A Family of Two-Grid Partially Penalized Immersed Finite Element Methods for Semi-linear Parabolic Interface Problems

In this paper, we present a family of two-grid algorithms for semi-linear parabolic interface problems based on Partially penalized immersed finite element discretizations. Optimal a priori error estimates are derived both in the energy norm and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} norm, under the standard piecewise H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2$$\end{document} regularity assumption for the exact solution. For the nonlinear right hand side, we investigate two-grid methods base on Newton method. The efficiency of the two-grid methods is confirmed theoretically and numerically.