Numerical homogenization for non-linear monotone elliptic problems
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In this work we introduce and analyze a new multiscale method for non-linear monotone elliptic equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. Both a Galerkin and a Petrov-Galerkin variant with only modified test functions are analyzed. The new method gives optimal a priori error estimates up to linearization errors beyond periodicity and scale separation and without assuming higher regularity of the solution. Several numerical examples including stationary Richards equation confirm the theory and underline the applicability of the method.
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