Adaptive Finite Element Methods for Multiscale Partial Differential Equations

Engineers rely on efficient simulations that provide them with reliable data in order to make proper engineering design decisions. The purpose of this thesis is to design adaptive numerical methods for multiscale problems in this spirit. We consider elliptic homogenization problems discretized by the finite element heterogeneous multiscale method (FE-HMM). Unlike standard (single-scale) finite element methods, our multiscale discretization scheme relies on coupled macro and micro finite elements. The framework of the HMM allows to design an algorithm that follows the classical finite element structure on the macro level. The fine scales of the multiscale problems are taken into account by replacing the element-wise numerical integration over unknown macroscopic data by a numerical integration over suitably averaged micro solutions. These micro solutions are obtained from micro FE problems on sampling domains within the macro elements. This thesis is divided into two parts. In the first part, we discuss a short and versatile FE implementation of the multiscale algorithm. The implementation is flexible, easy to use and to modify and can handle simplicial or quadrilateral FE and various macro-micro coupling conditions for the constrained micro problems. The implementation of time-dependent problems is also discussed. Numerical examples including three dimensional problems are presented and demonstrate the efficiency and the versatility of the computational strategy. In the second part (the main part of this thesis), we present an a posteriori error analysis for the FE-HMM. The a posteriori analysis enables us to estimate the accuracy of a numerical solution (and therefore its reliability) and further it allows for the design of adaptive numerical methods, which are the most efficient. The crucial component for the design of an adaptive multiscale method is the introduction of appropriate error indicators. As the error indicators depend on macroscopic data (such as the macroscopic diffusion tensor) that are not readily available, we construct error indicators that only depend on the available macro and micro FE solutions, available from previous computations. We provide a posteriori estimates for the upper and lower bound in the energy norm. The corresponding macroscopic mesh refinement strategy is therefore both reliable and efficient. The microscopic mesh is refined simultaneously and – under appropriate assumptions – optimally with the macroscopic mesh. This means that the strategy reduces the macro and micro error at the same rate. In the case of a uniformly oscillating tensor and exact micro computations, the standard a posteriori error estimates for the FEM applied to the homogenized problem are recovered. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method and demonstrate the optimality of the chosen macro-micro coupling. We extend the adaptive FE-HMM to higher order FE. We further derive a posteriori estimates for the error in quantities of interest that are needed to make certain design decisions; the quantity of interest is represented by a linear functional. We derive and analyze a multiscale counterpart to the classical dual-weighted residual method and design a corresponding goal-oriented adaptive multiscale method. The efficiency of the method is shown in numerical experiments.

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