FEM-Analysis on Graded Meshes for Turning Point Problems Exhibiting an Interior Layer

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.