HTL-WASI preconditioner for finite element discretization of complex ADR equation and its application

Abstract

The convection diffusion reaction (ADR) equation is widely used in engineering applications. In this paper, we focus on a complex ADR equation arising from Helmholtz equation with impedance boundary conditions, prove the coerceiveness and boundedness of the sesquilinear functional and discuss the fast solver of the discretization. Firstly, we design a Weighted Additive Schwarz with Impedance (WASI) preconditioner, in order to overcome the dependence of the WASI preconditioner on the number of subdomains and the overlapping width, an ideal coarse space is designed. Although it can be proved that the corresponding hybrid two-level preconditioner is the exact inverse of the ADR coefficient matrix, the dimension of this coarse space is too big. In order to overcome this deficiency, a smaller dimensional coarse space was constructed by introducing local generalized eigenvalue problems (GEP) on each subdomain, and the descent rate of the corresponding two-level preconditioned GMRES method was rigorously analyzed which does not depend on mesh size, overlapping width, wave number $k$, and the absorption parameter. Since the size of the coarse space is sensitive to $k$ and the GEP on each subdomain needs to be solved, the computational cost is too high. Therefore, we design an iterative method based on an economical two-level preconditioner, and establish a heuristic convergence theory that is supported by numerical results. Numerical experiments show the robustness of GMRES with the corresponding economical two-level preconditioner(HTLE-WASI-GMRES). Finally, for the Helmholtz finite element discretization, by using the Shifted Laplace technique and ADR equation, we design a novel fast solver combining with HTLE-WASI-GMRES for ADR equation. Numerical experiments have demonstrated the effectiveness of the algorithm.