A FFT-based DDSIIM solver for elliptic interface problems with discontinuous coefficients on arbitrary domains and its error analysis

Abstract

In this study, we propose a fast FFT-based domain decomposition simplified immersed interface method (DDSIIM) solver for addressing elliptic interface problems characterized by fully discontinuous coefficients on arbitrary domains. The method involves decomposing the original elliptic interface problem along the interfaces, resulting in sub-problems defined on subdomains embedded within larger regular domains. By utilizing a variety of novel solution extension schemes and augmented variable strategies, each sub-problem is transformed into a straightforward elliptic interface problem with constant coefficients on a regular domain, interconnected through augmented equations. The interconnected sub-interface problems are initially resolved by solving for the augmented variables using GMRES, which does not depend on mesh size, followed by the application of the fast FFT-based SIIM in each GMRES iteration. Rigorous error estimates are derived to ensure global second-order accuracy in both the discrete ${\text{L}}^2$-norm and the maximum norm. A large number of numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed DDSIIM solver.