation error, respectively) compact finite difference schemes for elliptic equations with intersecting interfaces. The approach uses the differential equation and the jump (interface) relations as additional identities which can be differentiated to eliminate higher order local truncation errors. Numerical experiments are carried out to demonstrate the high-order accuracy and to show that our method is effective to sharp contrast in the diffusion coefficients of the problems.