Catalcg ing-in-Publ ication Data applied forDie Deutsche Bibliothek -CIP-Einheitsaufnahme Roos, Hans-Görg: Nume rica\ meth ods for singularly perturbed diff erential equat io ns : con vecti on diffus ion and flow pr obl ems / H .-G.

A reaction-diffusion problem with a Caputo time derivative of order α ∈ (0, 1) is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time t = 0, and sharp pointwise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

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In convection-diffusion problems, transport processes dominate while diffusion effects are confined to a relatively small part of the domain. This state of affairs means that one cannot rely on the formal ellipticity of the differential operator to ensure the convergence of standard numerical algorithms. Thus new ideas and approaches are required.The survey begins by examining the asymptotic nature of solutions to stationary convection-diffusion problems. This provides a suitable framework for the understanding of these solutions and the difficulties that numerical techniques will face. Various numerical methods expressly designed for convection-diffusion problems are then presented and extensively discussed. These include finite difference and finite element methods and the use of special meshes.

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