Heterogeneous multiscale finite element methods for advection-diffusion and nonlinear elliptic multiscale problems

In this thesis we introduce a new version of a heterogeneous multiscale finite element method (HMM) for advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. We analyse the method under the restriction of periodicity, stating corresponding a-priori and a-posteriori error estimates. As a reference for the exact solution $u^{epsilon}$, we use the homogenized solution of the original advection-diffusion multiscale problem. We obtained this solution by a technique called 'two-scale homogenization with drift'. This technique was initially introduced by Maruv{s}i'{c}-Paloka and Piatnitski [Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection, 2005, J. London Math. Soc. (2), 72]. Finally, numerical experiments are given to validate the applicability of the method and the achieved error estimates in non-periodic scenarios. Furthermore, we also state a heterogeneous multiscale finite element method for nonlinear elliptic problems. In comparison to preceding works, the nonlinearity affects the gradient of the solution instead of the solution itself. Since this especially results in implementation problems, we present a general combination of the HMM with a Newton scheme. This combination produces new cell problems which must be solved. The implementation is realized with the software toolbox DUNE. In order to handle a-posteriori error estimation beyond the periodic setting, we identify an effective macro problem and show that the solution of this problem is equal to the $H^1$-limit of a sequence of HMM approximations. Using the localized constituents of the a-posteriori error estimate, we propose algorithms for an adaptive mesh refinement for the coarse macro-grid. Again, this is verified in numerical experiments.