Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver (Communications in Computational Physics, 2020)
This article is to discuss the bilinear and linear immersed finite element (IFE)solutions generated from the algebraic multigrid solver for both stationary and movinginterface problems. For the numerical methods based on finite difference formulationand a structured mesh independent of the interface, the stiffness matrix of the linearsystem is usually not symmetric positive-definite, which demands extra efforts to designefficient multigrid methods. On the other hand, the stiffness matrix arising fromthe IFE methods is naturally symmetric positive-definite. Hence the IFE-AMG algorithmis proposed to solve the linear systems of the bilinear and linear IFE methodsfor both stationary and moving interface problems. The numerical examples demonstratethe features of the proposed algorithms, including the optimal convergence inboth $L^2$ and semi-$H^1$ norms of the IFE-AMG solutions, the high efficiency with properchoice of the components and parameters of AMG, the influence of the tolerance andthe smoother type of AMG on the convergence of the IFE solutions for the interfaceproblems, and the relationship between the cost and the moving interface location.