[Paper Review] FEM-analysis on graded meshes for turning point problems exhibiting an interior layer

Simon Becher|arXiv (Cornell University)|Mar 15, 2016

Differential Equations and Numerical Methods4 references3 citations

TL;DR

This paper presents a finite element analysis on layer-adapted graded meshes for singularly perturbed two-point boundary value problems with an interior turning point and a cusp-type interior layer. Using higher-order finite elements on Liseikin's graded meshes, the authors prove $\varepsilon$-uniform error estimates in the energy norm and optimal $O(N^{-2})$ convergence in the $L^2$-norm for linear elements, establishing robust numerical stability and accuracy for problems with strong boundary and interior layers.

ABSTRACT

We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted graded meshes proposed by Liseikin. We prove $ε$-uniform error estimates in the energy norm. Furthermore, for linear elements we are able to prove optimal order $ε$-uniform convergence in the $L^2$-norm on these graded meshes.

Motivation & Objective

• To address the numerical challenge of solving singularly perturbed two-point boundary value problems with a simple interior turning point and a cusp-type interior layer.
• To develop and analyze a finite element method on layer-adapted graded meshes that ensures robust convergence independent of the perturbation parameter $\varepsilon$.
• To extend existing results beyond linear finite elements by proving $\varepsilon$-uniform convergence for higher-order finite elements on graded meshes.
• To establish optimal $L^2$-norm error estimates for linear finite elements on these meshes, confirming theoretical convergence rates.

Proposed method

• The study employs the finite element method (FEM) with higher-order Lagrange elements on graded meshes generated by Liseikin's mesh-generating function, which is designed to resolve interior layers.
• The mesh is constructed using a transformation $\varphi(\xi,\varepsilon)$ that ensures dense clustering near the turning point $x_0 = 0$, with grading parameter $\alpha \leq \lambda$ to match the layer thickness.
• The analysis uses a weighted energy norm $|||\cdot|||_{\varepsilon} = \left(\varepsilon|\cdot|_1^2 + \|\cdot\|^2\right)^{1/2}$ to measure error and prove $\varepsilon$-uniform convergence.
• For linear elements, a supercloseness result is derived and used to prove optimal $O(N^{-2})$ convergence in the $L^2$-norm via postprocessing and interpolation estimates.
• Key technical tools include mesh-dependent inverse inequalities, interpolation error estimates, and bounds on mesh ratios and second derivatives of the mesh function.
• Theoretical analysis relies on the equidistribution principle and majorant-based layer-damping transformations to control solution derivatives and mesh grading.

Experimental results

Research questions

• RQ1 Can higher-order finite elements on Liseikin's graded meshes achieve $\varepsilon$-uniform convergence for singularly perturbed problems with an interior turning point and cusp-type layer?
• RQ2 What is the optimal convergence rate in the $L^2$-norm for linear finite elements on these graded meshes?
• RQ3 How do the mesh grading parameter $\alpha$ and the layer parameter $\lambda$ affect the convergence behavior?
• RQ4 Can the supercloseness property be leveraged to derive optimal $L^2$-norm error estimates for linear elements?

Key findings

• The finite element method using higher-order elements on Liseikin's graded meshes achieves $\varepsilon$-uniform convergence in the energy norm of order $O(N^{-k})$ for elements of degree $k$.
• For linear finite elements ($k=1$), the method achieves optimal $\varepsilon$-uniform convergence in the $L^2$-norm at the rate $O(N^{-2})$.
• The error bounds are independent of the perturbation parameter $\varepsilon$, confirming robustness for small $\varepsilon$.
• The analysis confirms that the mesh grading parameter $\alpha$ must satisfy $\alpha \leq \lambda$ to preserve optimal convergence.
• Numerical experiments validate the theoretical convergence rates, showing agreement with predicted orders of magnitude.
• The supercloseness result enables the derivation of optimal $L^2$-norm error estimates by relating the discrete solution to the interpolation of the exact solution.