A numerical study of the higher-dimensional Gelfand-Bratu model

Abstract

In this article, a higher dimensional nonlinear boundary-value problem, viz., Gelfand-Bratu (GB) problem, is solved numerically. For the three-dimensional case, we present an accurate and efficient nonlinear multigrid (MG) approach and investigate multiplicities depending on the bifurcation parameter λ. We adopt a nonlinear MG approach Full approximation scheme (FAS) extended with a Krylov method as a smoother to handle the computational difficulties for obtaining the upper branches of the solutions. Further, we examine the numerical bifurcation behaviour of the GB problem in 3D and identify the existence of two new bifurcation points. Experiments illustrate the convergence of the numerical solutions and demonstrate the effectiveness of the proposed numerical strategy for all parameter values λ∈(0,λc]. For higher dimensions, we transform the GB problem, using n-dimensional spherical coordinates, to a nonlinear ordinary differential equation (ODE). The numerical solutions of this nonlinear ODE are computed by a shooting method for a range of values of the dimension parameter n. Numerical experiments show the existence of several types of solutions for different values of n and λ. These results confirm the bifurcation behaviour of the higher dimensional GB problem as predicted from theoretical results in literature.