# On the strength of nodal dependence in AMG for problems in linear elasticity

## MSc Erwin Karer

**June 3, 2008, 3:30 p.m. T 1010**

In the classical AMG framework an important task is the determination of the strength of connectivity between two so-called “algebraic vertices”. Thereby, in the classical sense, an algebraic vertex is identiﬁed by a single degree of freedom. These methods work well for problems stemming from scalar PDEs, but they often do not cope with vector-ﬁeld problems.

Mainly we are interested in problems stemming from ﬁnite element discretizations of PDEs. Preliminary the focus lies on problems in linear elasticity. Thereby, as it has been shown earlier (by J. Kraus in 2006) the computation of “edge matrices” provides a good starting point for standard coarsening and local energy minimizing interpolation. In our procedure the measure for the strength of connectivity is deﬁned via the CBS constant associated with the angle between the two subspaces spanned by the basis functions corresponding to the respective algebraic vertices. This can be seen as a generalization of a commonly used approach for scalar problems to vector-ﬁeld problems.

An edge matrix is a small-sized local matrix that represents the coupling of two algebraic vertices. The general construction is such that the global assembly of the edge matrices approximates the stiﬀness

matrix.

The talk will ﬁrst introduce some basic concepts of classical algebraic multigrid methods, such as coarsening and interpolation. Afterwards, we will concentrate on the method based on the edge matrix

concept.