A Second-Order Boundary Condition Capturing Method for Solving the Elliptic Interface Problems on Irregular Domains

A second-order boundary condition capturing method is presented for the elliptic interface problem with jump conditions in the solution and its normal derivative. The proposed method is an extension of the work in Liu et al. (J Comput Phys 160(1):151–178, 2000) to a higher order. The motivation of proposed method is that the approximated value at the interface can be reconstructed by proper interpolation based on the level set representation from Gibou et al. (J Comput Phys 176(1):205–227, 2002). A second-order accurate method is constructed, both in the solution and its gradient, using second-order finite difference approximation. Several numerical results demonstrate that the proposed method is indeed second-order accurate in the solution and its gradient in the 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} and L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document} norms.