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An efficient nonlinear multigrid scheme for 2D boundary value problems

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  • Iqbal, Sehar
  • Zegeling, Paul Andries

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.

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  • Iqbal, Sehar & Zegeling, Paul Andries, 2020. "An efficient nonlinear multigrid scheme for 2D boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    • Handle: RePEc:eee:apmaco:v:372:y:2020:i:c:s0096300319308902
    DOI: 10.1016/j.amc.2019.124898
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    1. Zegeling, Paul Andries & Iqbal, Sehar, 2018. "Nonstandard finite differences for a truncated Bratu–Picard model," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 266-284.
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    Cited by:

    1. Shahab, Muhammad Luthfi & Susanto, Hadi, 2024. "Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 483(C).

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