On fixed-point, Krylov, and 2×22\times 2 block preconditioners for nonsymmetric problems

The solution of matrices with $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained optimization problems, or the implicit or steady state treatment of any system of PDEs with multiple dependent variables. Often, these systems are solved iteratively using Krylov methods and some form of block preconditioner. Under the assumption that one diagonal block is inverted exactly, this paper proves a direct equivalence between convergence of $2\times2$ block preconditioned Krylov or fixed-point iterations to a given tolerance, with convergence of the underlying preconditioned Schur-complement problem. In particular, results indicate that an effective Schur-complement preconditioner is a necessary and sufficient condition for rapid convergence of $2\times 2$ block-preconditioned GMRES, for arbitrary relative-residual stopping tolerances. A number of corollaries and related results give new insight into block preconditioning, such as the fact that approximate block-LDU or symmetric block-triangular preconditioners offer minimal reduction in iteration over block-triangular preconditioners, despite the additional computational cost. Theoretical results are verified numerically on a nonsymmetric steady linearized Navier-Stokes discretization, which also demonstrate that theory based on the assumption of an exact inverse of one diagonal block extends well to the more practical setting of inexact inverses.Comment: Accepted to SIMA

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the ow of an electrically charged fuid under the in uence of an external electromagnetic eld with thermal coupling. This system of partial di erential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the di erent physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires e cient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a fexible and general software design for the code implementation of this type of preconditioners.Preprin

Matrix-based techniques for (flow-)transition studies

In this thesis, numerical techniques for the computation of flow transitions was introduced and studied. The numerical experiments on a variety of two- and three- dimensional multi-physics problems show that continuation approach is a practical and efficient way to solve series of steady states as a function of parameters and to do bifurcation analysis. Starting with a proper initial guess, Newton’s method converges in a few steps. Since solving the linear systems arising from the discretization takes most of the computational work, efficiency is determined by how fast the linear systems can be solved. Our home-made preconditioner Hybrid Multilevel Linear Solver(HYMLS) can compute three-dimensional solutions at higher Reynolds numbers and shows its robustness both in the computation of solutions as well as eigenpairs, due to the iteration in the divergence-free space. To test the efficiency of linear solvers for non-flow problems, we studied a well-known reaction-diffusion system, i.e., the BVAM model of the Turing problem. The application to the Turing system not only proved our program’s ability in doing nonlinear bifurcation analysis efficiently but also provided insightful information on two- and three- dimensional pattern formation