Numerical homogenization methods for advection-diffusion and nonlinear monotone problems with multiple scales

Abstract

Many applied problems, like transport processes in porous media or ferromagnetism in composite materials, can be modeled by partial differential equations (PDEs) with heterogeneous coefficients that rapidly vary at small scales. To capture the effective behavior at a scale of interest, standard numerical methods like the finite element method (FEM) however need to resolve the finest scale of the problem (scale resolution) and thus are prohibitively costly. Numerical homogenization methods based on effective models are an efficient alternative as they require scale resolution only in a small portion of the computational domain. In this thesis, we introduce numerical homogenization methods developed in the framework of heterogeneous multiscale methods (HMM) for two different classes of multiscale PDEs. In the first part, we consider linear parabolic advection-diffusion problems with highly oscillating data, large Peclet number and compressible velocity fields. We use a discontinuous Galerkin finite element method (DG-FEM) for the spatial discretization of an effective equation whose data is estimated from finite element simulations in microscopic domains. The numerical upscaling strategy appropriately models the effects of the highly oscillating velocity field on the effective diffusion (enhanced or depleted diffusion). Thanks to the favorable stability properties of DG-FEM, robust a priori error estimates with explicit convergence rates for the macro and micro spatial discretization errors can be shown. In the second part, we propose numerical homogenization methods for nonlinear monotone multiscale problems. For parabolic problems, we first combine the backward Euler method in time with a finite element heterogeneous multiscale method (FE-HMM) in space, which couples macro and micro FEMs. We prove convergence of the method in the general $L^p(W^{1,p})$ setting and provide fully discrete space-time a priori error estimates for $p=2$. The upscaling procedure however is based on nonlinear micro sampling problems, which can be computationally expensive. As a remedy, we propose a linearized method, which is only based on linear micro problems and is much more efficient as it avoids Newton iterations. We prove stability and optimal convergence rates of the linearized scheme in the classical $L^2(H^1)$ setting. Further, for elliptic problems, we extend the FE-HMM (based on nonlinear micro problems) to high order macro and micro FEMs and present fully discrete a priori error estimates.