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A modified graded mesh and higher order finite element method for singularly perturbed reaction–diffusion problems

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  • Kaushik, Aditya
  • Kumar, Vijayant
  • Sharma, Manju
  • Sharma, Nitika

Abstract

This paper presents a modified graded mesh for singularly perturbed reaction–diffusion problems. The mesh we offer is generated recursively using Newton’s algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on polynomials of degree p≥1. We prove parameter uniform convergence of optimal order in ϵ-weighted energy norm. Test examples are taken, and we present a rigorous comparative analysis with other adaptive meshes. Moreover, we compare the proposed method with others found in the literature.

Suggested Citation

  • Kaushik, Aditya & Kumar, Vijayant & Sharma, Manju & Sharma, Nitika, 2021. "A modified graded mesh and higher order finite element method for singularly perturbed reaction–diffusion problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 486-496.
    • Handle: RePEc:eee:matcom:v:185:y:2021:i:c:p:486-496
    DOI: 10.1016/j.matcom.2021.01.006
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    1. Aditya Kaushik & Manju Sharma, 2012. "Convergence Analysis of Weighted Difference Approximations on Piecewise Uniform Grids to a Class of Singularly Perturbed Functional Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 252-272, October.
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