Energy norm error estimate for singularly perturbed fourth-order differential equation with two parameters

Abstract

We consider a fourth-order reaction–diffusion-type singularly perturbed boundary value problem with two small parameters ${\epsilon }_1$ and ${\epsilon }_2$ multiplied to the fourth- and second-order derivative terms respectively. In this article, we restrict to a special case, where ${\epsilon }_1<<{\epsilon }_2^2$ and derive the finite element scheme using Ritz–Galerkin finite element method with lumping process. On discretizing the domain, the layer-adapted meshes like Standard-Shishkin, Bakhvalov-Shishkin and Modified-Bakhvalov-type meshes and piecewise quadratic polynomials are used and the error estimates are derived in $L_2$-norm and energy norm. The numerical experiments given in the article supports these theoretical findings.

Keywords:

• Singularly perturbation parameters
• boundary value problem
• reaction diffusion
• $L_2$-norm
• energy norm
• layer adapted meshes
• finite element method

AMS Mathematics Subject Classifications:

• 34D15
• 34E15
• 65L60

Disclosure statement

No potential conflict of interest was reported by the author(s).