An Efficient Solver to Helmholtz Equations by Recontruction Discontinuous Approximation

TL;DR

This paper introduces a novel discontinuous Galerkin-based solver for the Helmholtz equation that improves efficiency by reducing error, matrix sparsity, and computational cost, especially at higher approximation orders.

Contribution

The paper develops a new approximation space combined with a discontinuous Galerkin scheme and an optimal preconditioner, enhancing the efficiency of Helmholtz equation solutions.

Findings

Achieves lower error with same degrees of freedom

Produces sparser matrices reducing storage and computation

Preconditioner is proven optimal relative to mesh size

Abstract

In this paper, an efficient solver for the Helmholtz equation using a noval approximation space is developed. The ingradients of the method include the approximation space recently proposed, a discontinuous Galerkin scheme extensively used, and a linear system solver with a natural preconditioner. Comparing to traditional discontinuous Galerkin methods, we refer to the new method as being more efficient in the following sense. The numerical performance of the new method shows that: 1) much less error can be reached using the same degrees of freedom; 2) the sparse matrix therein has much fewer nonzero entries so that both the storage space and the solution time cost for the iterative solver are reduced; 3) the preconditioner is proved to be optimal with respect to the mesh size in the absorbing case. Such advantage becomes more pronounced as the approximation order increases.