A class of nonsymmetric preconditioners for saddle point problems

For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric

Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations

Multigrid methods belong to the best-known methods for solving linear systems arising from the discretization of elliptic partial differential equations. The main attraction of multigrid methods is that they have an asymptotically meshindependent convergence behavior. Multigrid with Vanka (or local multilevel pressure Schur complement method) as smoother have been frequently used for the construction of very effcient coupled monolithic solvers for the solution of the stationary incompressible Navier-Stokes equations in 2D and 3D. However, due to its innate Gauß-Seidel/Jacobi character, Vanka has a strong influence of the underlying mesh, and therefore, coupled multigrid solvers with Vanka smoothing very frequently face convergence issues on meshes with high aspect ratios. Moreover, even on very nice regular grids, these solvers may fail when the anisotropies are introduced from the differential operator. In this thesis, we develop a new class of robust and efficient monolithic finite element multilevel Krylov subspace methods (MLKM) for the solution of the stationary incompressible Navier-Stokes equations as an alternative to the coupled multigrid-based solvers. Different from multigrid, the MLKM utilizes a Krylov method as the basis in the error reduction process. The solver is based on the multilevel projection-based method of Erlangga and Nabben, which accelerates the convergence of the Krylov subspace methods by shifting the small eigenvalues of the system matrix, responsible for the slow convergence of the Krylov iteration, to the largest eigenvalue. Before embarking on the Navier-Stokes equations, we first test our implementation of the MLKM solver by solving scalar model problems, namely the convection-diffusion problem and the anisotropic diffusion problem. We validate the method by solving several standard benchmark problems. Next, we present the numerical results for the solution of the incompressible Navier-Stokes equations in two dimensions. The results show that the MLKM solvers produce asymptotically mesh-size independent, as well as Reynolds number independent convergence rates, for a moderate range of Reynolds numbers. Moreover, numerical simulations also show that the coupled MLKM solvers can handle (both mesh and operator based) anisotropies better than the coupled multigrid solvers

Preconditioners for Generalized Saddle-Point Problems

Generalized saddle point problems arise in a number of applications, ranging from optimization and metal deformation to fluid flow and PDE-governed optimal control. We focus our discussion on the most general case, making no assumption of symmetry or definiteness in the matrix or its blocks. As these problems are often large and sparse, preconditioners play a critical role in speeding the convergence of Krylov methods for these problems. We first examine two types of preconditioners for these problems, one block-diagonal and one indefinite, and present analyses of the eigenvalue distributions of the preconditioned matrices. We also investigate the use of approximations for the Schur complement matrix in these preconditioners and develop eigenvalue analysis accordingly. Second, we examine new developments in probing methods, inspired by graph coloring methods for sparse Jacobians, for building approximations to Schur complement matrices. We then present an analysis of these techniques and their accuracy. In addition, we provide a mathematical justification for their use in approximating Schur complements and suggest the use of approximate factorization techniques to decrease the computational cost of applying the inverse of the probed matrix. Finally, we consider the effect of our preconditioners on four applications. Two of these applications come from the realm of fluid flow, one using a finite element discretization and the other using a spectral discretization. The third application involves the stress relaxation of aluminum strips at low stress levels. The final application involves mesh parameterization and flattening. For these applications, we present results illustrating the eigenvalue bounds on our preconditioners and demonstrating the theoretical justification of these methods. We also present convergence and timing results, showing the effectiveness of our methods in practice. Specifically the use of probing methods for approximating the Schur compliment matrices in our preconditioners is empirically justified. We also investigate the $h$-dependence of our preconditioners one model fluid problem, and demonstrate empirically that our methods do not suffer from a deterioration in convergence as the problem size increases

An algebraic multigrid method for Q2−Q1Q_2-Q_1 mixed discretizations of the Navier-Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Specifically, we investigate a $Q_2-Q_1$ mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches leveraging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the $Q_2-Q_1$ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the finest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

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