Super-localized Orthogonal Decomposition for high-frequency Helmholtz problems

Abstract

We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber $\kappa$. On a coarse mesh of width $H$, the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width $\ell H$ and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multi-scale methods, the localization error decays super-exponentially as the oversampling parameter $\ell$ is increased. This implies that optimal convergence is observed under the substantially relaxed oversampling condition $\ell \gtrsim (\log \frac{\kappa }{H}{)}^{(d-1)/d}$ with $d$ denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.

arXiv comment21 pages, 7 figures

References in corpus (9)

• Oversampling for the Multiscale Finite Element Method
• An analysis of a class of variational multiscale methods based on subspace decomposition
• A Survey of Trefftz Methods for the Helmholtz Equation
• Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation
• Stable Multiscale Petrov-Galerkin Finite Element Method for High Frequency Acoustic Scattering
• On stability of discretizations of the Helmholtz equation (extended version)
• Randomized Local Model Order Reduction
• Stability estimate for the Helmholtz equation with rapidly jumping coefficients
• A Super-Localized Generalized Finite Element Method