This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an $L^2$-projection. We derive explicit convergence rates of the method in the $L^{\infty}(L^2)$-, $W^{1,\infty}(L^2)$- and $L^{\infty}(H^1)$-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.
Efficient implementation of the Localized Orthogonal Decomposition method
University of Münster（明斯特大学）；KTH Royal Institute of Technology（皇家理工学院）；Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）；University of Augsburg（奥格斯堡大学）
A Localized Orthogonal Decomposition Method for Heterogeneous Mixed-Dimensional Problems
Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院应用与数值数学研究所）；Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学数学科学系）
We introduce a novel multi-resolution Localized Orthogonal Decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets (gamblets) and proves its applicability for a class of complex-valued, non-hermitian and indefinite problems. It computes hierarchical bases that block-diagonalize the Helmholtz operator and thereby decouples the discretization scales. Sparsity is preserved by a novel localization strategy that improves stability properties even in the elliptic case. We pre…
This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both meth…
This paper presents a multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The application of the solution operator to piecewise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the $L^2$-norm, the Galerkin projection onto this generalized finite element space even yields $\varepsilon$-independent error bounds, $\varepsilon$ being the singular perturbation parameter. By constructing an approximate local bas…
