Subspace correction methods for indefinite problems

In this project we plan to develop and analyze new subspace correction (SC) methods for the numerical solution of coupled systems of partial differential equations (PDE). The focus is on nearly singular symmetric positive definite (SPD) and on indefinite problems. We propose an integrated approach in which it is essential to use discretization techniques that preserve certain conservation laws, and to combine them with an adaptive solution process. In this way, one can design methods that perform optimally with respect to: (i) accurate approximation of the unknown quantities; (ii) obtaining the numerical solution in optimal time; and (iii) scalability with respect to both, problem size and advances in computer hardware. The present project has the following three interrelated Components (C1)--(C3) with a main emphasis on systems with highly oscillatory coefficients: (C1): SC methods for nearly incompressible elasticity and Stokes flow. (C2): SC methods for total variation minimization of discrete functionals arising in sparse data recovery. (C3): Auxiliary space and SC methods for elliptic problems with highly oscillatory coefficients. The primary goal of the proposed research work is to contribute to extending the theory and applicability of subspace correction methods to the above-mentioned classes of problems. Starting point of the research plan is the use and interplay of stable and accurate finite element schemes and of the efficient preconditioning of the related discrete problems. In the present setting nonconforming and in particular discontinuous Galerkin (DG) finite element methods provide adequate discretization tools. Some of their most attractive properties and practical advantages over conforming methods are that a) it is easy to extended DG methods to higher approximation order; b) they are well suited to treat complex geometries in combination with unstructured and hybrid meshes; c) they can be combined with any element type where the grids are also allowed to have hanging nodes; d) they can easily handle adaptive strategies; e) they have favorable properties in view of parallel computing. A main disadvantage of DG discretizations is that they produce an excess of degrees of freedom (as compared to conforming methods of the same approximation order) which in general makes the solution of the arising linear systems more difficult and more time consuming. We therefore put strong efforts on devising new efficient and robust solution methods, covering wider classes of problems (see (C1)--(C3)) that arise from nonconforming and discontinuous Galerkin discretizations. The final aim is to adapt our methods to and to test them on industrial and multiphysics applications, e.g., in reservoir engineering, or in life science. Some of the problems in which we are particularly interested stem from micro-mechanics modeling of heterogeneous media, e.g., the modeling of fluid flow in porous media, the determination of the bio-mechanical properties of bones, or the reconstruction of (medical) images. Typically such problems involve parameters that lead to highly ill-conditioned systems of linear algebraic equations.

The scope of this project was the development and investigation of new subspace correction methods for the numerical solution of coupled systems of partial differential equations (PDE) as they play a key role in industrial applications, e.g., in the area of reservoir engineering, or in medical applications, e.g., in the area of biomechanics, or for the optimal control of flow and filtration processes. Especially challenging are models describing nearly incompressible materials or highly heterogeneous media, which usually lead to extremely ill-conditioned systems of linear algebraic equations. The present project had the following three components (C1)(C3): (C1): Stable discretizations and optimal iterative solution methods for elasticity problems in case of nearly incompressible materials and for the Stokes problem. (C2): Stable discretizations and fast solvers for convection-diffusion problems. (C3): Robust subspace correction methods for elliptic problems with highly oscillatory coefficients. The primary goal of the performed research work was to contribute substantially to the extension of the theory as well as to the applicability of subspace correction methods to the above-mentioned problem classes. The starting point was the utilization and the interplay of problem-oriented finite element methods (stable discretizations) with efficient and robust preconditioning and iterative solution methods for the related discrete problems. In this context nonconforming and discontinuous Galerkin (DG) methods provide attractive discretization tools. The main results in the area of component (C1) of the research plan were the development of robust preconditioners for linear elasticity problems discretized by DG methods as well as a complete theory of optimal multigrid methods for Stokes and Brinkman equations, again for DG discretizations. Regarding component (C2) new a priori error estimates could be derived for a stable monotone discretization scheme. The latter provides the starting point for the construction of fast solvers. In view of component (C3) a novel multigrid method was developed based on the ideas of auxiliary space correction and additive Schur complement approximation. This auxiliary space multigrid (ASMG) method exhibits favorable robustness properties with respect to highly oscillatory coefficients.