An efficient nonlinear multigrid scheme for 2D boundary value problems

Abstract

In this article, a two-dimensional nonlinear boundary value problem which is strongly related to the well-known Gelfand–Bratu model is solved numerically. The numerical results are obtained by employing three different numerical strategies namely: finite difference based method, a Newton multigrid method and a nonlinear multigrid full approximation storage (FAS). We are able to handle the difficulty of unstable convergence behaviour by using MINRES method as a relaxation smoother in multigrid approach with an appropriate sinusoidal approximation as an initial guess. A comparison, in terms of convergence, accuracy and efficiency among the three numerical methods demonstrate an improvement for the values of λ ∈ (0, λc]. Numerical results illustrate the performance of the proposed numerical methods wherein FAS-MG method is shown to be the most efficient. Further, we present the numerical bifurcation behaviour for two-dimensional Gelfand-Bratu models and find new multiplicity of solutions in the case of a quadratic and cubic approximation of the nonlinear exponential term. Numerical experiments confirm the convergence of the solutions for different values of λ and prove the effectiveness of the nonlinear FAS-MG scheme.