8 research outputs found
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
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents
for better readability, part on wavelet preconditioner adde
Local Fourier analysis for saddle-point problems
The numerical solution of saddle-point problems has attracted considerable interest in
recent years, due to their indefiniteness and often poor spectral properties that make
efficient solution difficult. While much research already exists, developing efficient
algorithms remains challenging. Researchers have applied finite-difference, finite element,
and finite-volume approaches successfully to discretize saddle-point problems,
and block preconditioners and monolithic multigrid methods have been proposed for
the resulting systems. However, there is still much to understand.
Magnetohydrodynamics (MHD) models the flow of a charged fluid, or plasma, in
the presence of electromagnetic fields. Often, the discretization and linearization of
MHD leads to a saddle-point system. We present vector-potential formulations of
MHD and a theoretical analysis of the existence and uniqueness of solutions of both
the continuum two-dimensional resistive MHD model and its discretization.
Local Fourier analysis (LFA) is a commonly used tool for the analysis of multigrid
and other multilevel algorithms. We first adapt LFA to analyse the properties of
multigrid methods for both finite-difference and finite-element discretizations of the
Stokes equations, leading to saddle-point systems. Monolithic multigrid methods,
based on distributive, Braess-Sarazin, and Uzawa relaxation are discussed. From
this LFA, optimal parameters are proposed for these multigrid solvers. Numerical
experiments are presented to validate our theoretical results. A modified two-level
LFA is proposed for high-order finite-element methods for the Lapalce problem, curing
the failure of classical LFA smoothing analysis in this setting and providing a reliable
way to estimate actual multigrid performance. Finally, we extend LFA to analyze the
balancing domain decomposition by constraints (BDDC) algorithm, using a new choice
of basis for the space of Fourier harmonics that greatly simplifies the application of
LFA. Improved performance is obtained for some two- and three-level variants
Bilinear Immersed Finite Elements for Interface Problems
In this dissertation we discuss bilinear immersed finite elements (IFE) for solving interface problems. The related research works can be categorized into three aspects: (1) the construction of the bilinear immersed finite element spaces; (2) numerical methods based on these IFE spaces for solving interface problems; and (3) the corresponding error analysis. All of these together form a solid foundation for the bilinear IFEs.
The research on immersed finite elements is motivated by many real world applications, in which a simulation domain is often formed by several materials separated from each other by curves or surfaces while a mesh independent of interface instead of a body-fitting mesh is preferred. The bilinear IFE spaces are nonconforming finite element spaces and the mesh can be independent of interface. The error estimates for the interpolation of a Sobolev function in a bilinear IFE space indicate that this space has the usual approximation capability expected from bilinear polynomials, which is O(h2) in L2 norm and O(h) in H1 norm. Then the immersed spaces are applied in Galerkin, finite volume element (FVE) and discontinuous Galerkin (DG) methods for solving interface problems. Numerical examples show that these methods based on the bilinear IFE spaces have the same optimal convergence rates as those based on the standard bilinear finite element for solutions with certain smoothness. For the symmetric selective immersed discontinuous Galerkin method based on bilinear IFE, we have established its optimal convergence rate. For the Galerkin method based on bilinear IFE, we have also established its convergence.
One of the important advantages of the discontinuous Galerkin method is its flexibility for both p and h mesh refinement. Because IFEs can use a mesh independent of interface, such as a structured mesh, the combination of a DG method and IFEs allows a flexible adaptive mesh independent of interface to be used for solving interface problems. That is, a mesh independent of interface can be refined wherever needed, such as around the interface and the singular source. We also develop an efficient selective immersed discontinuous Galerkin method. It uses the sophisticated discontinuous Galerkin formulation only around the locations needed, but uses the simpler Galerkin formulation everywhere else. This selective formulation leads to an algebraic system with far less unknowns than the immersed DG method without scarifying the accuracy; hence it is far more efficient than the conventional discontinuous Galerkin formulations