A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Helmholtz

TL;DR

This paper presents a novel least-squares weak Galerkin finite element method for the ill-posed Helmholtz Cauchy problem, offering rigorous analysis, optimal error estimates, and validated numerical performance.

Contribution

It introduces a flexible LS-WG scheme for complex boundary conditions, proving uniqueness and optimal error estimates for the Helmholtz Cauchy problem.

Findings

Numerical experiments confirm theoretical convergence rates.

The scheme demonstrates robustness and efficiency.

Optimal error estimates are established.

Abstract

This paper introduces and rigorously analyzes a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem associated with the Helmholtz equation. By utilizing a weak Laplacian operator defined on a space of discontinuous functions, the proposed framework facilitates the seamless treatment of complex boundary conditions and internal interfaces. We emphasize the geometric flexibility of the LS-WG scheme on general polygonal and polyhedral partitions. Furthermore, we prove the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a specifically designed discrete energy norm. Extensive numerical experiments validate the theoretical convergence rates and demonstrate the algorithm's robustness and efficiency over traditional Galerkin approaches.