A Uniform Convergent Petrov-Galerkin method for a Class of Turning Point Problems

TL;DR

This paper introduces a Petrov-Galerkin finite element method for one-dimensional turning point problems, providing theoretical convergence analysis and demonstrating efficiency through numerical experiments.

Contribution

The paper develops a novel PGFEM approach for turning point problems, including a priori estimates and convergence proofs, with validation via numerical results.

Findings

First-order convergence in $L^ abla$ and energy norms

Method effectively handles singularities in turning point problems

Numerical results confirm theoretical convergence rates

Abstract

In this paper, we propose a numerical method for turning point problems in one dimension based on Petrov-Galerkin finite element method (PGFEM). We first give a priori estimate for the turning point problem with a single boundary turning point. Then we use PGFEM to solve it, where test functions are the solutions to piecewise approximate dual problems. We prove that our method has a first-order convergence rate in both $L^\infty$ norm and an energy norm when we select the exact solutions to dual problems as test functions. Numerical results show that our scheme is efficient for turning point problems with different types of singularities, and the convergency coincides with our theoretical results.