77 research outputs found
Low-Rank Iterative Solvers for Large-Scale Stochastic Galerkin Linear Systems
Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Dissertation, 2016von Dr. rer. pol. Akwum Agwu OnwuntaLiteraturverzeichnis: Seite 135-14
Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization
Topology optimization problems generally support multiple local minima, and
real-world applications are typically three-dimensional. In previous work [I.
P. A. Papadopoulos, P. E. Farrell, and T. M. Surowiec, Computing multiple
solutions of topology optimization problems, SIAM Journal on Scientific
Computing, (2021)], the authors developed the deflated barrier method, an
algorithm that can systematically compute multiple solutions of topology
optimization problems. In this work we develop preconditioners for the linear
systems arising in the application of this method to Stokes flow, making it
practical for use in three dimensions. In particular, we develop a nested block
preconditioning approach which reduces the linear systems to solving two
symmetric positive-definite matrices and an augmented momentum block. An
augmented Lagrangian term is used to control the innermost Schur complement and
we apply a geometric multigrid method with a kernel-capturing relaxation method
for the augmented momentum block. We present multiple solutions in
three-dimensional examples computed using the proposed iterative solver
Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems
In this paper we develop a family of preconditioners for the linear algebraic
systems arising from the arbitrary Lagrangian-Eulerian discretization of some
fluid-structure interaction models. After the time discretization, we formulate
the fluid-structure interaction equations as saddle point problems and prove
the uniform well-posedness. Then we discretize the space dimension by finite
element methods and prove their uniform well-posedness by two different
approaches under appropriate assumptions. The uniform well-posedness makes it
possible to design robust preconditioners for the discretized fluid-structure
interaction systems. Numerical examples are presented to show the robustness
and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2.
Rerun all the numerical tests 3. changed title, abstract and corrected lots
of typos and inconsistencies 4. added reference
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
- …