A Large Jump Asymptotic Framework for Solving Elliptic and Parabolic Equations with Interfaces and Strong Coefficient Discontinuities

We address the solution of multiregion elliptic and parabolic problems with strongly discontinuous coefficients across interfaces. The underlying idea is, roughly, to perturb around the infinitely discontinuous coefficient solution, that is, to perturb the solution where one region is a perfect conductor. The result is partial decoupling of the interface conditions. The algorithm requires a small number of well-conditioned solves in the individual regions (in many cases only a single solve over each region is necessary) that are then assembled into an accurate global solution. The error from the assembly step is asymptotically small in the ratio of the discontinuous coefficients. Further, the framework can be extended to problems with moderately discontinuous coefficients using a series expansion in the discontinuity ratio in a manner similar to a Schwarz alternating method.