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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… (More)

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… (More)

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 d×d sym) with uniform spectral bounds without any assumption on scale separation or periodicity. The multiscale method uses a corrected basis that is computed on… (More)

This paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi-optimal complexity for the Stokes equations with respect to natural approximation classes. The proof does not explicitly involve the pressure variable and follows from a novel discrete Helmholtz decomposition of devi-atoric functions.