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Uniform Norm Error Estimates for 2D Turning Point Problem

Shallu, Sudipto Chowdhury, Vikas Gupta

TL;DR

This work analyzes a two-dimensional singularly perturbed convection–diffusion turning point problem and develops a finite element method on a layer-adapted Shishkin mesh. It leverages a discrete Green’s function framework to derive pointwise error estimates and establishes parameter-uniform convergence in the coarse and $x$-layer regions, with a convergence rate of order $N^{-1}\log^{3}(1/\\varepsilon^{3/2})$. Theoretical results are complemented by numerical experiments using a turning-point example and the double-mesh principle, which confirm the predicted behavior in the targeted regions while highlighting limitations in the $y$-layer areas. Overall, the paper advances pointwise error analysis for 2D turning-point SPDEs and demonstrates robust layer-resolving performance of FEM on Shishkin meshes.

Abstract

This work presents error analysis for a finite element method applied to a two-dimensional singularly perturbed convection-diffusion turning point problem. Utilizing a layer-adapted Shishkin mesh, we prove uniform convergence in the maximum norm in the coarse layer and x-layer regions. The analysis, critically based on the properties of a discrete Green's function, guarantees the method's robustness and accuracy in capturing sharp solution layers. There is no work for the coarse region for two-dimensional singularly perturbed turning point problems, and also, we are getting a linear order of convergence better than Stynes' article.

Uniform Norm Error Estimates for 2D Turning Point Problem

TL;DR

This work analyzes a two-dimensional singularly perturbed convection–diffusion turning point problem and develops a finite element method on a layer-adapted Shishkin mesh. It leverages a discrete Green’s function framework to derive pointwise error estimates and establishes parameter-uniform convergence in the coarse and -layer regions, with a convergence rate of order . Theoretical results are complemented by numerical experiments using a turning-point example and the double-mesh principle, which confirm the predicted behavior in the targeted regions while highlighting limitations in the -layer areas. Overall, the paper advances pointwise error analysis for 2D turning-point SPDEs and demonstrates robust layer-resolving performance of FEM on Shishkin meshes.

Abstract

This work presents error analysis for a finite element method applied to a two-dimensional singularly perturbed convection-diffusion turning point problem. Utilizing a layer-adapted Shishkin mesh, we prove uniform convergence in the maximum norm in the coarse layer and x-layer regions. The analysis, critically based on the properties of a discrete Green's function, guarantees the method's robustness and accuracy in capturing sharp solution layers. There is no work for the coarse region for two-dimensional singularly perturbed turning point problems, and also, we are getting a linear order of convergence better than Stynes' article.

Paper Structure

This paper contains 9 sections, 4 theorems, 137 equations, 3 figures, 6 tables.

Table of Contents

  1. INTRODUCTION
  2. The continuous problem
  3. A classical discretization on a Shishkin mesh
  4. Estimates for Discrete Green’s function
  5. Estimates for convection and reaction terms involving discrete Green's function
  6. Numerical results and discussion
  7. Numerical implementation
  8. Validation
  9. Conclusions

Key Result

Theorem 1

If we assume $c(x,y) -\frac{1}{2}\frac{\partial b_1(x,y)}{\partial x}\geq \gamma_1>0 \;\forall\; (x,y) \in\bar{\Omega}$ then for B(.,.) given in (weak) the following holds Henceforth, we denote the positive function $c - \frac{1}{2} \frac{\partial b_1}{\partial x}$ = $c_0^2$.

Figures (3)

  • Figure 1: Shishkin mesh
  • Figure 2: Domain $\Omega$ and subdomains
  • Figure 3: FEM solution $U_N$(N=512,$\varepsilon$=1.0e-06)

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4