p and hp Spectral Element Methods for Elliptic Boundary Layer Problems

TL;DR

This paper introduces p and hp spectral element methods for one-dimensional elliptic boundary layer problems, providing stability analysis, residual minimization schemes, and demonstrating uniform error estimates and convergence rates through numerical experiments.

Contribution

It develops novel p and hp spectral element methods with proven stability and uniform error estimates for elliptic boundary layer problems, including convergence rates and numerical validation.

Findings

Robust uniform error estimates independent of boundary layer parameters.

Convergence rate of O(sqrt(log W)/W) for p-version methods.

Exponential convergence rate O(e^(-W/logW)) for hp-version methods.

Abstract

In this article, we propose p and hp least-squares spectral element methods for one-dimensional elliptic boundary layer problems. Stability estimates are derived and we design numerical schemes based on minimizing the residuals in the sense of least-squares in appropriate Sobolev norms. We prove parameter robust uniform error estimates i.e. error in the approximation is independent of the boundary layer parameter. For the p-version we prove a robust uniform convergence rate of O(sqrt(log W)/W), where W denotes the polynomial order used in approximation and for the hp-version the convergence rate is shown to be O(e^(-W/logW)). Numerical results are presented for a number of model elliptic boundary layer problems confirming the theoretical estimates and uniform convergence results for the p and hp versions.