On stability of discretizations of the Helmholtz equation (extended version)
Provides rigorous stability guarantees for numerical solutions of the Helmholtz equation at high wavenumbers, benefiting computational wave propagation and scattering problems.
The paper develops a complete k-explicit stability and convergence theory for high-order finite element methods for the Helmholtz equation at large wavenumbers, showing quasi-optimality when mesh size and approximation order satisfy specific conditions. It also reviews stability properties of least squares and discontinuous Galerkin methods.
We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.