A Uniform Convergent Petrov-Galerkin Method for a Class of Turning Point Problems (Journal of Computational Mathematics, 2024)
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 prioriestimate for the turning point problem with a single boundary turning point. Then we usePGFEM to solve it, where test functions are the solutions to piecewise approximate dualproblems. We prove that our method has a first-order convergence rate in both $L^∞_h$ normand a discrete energy norm when we select the exact solutions to dual problems as testfunctions. Numerical results show that our scheme is efficient for turning point problemswith different types of singularities, and the convergency coincides with our theoreticalresults.
Keywords: Turning point problem; Petrov-Galerkin finite element method; Uniform convergency.