A Novel T-G IFE Method for Two Dimensional Semi-Linear Elliptic Interface Problems Based on Coarse Grid Correction

Abstract

In this paper, a novel Two-Grid (T-G) algorithm is proposed and analyzed for semi-linear interface problems in two dimension. To linearize the Immersed Finite Element Method (IFEM) equations, a T-G method based on some Newton iteration approach and correction method is investigated. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H = \mathcal {O}(h^{1/3})$ in $L^{p}$ norm (for $H^{1}$ norm, it even suffices to take $H = \mathcal {O}(h^{1/5})$ ). As a result, solving such a large class of nonlinear equation will not be much more difficult than solving one linearized equation.

Keywords

• Interface problem
• coarse gird correction
• immersed finite element method
• nonlinear problem
• Newton iteration