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 coefficients. This result motivates new and justifies old classes of variational multiscale methods.

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 but including coarse finite element functions. We suggest, by contrast, to perform local computations with the additional constraint that trial and test functions are linear independent from coarse finite element functions. This approach re-interprets 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 a MsFEM with oversampling.

Abstract. We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number κ in bounded domains in R d . The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale correction in the spirit of numerical homogenization. The precomputation of the correction involves the solution of coercive cell problems on localized subdomains of size ℓH; H being the mesh size and ℓ being the oversampling parameter. If the mesh size and the oversampling parameter are such that Hκ and log(κ)/ℓ fall below some generic constants and if the cell problems are solved sufficiently accurate on some finer scale of discretization, then the method is stable and its error is proportional to H; pollution effects are eliminated in this regime.

We present an efficient adaptive refinement procedure that preserves analysis-suitability of the T-mesh, this is, the linear independence of the T-spline blending functions. We prove analysis-suitability of the overlays and boundedness of their cardinalities, nestedness of the generated T-spline spaces, and linear computational complexity of the refinement procedure in terms of the number of marked and generated mesh elements.Comment: We now account for T-splines of arbitrary polynomial degree. We replaced the proof of Dual-Compatibility by a proof of Analysis-suitability, added a section where we address nestedness of the corresponding T-spline spaces, and removed the section on finite overlap the spline supports. 24 pages, 9 Figure

We present and analyze a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number κ as a variant of Peterseim (2014). We use standard continuous Q 1 finite elements at a coarse discretization scale H as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale h. The diameter of the support of the test functions behaves like m H for some oversampling parameter m. Provided m is of the order of log(κ) and h is sufficiently small, the resulting method is stable and quasi-optimal in the regime where H is proportional to κ −1 . In homogeneous (or more general periodic) media, the fine scale test functions depend only on local mesh-configurations. Therefore, the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes. We present numerical experiments in two and three space dimensions.

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present a class of such methods that are closely related to the methods that have recently been proposed by Målqvist and Peterseim [Math. Comp. 83, 2014, pp. 2583-2603. Like these methods, the new methods do not make explicit or implicit use of a scale separation. Their comparatively simple analysis is based on the theory of additive Schwarz or subspace decomposition methods.

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This paper reviews the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L 2 approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.