A two-grid method for semi-linear elliptic interface problems by partially penalized immersed finite element methods

In this paper, we present a two-grid partially penalized immersed finite element (IFE) scheme for the approximation of semi-linear elliptic interface problems. Extra stabilization terms are introduced at interface edges for penalizing the discontinuity in IFE functions. Optimal error estimates in both H-1 and L-P norms are obtained for IFE discretizations. To linearize the IFE equations, two-grid algorithm based on some Newton iteration approach is applied. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h(1/4)). As a result, solving such a large class of non-linear equation will not be much more difficult than solving one single linearized equation. (C) 2019 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.Y. All rights reserved.