Regularity and approximation of a hyperbolic-elliptic coupled problem

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, we investigate the regularity and approximation of a hyperbolic-elliptic coupled problem. In particular, we consider the Poisson and the transport equation where both are assigned nonhomogeneous Dirichlet boundary conditions. The coupling of the two problems is executed as follows. The right hand side function of the Poisson equation is the solution ρ of the transport equation whereas the gradient field E = −∇u, with u being solution of the Poisson problem, is the convective field for the transport equation. The analysis is done throughout on a nonconvex, not simply connected domain that is supposed to be homeomorph to an annular domain. In the first part of this thesis, we will focus on the existence and uniqueness of a classical solution to this highly nonlinear problem using the framework of Hölder continuous functions. Herein, we distinguish between a time dependent and time independent formulation. In both cases, we investigate the streamline functions defined by the convective field E. These are used in the time dependent case to derive an operator equation whose fixed point is the streamline function to the gradient of the classical solution u. In the time independent setting, we formulate explicitly the solution operators L for the Poisson and T for the transport equation and show with a fixed point argument the existence and uniqueness of a classical solution (u,ρ). The second part of this thesis deals with the approximation of the coupled problem in Sobolev spaces. First, we show that the nonlinear transport equation can be formulated equivalently as variational inequality and analyse its Galerkin finite element discretization. Due to the nonlinearity of the coupled problem, it is necessary to use iterative solvers. We will introduce the staggered algorithm which is an iterative method solving alternating the Poisson and transport equation until convergence is obtained. Assuming that LοT is a contraction in the Sobolev space H1(Ω), we will investigate the convergence of the discrete staggered algorithm and obtain an error estimate. Subsequently, we present numerical results in two and three dimensions. Beside the staggered algorithm, we will introduce other iterative solvers that are based on linearizing the coupled problem by Newton’s method. We illustrate that all iterative solvers converge satisfactorily to the solution (u, ρ).Department of Mathematical Sciences, Brunel Universit

Deflation for the off-diagonal block in symmetric saddle point systems

Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub-Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method like MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.Comment: 26 pages, 12 figure

L²(H¹γ) finite element convergence for degenerate isotropic Hamilton–Jacobi–Bellman equations

In this paper we study the convergence of monotone P1 finite element methods for fully nonlinear Hamilton–Jacobi–Bellman equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in a weighted Sobolev space L²(H¹γ(Ω)) to the viscosity solution without assuming uniform parabolicity of the HJB operator

Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy

We study an inexact inner-outer generalized Golub-Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.Comment: 25 pages, 9 figure

Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications

In the context of Tokamak fusion plasma, the gyrokinetic Poisson equation has to be solved on disk-like domains which correspond to the poloidal cross-section of the Tokamak geometry. In its simpliest form, this cross section takes a circular form but deformed geometries were found to be advantageous and are more realistic. We propose a taylored solver for the gyrokinetic Poisson equation on disklike geometries described by curvilinear coordinates. Our solver also copes with an anisotropic, locally refined mesh to model the edge part of the Tokamak and a rapidly dropping density profile. Multigrid methods can achieve optimal complexity for many problems and are among the most efficient solvers for elliptic model problems. Multigrid methods for geometries described by curvilinear coordinates are however less common. We present a taylored geometric multigrid algorithm using optimized line smoothers to enable parallel scalability. We propose an implicit extrapolation scheme for our algorithm to increase the order of convergence by using nonstandard numerical integration rules for P1 finite elements. We also use problem-specific finite differences, which numerically show the same order of convergence as their finite elements' counterpart, enabling a matrix-free implementation with a low memory footprint

Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications

The gyrokinetic Poisson equation arises as a subproblem of Tokamak fusion reactor simulations. It is often posed on disk-like cross sections of the Tokamak that are represented in generalized polar coordinates. On the resulting curvilinear anisotropic meshes, we discretize the differential equation by finite differences or low order finite elements. Using an implicit extrapolation technique similar to multigrid tau-extrapolation, the approximation order can be increased. This technique can be naturally integrated in a matrix-free geometric multigrid algorithm. Special smoothers are developed to deal with the mesh anisotropy arising from the curvilinear coordinate system and mesh grading

A numerical scheme for the early steps of nucleation-aggregation Models

International audienceIn the formation of large clusters out of small particles, the initializing step is called the nucleation, and consists in the spontaneous reaction of agents which aggregate into small and stable polymers called nucleus. After this early step, the polymers are involved into a bunch of reactions such as polymerization, fragmentation and coalescence. Since there may be several orders of magnitude between the size of a particle and the size of an aggregate, building efficient numerical schemes to capture accurately the kinetics of the reaction is a delicate step of key importance. In this article, we propose a conservative scheme, based on finite volume methods on an adaptive grid, which is able to render out the early steps of the reaction as well as the later chain reactions

Energy-Minimizing, Symmetric Discretizations for Anisotropic Meshes and Energy Functional Extrapolation

Self-adjoint differential operators often arise from variational calculus on energy functionals. In this case, a direct discretization of the energy functional induces a discretization of the differential operator. Following this approach, the discrete equations are naturally symmetric if the energy functional was self-adjoint, a property that may be lost when using standard difference formulas on nonuniform meshes or when the differential operator has varying coefficients. Low order finite difference or finite element systems can be derived by this approach in a systematic way and on logically structured meshes they become compact difference formulas. Extrapolation formulas used on the discrete energy can then lead to higher oder approximations of the differential operator. A rigorous analysis is presented for extrapolation used in combination with nonstandard integration rules for finite elements. Extrapolation can likewise be applied on matrix-free finite difference stencils. In our applications, both schemes show up to quartic order of convergence