Multilevel Schwarz Methods for Porous Media Problems

In this thesis, efficient overlapping multilevel Schwarz preconditioners are used to iteratively solve Hdiv-conforming finite element discretizations of models in poroelasticity, and an innovative two-scale multilevel Schwarz method is developed for the solution of pore-scale porous media models. The convergence of two-level Schwarz methods is rigorously proven for Biot’s consolidation model, as well as a Biot-Brinkman model by utilizing the conservation property of the discretization. The numerical performance of the proposed multiplicative and hybrid two-level Schwarz methods is tested in different problem settings by covering broad ranges of the parameter regimes, showing robust results in variations of the parameters in the system that are uniform in the mesh size. For extreme parameters a scaling of the system yields robustness of the iteration counts. Optimality of the relaxation factor of the hybrid method is investigated and the performance of the multilevel methods is shown to be nearly identical to the two-level case. The additional diffusion term in the Biot-Brinkman model yields a stabilization for high permeabilities. Additionally, a homogenizing two-scale multilevel Schwarz preconditioner is developed for the iterative solution of high-resolution computations of flow in porous media at the pore scale, i.e., a Stokes problem in a periodically perforated domain. Different homogenized operators known from the literature are used as coarse-scale operators within a multilevel Schwarz preconditioner applied to Hdiv-conforming discretizations of an extended model problem. A comparison in the numerical performance tests shows that an operator of Brinkman type with optimized effective tensor yields the best performance results in an axisymmetric configuration and a moderately anisotropic geometry of the obstacles, outperforming Darcy and Stokes as coarse-scale operators, as well as a standard multigrid method, that serves as a benchmark test

Geometric Multigrid Methods for Flow Problems in Highly Heterogeneous Porous Media

In this dissertation, we develop geometric multigrid methods for the finite element approximation of flow problems (e:g:, Stokes, Darcy and Brinkman models) in highly heterogeneous porous media. Our method is based on H^(div)-conforming discontinuous Galerkin methods and the Arnold-Falk-Winther (AFW) type smoothers. The main advantage of using H^(div)-conforming elements is that the discrete velocity field will be globally divergence-free for incompressible fluid flows. Besides, the smoothers used are of overlapping domain decomposition Schwarz type and employ a local Helmholtz decomposition. Our flow solvers are the combination of multigrid preconditioners and classical iterative solvers. The proposed preconditioners act on the combined velocity and pressure space and thus does not need a Schur complement approximation. There are two main ingredients of our preconditioner: first, the AFW type smoothers can capture a meaningful basis on local divergence free subspace associated with each overlapping patch; second, the grid operator does not increase the divergence from the coarse divergence free subspace to the fine one as the divergence free spaces are nested. We present the convergence analysis for the Stokes' equations and Brinkman's equations ( with constant permeability field ), as well as extensive numerical experiments. Some of the numerical experiments are given to support the theoretical results. Even though we do not have analysis work for the highly heterogeneous and highly porous media cases, numerical evidence exhibits strong robustness, efficiency and unification of our algorithm

Multilevel Schwarz Methods for Incompressible Flow Problems

In this thesis, we address coupled incompressible flow problems with respect to their efficient numerical solutions. These problems are modeled by the Oseen equations, the Navier-Stokes equations and the Brinkman equations. For numerical approximations of these equations, we discretize these systems by Hdiv-conforming discontinuous Galerkin method which globally satisfy the divergence free velocity constraint on discrete level. The algebraic systems arising from discretizations are large in size and have poor spectral properties which makes it challenging to solve these linear systems efficiently. For efficient solution of these algebraic system, we develop our solvers based on classical iterative solvers preconditioned with multigrid preconditioners employing overlapping Schwarz smoothers of multiplicative type. Multigrid methods are well known for their robustness in context of self-adjoint problems. We present an overview of the convergence analysis of multigrid method for symmetric problems. However, we extend this method to non self-adjoint problems, like the Oseen equations, by incorporating the downwind ordering schemes of Bey and Hackbusch and we show the robustness of this method by empirical results. Furthermore, we extend this approach to non-linear problems, like the Navier-Stokes and the non-linear Brinkman equations, by using a Picard iteration scheme for linearization. We investigate extensively by performing numerical experiment for various examples of incompressible flow problems and show by empirical results that the multigrid method is efficient and robust with respect to the mesh size, the Reynolds number and the polynomial degree. We also observe from our numerical results that in case of highly heterogeneous media, multigrid method is robust with respect to a high contrast in permeability

Discretizations & Efficient Linear Solvers for Problems Related to Fluid Flow

Numerical solutions to fluid flow problems involve solving the linear systems arising from the discretization of the Stokes equation or a variant of it, which often have a saddle point structure and are difficult to solve. Geometric multigrid is a parallelizable method that can efficiently solve these linear systems especially for a large number of unknowns. We consider two approaches to solve these linear systems using geometric multigrid: First, we use a block preconditioner and apply geometric multigrid as in inner solver to the velocity block only. We develop deal.II tutorial step-56 to compare the use of geometric multigrid to other popular alternatives. This method is found to be competitive in serial computations in terms of performance and memory usage. Second, we design a special smoother to apply multigrid to the whole linear system. This smoother is analyzed as a Schwarz method using conforming and inf-sup stable discretization spaces. The resulting method is found to be competitive to a similar multigrid method using non-conforming finite elements that were studied by Kanschat and Mao. This approach has the potential to be superior to the first approach. Finally, expanding on the research done by Dannberg and Heister, we explore the analysis of a three-field Stokes formulation that is used to describe melt migration in the earth\u27s mantle. Multiple discretizations were studied to find the best one to use in the ASPECT software package. We also explore improvements to ASPECT\u27s linear solvers for this formulation utilizing block preconditioners and algebraic multigrid

A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier--Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for GMRES iterations. Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the \emph{deal.II} finite element library. The concepts are flexible and can be transferred to similar software platforms.Comment: Key updates of this revision: - Added Subsection 5.2 "Parallel scaling", in which a strong scaling benchmark is performed - Added Subsection 5.3 "Parameter robustness regarding v", where the robustness of the proposed numerical scheme, regarding changes in the viscosity, is computationally analyze

Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed