Finite elements on layer-adapted meshes for convection-diffusion problems and applications

• Applicant: Professor Dr. Hans-Görg Roos
• Subject Area: Mathematics
• Term: from 2010 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 166224747

The numerical solution of convection-diffusion problems in the convection dominating regime with finite elements leads to well known difhculties if a Galerkin technique is used on standard meshes. Therefore stabilization techniques were developed and even combined with layer-adapted meshes if it is necessary to resolve layer structures, see [43]. So far the numerical analysis of stabilized or non-stabilized finite element methods on layer adapted meshes is restricted to single convection-diffusion problems (for some first progress for systems, see [33]) with a simple solution structure: the solution is assumed to be the sum of a smooth component and layer terms; singularities are excluded. In the project proposed we concentrate our attention on more realistic situation and want to handle as well layers as singularities. We look for optimal meshes which are adapted to the singularities and to the existing layers. Moreover, as an application we want to study optimal control problems governed by convection diffusion problems in the first application period. Here new difficulties arise because the optimality conditions generate a system of convection-diffusion problems and so far not much is known on the layer structure of the solution of that system.

• DFG Programme: Research Grants