Remarks on boundary layers in singularly perturbed Caputo fractional boundary value problems

Almost nothing is known about the layer structure of solutions to singularly perturbed Caputo fractional boundary value problems. We discuss simple convection-diffusion and reaction-diffusion problems

Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

Error estimates for linear finite elements on Bakhvalov-type meshes

summary:For convection-diffusion problems with exponential layers, optimal error estimates for linear finite elements on Shishkin-type meshes are known. We present the first optimal convergence result in an energy norm for a Bakhvalov-type mesh

Numerical analysis of a singularly perturbed convection diffusion problem with shift in space

We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.Comment: 17 pages, 1 figur

Superconvergence for convection-diffusion problems with low regularity

summary:The finite element method is applied to a convection-diffusion problem posed on the unite square using a tensor product mesh and bilinear elements. The usual proofs that establish superconvergence for this setting involve a rather high regularity of the exact solution - typically \(u \in H^3(\Omega)\), which in many cases cannot be taken for granted. In this paper we derive superconvergence results where the right hand side of our a priori estimate no longer depends on the \(H^3\) norm but merely requires finiteness of some weaker functional measuring the regularity. Moreover, we consider the streamline diffusion stabilization method and how superconvergence is affected by the regularity of the solution. Finally, numerical experiments for both discretizations support and illustrate the theoretical results

Moderne Methoden zur Behandlung von Konvektions-Diffusions-Gleichungen

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