Open Access

Roland Maier

• In this thesis, we consider the numerical approximation of solutions of partial differential equations that exhibit some kind of multiscale features. Such equations describe, for instance, the deformation of porous media, diffusion processes, or wave propagation and the multiscale behavior of corresponding solutions is typically the result of material coefficients that include variations on some fine scale. To avoid global computations on scales that resolve the microscopic quantities, the aim is to provide suitable approximations on some coarse discretization level while taking into account these fine-scale characteristics of underlying coefficients. To this end, we employ the framework of Localized Orthogonal Decomposition that is able to cope with general heterogeneous coefficients without the requirement for structural assumptions such as periodicity or an explicit characterization of a fine scale. The approach provides adapted finite element functions with improved approximationIn this thesis, we consider the numerical approximation of solutions of partial differential equations that exhibit some kind of multiscale features. Such equations describe, for instance, the deformation of porous media, diffusion processes, or wave propagation and the multiscale behavior of corresponding solutions is typically the result of material coefficients that include variations on some fine scale. To avoid global computations on scales that resolve the microscopic quantities, the aim is to provide suitable approximations on some coarse discretization level while taking into account these fine-scale characteristics of underlying coefficients. To this end, we employ the framework of Localized Orthogonal Decomposition that is able to cope with general heterogeneous coefficients without the requirement for structural assumptions such as periodicity or an explicit characterization of a fine scale. The approach provides adapted finite element functions with improved approximation properties based on localized corrections of classical finite element functions. We introduce the method in an abstract stationary setting and rigorously analyze its convergence behavior in terms of theoretical and numerical investigations. We also present a higher-order generalization of the approach based on non-conforming spaces and study the interplay between the mesh parameter, the polynomial degree, and the localization parameter. We provide convergence results with explicit dependencies on the above-mentioned parameters and present numerical experiments. Further, we consider an inverse problem of recovering information about an underlying diffusion coefficient from given coarse-scale measurements. Instead of reconstructing the actual coefficient, we follow the idea of finding a coarse model in the spirit of general numerical homogenization methods that is able to satisfactorily reproduce the given data. Although this is a seemingly very different setting, the results of the inverse procedure provide a justification of general (forward) numerical homogenization methods (as, e.g., the Localized Orthogonal Decomposition) and therefore solidify the approach from a different point of view. Beyond these stationary problems, we apply the Localized Orthogonal Decomposition method to the wave equation and the multiphysics problem of linear poroelasticity. We provide rigorous convergence studies and numerical examples. The approach displays its full potential in these time-dependent settings in the sense of an overall complexity reduction. In the context of the wave equation, we focus on an explicit time stepping scheme and the effect of the method on the time step restriction. For the poroelastic problem, we use an implicit scheme and introduce an alternative approach that exploits the saddle point structure which arises if the system is first discretized in time.…