Numerical Methods for Nonlinear Elliptic Boundary Value Problems with Parameter Dependence

Abstract

This thesis discusses numerical methods for nonlinear elliptic partial differential equations with parameter dependence such as the Gelfand-Bratu model and singularly perturbed convection-diffusion-reaction equations. In numerical investigations, more accurate and efficient nonstandard finite difference and multigrid methods are adopted to solve parameter dependent elliptic boundary-value problems. Aim and objectives of the research are given in chapter 1. Further, it describes the background of Gelfand-Bratu and singularly perturbed problems. In addition, an overview of nonstandard finite difference and multigrid methods is given followed by an outline of the thesis. In chapter 2, standard and nonstandard finite difference approximations are employed to find the numerical solutions of the one-dimensional truncated Bratu-Picard (BP) model. Numerical results show the existence of infinitely many solutions, which are calculated numerically (for a large, but finite, set of solutions). These new types of solutions are either periodic or semi-periodic. We observe that the nonstandard finite difference schemes provide more accurate and efficient results than standard finite difference schemes. In chapter 3, we propose a higher order non-uniform finite difference grid, to solve singularly perturbed boundary value problems with steep boundary-layers. Theoretical properties concerning the extremum values and the asymptotic value at the right boundary point are presented. Several examples are provided, which demonstrate the effectiveness of the proposed numerical strategy. We establish numerically, not only 4th-order but also a 6th-order of accuracy by considering only three-point central non-uniform finite differences. Numerical results illustrate that to achieve the 6th-order of accuracy, the proposed method needs approximately a factor of 5-10 fewer grid points than the uniform case. In chapter 4, we propose three numerical methods, viz, a finite-difference approximation and two multigrid (MG) approaches: Newton-MG and Full Approximation Storage (FAS). A comparison, in terms of convergence, accuracy and efficiency among the three numerical methods demonstrate improvement for the whole parameter range λ ∈ (0, λc]. Further, we investigate the bifurcation behaviour of solutions and find new multiplicity of solutions in the case of a cubic approximation of the nonlinear exponential term. We demonstrate that the convergence of all solutions namely, unique, lower, upper, periodic and semi-periodic is obtained for small values of the parameter λ. Particularly, FAS-MG is found to be more efficient than the other two methods. In chapter 5, we present a numerical study of the Gelfand-Bratu model for higher dimensions. For three dimensions, we adopt an accurate and efficient nonlinear multigrid approach, namely, FAS-MG extended with a Krylov method as a smoother. New types of solutions are obtained or specific values of the bifurcation parameter. Further, the numerical bifurcation curve of the Gelfand-Bratu problem in three dimensions shows the existence of two new turning points. Numerical results confirm the convergence of all types of solutions and demonstrate the effectiveness of the proposed numerical strategy. For even higher dimensions, numerical experiments show the existence of several types of solutions. Bifurcation curves confirm the theoretical results of the higher-dimensional Gelfand-Bratu problem as presented in the literature.