An efficient nonlinear multigrid scheme for 2D boundary value problems

Sehar Iqbal and Paul Andries Zegeling

Applied Mathematics and Computation, 2020, vol. 372, issue C

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.

Keywords: Nonlinear boundary value problems; Gelfand-Bratu problem; Finite differences; Newton multigrid method; Bifurcation diagram; Nonlinear multigrid; Multiple solutions (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:372:y:2020:i:c:s0096300319308902

DOI: 10.1016/j.amc.2019.124898

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