FEM-analysis on graded meshes for turning point problems exhibiting an interior layer
Provides rigorous convergence guarantees for numerical methods on layer-adapted meshes for interior layer problems, an incremental advance in numerical analysis.
The authors prove ε-uniform error estimates in the energy norm for higher-order finite elements on graded meshes for singularly perturbed problems with interior layers, and achieve optimal L² convergence for linear elements.
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.