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An Efficient Solver to Helmholtz Equations by Recontruction Discontinuous Approximation

Shuhai Zhao

TL;DR

The paper tackles efficient numerical solution of the high-frequency Helmholtz equation by introducing the Reconstructed Discontinuous Approximation (RDA) space, which uses one DOF per element and a local least-squares reconstruction to achieve high-order accuracy with reduced memory. Coupled with a discontinuous Galerkin formulation and a natural, order-independent preconditioner, the method yields sparser matrices and faster solves than traditional DG methods. The authors develop a rigorous error analysis via elliptic projections and provide a robust preconditioning strategy, including a geometric multigrid solver for the preconditioner, ensuring GMRES convergence that is effectively mesh- and frequency-insensitive. Numerical experiments in 2D and 3D demonstrate superior accuracy per DOF, reduced nonzeros, and strong performance gains over standard DG and competitive preconditioners, including large wavenumbers. Overall, the work delivers a scalable, memory-efficient approach for high-frequency Helmholtz simulations with clear practical impact for large-scale wave problems.

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.

An Efficient Solver to Helmholtz Equations by Recontruction Discontinuous Approximation

TL;DR

The paper tackles efficient numerical solution of the high-frequency Helmholtz equation by introducing the Reconstructed Discontinuous Approximation (RDA) space, which uses one DOF per element and a local least-squares reconstruction to achieve high-order accuracy with reduced memory. Coupled with a discontinuous Galerkin formulation and a natural, order-independent preconditioner, the method yields sparser matrices and faster solves than traditional DG methods. The authors develop a rigorous error analysis via elliptic projections and provide a robust preconditioning strategy, including a geometric multigrid solver for the preconditioner, ensuring GMRES convergence that is effectively mesh- and frequency-insensitive. Numerical experiments in 2D and 3D demonstrate superior accuracy per DOF, reduced nonzeros, and strong performance gains over standard DG and competitive preconditioners, including large wavenumbers. Overall, the work delivers a scalable, memory-efficient approach for high-frequency Helmholtz simulations with clear practical impact for large-scale wave problems.

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.

Paper Structure

This paper contains 10 sections, 14 theorems, 85 equations, 10 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reconstructed Discontinuous Space
  4. An Intuitive Analysis of Approximation Efficiency for 1D Reconstruction
  5. Numerical Analysis
  6. Elliptic Projection
  7. Error Analysis
  8. Preconditioner
  9. Numerical Results
  10. Conclusion

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^d$ be a smooth domain that is star-shaped with respect to a ball. Assume the coefficient satisfies $\epsilon \lesssim k^2$, and let $f \in L^2(\Omega)$, $g \in H^{1/2}(\partial\Omega)$. Then the solution $u$ to the Helmholtz problem belongs to $H^2(\Omega)$. Moreover,

Figures (10)

  • Figure 1: Eigenvalues for $\epsilon=0$. Left: $A_{\epsilon}$, right: $P^{-1}A_{\epsilon}$.
  • Figure 2: Eigenvalues for $\epsilon=k^2$. Left: $A_{\epsilon}$, right: $P^{-1}A_{\epsilon}$.
  • Figure 3: 2D triangular partition with $h = 1/10$ (left) and 3D tetrahedral partition with $h = 1/4$ (right).
  • Figure 4: 2D accuracy test, $k=5$.
  • Figure 5: 2D accuracy test, $k=10$.
  • ...and 5 more figures

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 3
  • Theorem 4
  • proof
  • ...and 12 more