A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions (Advances in Applied Mathematics and Mechanics, 2021)
An interesting discretization method for Helmholtz equations wasintroduced in B. Després [1]. This method is based on theultra weak variational formulation (UWVF) and the wave shapefunctions, which are exact solutions of the governing Helmholtzequation. In this paper we are concerned with fast solver for thesystem generated by the method in [1]. We propose a newpreconditioner for such system, which can be viewed as a combinationbetween a coarse solver and the block diagonal preconditionerintroduced in [13]. In our numerical experiments, thispreconditioner is applied to solve both two-dimensional andthree-dimensional Helmholtz equations, and the numerical resultsillustrate that the new preconditioner is much more efficient thanthe original block diagonal preconditioner.