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We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for addressing time-harmonic scattering problems of Helmholtz type with high wavenumber k. This method operates on a coarse mesh of width HH and identifies local finite element source terms that produce rapidly decaying responses under the solution operator. These source terms can be constructed with high accuracy from independent local snapshot solutions on patches of width ℓH, and they are used as problem-adapted basis functions. Compared to classical LOD and other state-of-the-art multiscale methods, our approach demonstrates that the localization error decays super-exponentially as the oversampling parameter ℓ increases. This indicates that optimal convergence is achieved under a substantially relaxed oversampling condition of ℓ≳(log⁡κ/H)(d−1)/d, where d represents the spatial dimension. Numerical experiments highlight the significant improvements in both offline and online performance of the method, even in the presence of heterogeneous media and perfectly matched layers. [ABSTRACT FROM AUTHOR]

This article considers the computational (acoustic) wave propagation in strongly heterogeneous structures beyond the assumption of periodicity. A high contrast between the constituents of microstructured multiphase materials can lead to unusual wave scattering and absorption, which are interesting and relevant from a physical viewpoint, for instance, in the case of crystals with defects. We present a computational multiscale method in the spirit of the Localized Orthogonal Decomposition and provide its rigorous a priori error analysis for two-phase diffusion coefficients that vary between 1 and very small values. Special attention is paid to the extreme regimes of high frequency, high contrast, and their previously unexplored coexistence. A series of numerical experiments confirms the theoretical results and demonstrates the ability of the multiscale approach to efficiently capture relevant physical phenomena. [ABSTRACT FROM AUTHOR]

This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size H, patches of diameter H log(1/H) are sufficient to preserve a linear rate of convergence in H without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H log (H) where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space. [ABSTRACT FROM AUTHOR]

This work presents a new methodology for computing ground states of Bose-Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a preprocessing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties. The nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without significant loss of accuracy when compared with the solution of the full fine scale problem. The preprocessing step is independent of the types and numbers of bosons. A postprocessing step further improves the accuracy of the method. We present rigorous a priori error estimates that predict convergence rates H³ for the ground state eigenfunction and H4 for the corresponding eigenvalue without pre-asymptotic effects; H being the coarse scale discretization parameter. Numerical experiments indicate that these high rates may still be pessimistic. [ABSTRACT FROM AUTHOR]

We present a discontinuous Galerkin multiscale method for second order elliptic problems and prove convergence. We consider a heterogeneous and highly varying diffusion coefficient in L∞ (Ω, ℝ dxdsym) with uniform spectral bounds without any assumption on scale separation or periodicity. The multiscale method uses a corrected basis that is computed on patches/subdomains. The error, due to truncation of the corrected basis, decreases exponentially with the size of the patches. Hence, to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size to a reference solution, it is sufficient to choose the patch sizes O(H| log H|). We also discuss a way to further localize the corrected basis to elementwise support. Improved convergence rate can be achieved depending on the piecewise regularity of the forcing function. Linear convergence in energy norm and quadratic convergence in the L²-norm is obtained independently of the forcing function. A series of numerical experiments confirms the theoretical rates of convergence. [ABSTRACT FROM AUTHOR]

This paper reviews standard oversampling strategies as performed in the multiscale finite element method (MsFEM). Common to those approaches is that the oversampling is performed in the full space restricted to a patch including coarse finite element functions. We suggest, by contrast, performing local computations with the additional constraint that trial and test functions be linear independent from coarse finite element functions. This approach reinterprets the variational multiscale method in the context of computational homogenization. This connection gives rise to a general fully discrete error analysis for the proposed multiscale method with constrained oversampling without any resonance effects. In particular, we are able to give the first rigorous proof of convergence for an MsFEM with oversampling. [ABSTRACT FROM AUTHOR]

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 methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side. [ABSTRACT FROM AUTHOR]