Robust Preconditioning and Upscaling of Problems with Inseparable Scales

Many important processes in science and engineering are governed by partial differential equations (PDEs). In most practical situations analytical solutions to these PDEs are not available. Thus, one is interested in computing approximate solutions by solving suitable discretizations of the PDEs under consideration. In this research project we will in particular focus on the equations of linear elasticity and Maxwell`s equations. As the linear systems resulting from the discretizations are typically too large to be solved by direct methods, the design of efficient (multilevel) iterative schemes is imperative. In particular the convergence rates of the iterative methods should be independent of the problem sizes and problem parameters. Important instances of such problem parameters are highly varying coefficients. We are in particular interested in the situation when these coefficient variations happen on multiple scales that are not clearly separated. It is well-known that in the design of robust multilevel methods the specific coarsening strategies are of central importance. In the proposed research project we intend to investigate a spectral construction of increasingly coarser spaces based on local generalized eigenvalue problems, which has previously been successfully applied to the stationary diffusion equation with highly varying coefficients. As the dimensions of these coarse spaces are inherently problem dependent, we plan to put particular emphasis on choosing the components used in their construction in such a way that the dimensions are small. This is important for the overall computational complexity of the method. Also relating to the issue of numerical costs we intend to investigate the possibilities to supersede the actual computation of local generalized eigenpairs by constructing the coarse spaces using approximations of the eigenfunctions, obtained by analytical considerations and analyzing the coefficients. For many multiscale problems discretizations that resolve even the finest scales can easily become too large for even modern computer architectures. Furthermore, one is typically not interested in resolving solutions to the finest scales but in obtaining reasonable coarse approximations. Thus, besides the design of efficient and robust multilevel solvers the second main objective of the proposed research project is to develop numerical upscaling schemes for computing coarse scale discretizations whose solutions are close to the corresponding solutions of discretizations resolving even the finest scales. Since for geometries with inseparable scales standard approaches such as homogenization theory are known to be limited in their applicability, we intend to set up an upscaled discretization using an ansatz space based on local low energy generalized eigenfunctions. It can be expected that the viability of this approach depends in particular on certain assumptions on the right hand sides in the problems under consideration. Thus, the validity of these assumptions in problem settings of practical importance will be part of our investigations.