A Staggered Grid Multi-Level ILU for steady incompressible flows

We present a parallel fully coupled multi-level incomplete factorization preconditioner for the 3D stationary incompressible Navier-Stokes equations on a structured grid. The algorithm and software are based on the robust two-level method developed in [1]. In this paper, we identify some of the weak spots of the two-level scheme and propose remedies such as a different domain partitioning and recursive application of the method. We apply the method to the wellknown 3D lid-driven cavity benchmark problem, and demonstrate its superior robustness by comparing with a segregated SIMPLE-type preconditioner

A staggered‐grid multilevel incomplete LU for steady incompressible flows

Algorithms for studying transitions and instabilities in incompressible flows typically require the solution of linear systems with the full Jacobian matrix. Other popular approaches, like gradient-based design optimization and fully implicit time integration, also require very robust solvers for this type of linear system. We present a parallel fully coupled multilevel incomplete factorization preconditioner for the 3D stationary incompressible Navier-Stokes equations on a structured grid. The algorithm and software are based on the robust two-level method developed by Wubs and Thies. In this article, we identify some of the weak spots of the two-level scheme and propose remedies such as a different domain partitioning and recursive application of the method. We apply the method to the well-known 3D lid-driven cavity benchmark problem, and demonstrate its superior robustness by comparing with a segregated SIMPLE-type preconditioner

Exploiting multilevel preconditioning techniques in eigenvalue computations

In the Davidson method, any preconditioner can be exploited for the iterative computation of eigenpairs. However, the convergence of the eigenproblem solver may be poor for a high quality preconditioner. Theoretically, this counter-intuitive phenomenon with the Davidson method is remedied by the Jacobi–Davidson approach, where the preconditioned system is restricted to appropriate subspaces of codimension one. However, it is not clear how the restricted system can be solved accurately and efficiently in the case of a good preconditioner. The obvious approach introduces instabilities that hamper convergence. In this paper, we show how incomplete decomposition based on multilevel approaches can be used in a stable way. We also show how these preconditioners can be efficiently improved when better approximations for the eigenvalue of interest become available. The additional costs for updating the preconditioners are negligible. Furthermore, our approach leads to a good initial guess for the wanted eigenpair and to high quality preconditioners for nearby eigenvalues. We illustrate our ideas for the MRILU preconditioner

Exploiting multilevel preconditioning techniques in eigenvalue computations

Abstract. In the Davidson method, any preconditioner can be exploited for the iterative computation of eigenpairs. However, the convergence of the eigenproblem solver may be poor for a high quality preconditioner. Theoretically, this counter-intuitive phenomenon with the Davidson method is remedied by the Jacobi–Davidson approach, where the preconditioned system is restricted to appropriate subspaces of co-dimension one. However, it is not clear how the restricted system can be solved accurately and efficiently in case of a good preconditioner. The obvious approach introduces instabilities that hamper convergence. In this paper, we show how incomplete decomposition based on multilevel approaches can be used in a stable way. We also show how these preconditioners can be efficiently improved when better approximations for the eigenvalue of interest become available. The additional costs for updating the preconditioners are negligible. Furthermore, our approach leads to a good initial guess for the wanted eigenpair and to high quality preconditioners for nearby eigenvalues. We illustrate our ideas for the MRILU preconditioner

Exploiting Multilevel Preconditioning Techniques in Eigenvalue Computations

In the Davidson method, any preconditioner can be exploited for the iterative computation of eigenpairs. However, the convergence of the eigenproblem solver may be poor for a high quality preconditioner. Theoretically, this counter-intuitive phenomenon with the Davidson method is remedied by the Jacobi--Davidson approach, where the preconditioned system is restricted to appropriate subspaces of co-dimension one. However, it is not clear how the restricted system can be solved accurately and e#ciently in case of a good preconditioner. The obvious approach introduces instabilities that hamper convergence

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