Multigrid Preconditioning for FD-DLM Method in Elliptic Interface Problems

TL;DR

The paper addresses elliptic interface problems formulated via the FD-DLM method, which yields a saddle-point linear system with blocks including $A_1$, $A_2$, and coupling terms through $C_1$, $C_2$. It introduces three block preconditioners $P_1$, $P_2$, $P_3$ whose inverses involve $A_1^{-1}$ and $B^{-1}$, and designs practical multigrid approximations $ ilde{A_1^{-1}}$ and $ ilde{B^{-1}}$ (with $ ilde{A_1^{-1}}$ obtained by a single $V$-cycle using SOR, and $ ilde{B^{-1}}$ via Vanka preconditioners). Numerical experiments in 2D on a domain with $eta=1$, $eta_2=10$, and unit sources demonstrate that multigrid preconditioning substantially improves conditioning and often yields iteration counts that are mesh-size independent; among configurations, the ``md'' strategy with P1 or P2 tends to be robust, while the diagonal P3 is less effective in some cases. The results support multigrid-based preconditioning as a viable approach for FD-DLM saddle-point systems and point to avenues for theoretical convergence analysis and more advanced 3×3 block preconditioners.

Abstract

We investigate the performance of multigrid preconditioners for solving linear systems arising from finite element discretizations of elliptic interface problems using the Fictitious Domain with Distributed Lagrange Multipliers (FD-DLM) formulation. Numerical experiments are conducted using continuous and discontinuous finite element spaces for the Lagrange multiplier. Results indicate that multigrid is a promising preconditioner for problems in the FD-DLM formulation.