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This note constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying diffusion tensor. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially w.r.t. the number of element layers in the patches. Hence, on a uniform mesh of size H, patches of diameter log(1/H) are sufficient to preserve the convergence rates of the classical P1-FEM for the Poisson problem.
The analysis does not rely on regularity of the solution or scale separation in the coefficient.
The result justifies the use of the class of variational multiscale methods, introduced in [Comput. Methods Appl. Mech. Engrg., 196:2313--2324, 2007].
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coeffcient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear finescale computations in 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 coeffcients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh
size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
We discuss the possibility of computing eigenpairs of some prototypical linear second-order self-adjoint elliptic partial differential operator (or its high-resolution finite element discretization) by numerical upscaling techniques. We compute a low-dimensional generalized finite element space that preserves small eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of H1 by means of a certain Clement-type quasi-interpolation operator.