Local multigrid methods on hierarchical meshes

This note summarizes the results in [1, 3, 2] on the construction of parallel local multigrid methods. In addition, we present an extended formalization of the adaptive data structure in order to describe precisely the requirements for adaptive multigrid methods with local smoothing for general finite element discretizations. Based on this formalization algorithms can be defined, developed, and the correctness can be verified. This may serve as a short reference for the adaptive model with is implemented in the software system UG. Finally, we illustrate the behavior of adaptive local multigrid methods by various examples for model problems including curved boundaries, multigrid convergence for different local refinements for problems with singularities, and a comparison of local and global smoothing for hybrid mixed finite elements. 1 Linear algebra on local mesh hierarchies In the same way as we presented a model for a parallel linear algebra in [3], we formalized here an abstract model for linear algebra operations which are suitable for implementing local adaptive multigrid methods. In addition, we consider the requirements on the geometry representation in case of curved boundaries and for domains which cannot be resolved by the coarse mesh. Note that a formalization – as an intermediate step between developing and implementing a method – is not necessarily required for realizing a prototype for local multigrid methods for simple piecewise linear conforming discretizations on triangles. Here, we focus on a model which allows the parallel implementation of more advanced applications including mixed type meshes including hexahedra, prisms, pyramids, and tetrahedra, and conforming, nonconforming and mixed discretizations. Then, a precise model is absolutely required for developing, extending, debugging, and documenting the code.