Simon Becher

Simon Becher

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.

Simon Becher

We consider a singularly perturbed semilinear boundary value problem of a general form that allows various types of turning points. A solution decomposition is derived that separates the potential exponential boundary layer terms. The problem is discretized using higher order finite elements on suitable constructed layer-adapted meshes. Finally, error estimates uniform with respect to the singular perturbation parameter $\varepsilon$ are proven in the energy norm.