# Auxiliary space preconditioner for almost incompressible elasticity problems

## MSc Erwin Karer

**Dec. 1, 2009, 2:30 p.m. P 215**

We consider problems in linear elasticity. Thereby we are mainly interested in almost incompressible materials. When dealing with the pure displacement formulation we suffer from locking effects when nodal finite elements up to the order 3 are used. Falk proposed an adapted formulation using order 1 elements, which is of optimal order in 2D.

We will consider this bilinear form. Further, we want to come up with a preconditioner for the arising linear system by exploiting the auxiliary space method. The auxiliary space method can be viewed as a two-level method with a smoother on the original space, which is determined by a biliniear form that is less expensive to invert. Moreover, we do have a coarse grid part which acts on less degrees of freedom.

The auxiliary space method aims in a block diagonal preconditioner on the product space of the nodal piecewise linear space and the coarse space. Therefore, we cannot apply the auxiliary space technique directly and thus we propose an equivalent biliniear form which we will then precondition.

This presentation results from joint work with J. Kraus and L. Zikatanov.