Abstract

Mesh generation is an important part of the numerical solution of many PDEs. If the PDE has evolving structure on small scales then it is often essential that the mesh adapts to resolve these scales. One way of doing this is to move the mesh points into regions where greater resolution is needed. Such movement can be regarded as the action of a map from a regular mesh into an adapted mesh. Thus mesh generation can be studied in terms of the properties of this map and the differential equations that it satisfies.

In this talk I will look at a class of such maps derived from solutions of the (fully nonlinear) Monge Ampere equation. I will show that such maps can be generated easily and moreover the global regularity of the mesh can be understood in terms of the regularity of the solution of the Monge Ampere equation. In particular I will demonstrate that such meshes align themselves very well with underlying features of the solution and thus are effective in approximating these features.