THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D

doi:10.1007/s10473-024-0421-7

Abstract

Abstract: For singularly perturbed convection-diffusion problems, supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes, especially in the case of 2D. The difficulties arise from the width of the mesh in the layer adjacent to the transition point, resulting in a suboptimal estimate for convergence. Existing analysis techniques cannot handle these difficulties well. To fill this gap, here a novel interpolation is designed delicately for the smooth part of the solution, bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method. Our theoretical result is uniform in the singular perturbation parameter $\varepsilon$ and is supported by the numerical experiments.

Key words: singularly perturbed, convection-diffusion, finite element method, supercloseness, Bakhvalov-type mesh

CLC Number:

65N12

Chunxiao Zhang, Jin Zhang. THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D[J].Acta mathematica scientia,Series B, 2024, 44(4): 1572-1593.

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