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Optimal order uniform convergence in energy and balanced norms of weak Galerkin finite element method on Bakhvalov-type meshes for nonlinear singularly perturbed problems

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Abstract

In this paper, we propose a weak Galerkin finite element method (WG-FEM) for solving nonlinear boundary value problems of reaction–diffusion type on a Bakhvalov-type mesh. A robust optimal order of uniform convergence on a Bakhvalov-type mesh has been proved in the energy norm and a stronger balanced norm using piecewise polynomials of degree \(k\ge 1\) on interior of the elements and piecewise constant on the boundary of each element. The proposed finite element scheme is independent of parameter and since the interior degree of freedom can be eliminated efficiently from the resulting discrete system, number of unknowns of the proposed method is comparable with the standard finite element methods. For the first time, a uniform error estimate has been established in the energy norm and in a balanced norm using higher order polynomials on a Bakhvalov-type mesh. Finally, numerical experiments are given to support the theoretical findings and show the efficiency of the proposed method.

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Authors and Affiliations

  1. Faculty of Engineering, Department of Computer Science, Artvin Çoruh University, Seyitler, Artvin, 08100, Turkey

    Şuayip Toprakseven

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  1. Şuayip Toprakseven
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Correspondence to Şuayip Toprakseven.

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Communicated by Frederic Valentin.

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Toprakseven, Ş. Optimal order uniform convergence in energy and balanced norms of weak Galerkin finite element method on Bakhvalov-type meshes for nonlinear singularly perturbed problems. Comp. Appl. Math. 41, 377 (2022). https://doi.org/10.1007/s40314-022-02090-z

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  • DOI: https://doi.org/10.1007/s40314-022-02090-z

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