Uniform error estimates for general semilinear turning point problems on layer-adapted meshes

TL;DR

This paper develops uniform error estimates for higher order finite element solutions of general semilinear boundary value problems with turning points, using layer-adapted meshes to handle boundary layers effectively.

Contribution

It introduces a solution decomposition and proves uniform error estimates in the energy norm for semilinear turning point problems discretized with higher order finite elements on layer-adapted meshes.

Findings

Error estimates are uniform with respect to the perturbation parameter.

Higher order finite element methods effectively approximate solutions with boundary layers.

The approach handles various types of turning points in singularly perturbed problems.

Abstract

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.