A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Helmholtz
It provides a novel numerical method for solving ill-posed Cauchy problems in Helmholtz, which is important for applications like inverse scattering and non-destructive testing.
This paper introduces a least-squares weak Galerkin finite element method for the Cauchy problem in Helmholtz, proving uniqueness and optimal error estimates, with numerical experiments showing robustness and efficiency.
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.