# Subspace correction method for a locking-free finite element approximation of the linear elasticity equations

## MSc Erwin Karer

**March 29, 2011, 2 p.m. HS 14**

In this talk we consider a finite element discretization of the equations of linear elasticity, introduced in [Falk, Math. of Comp, 1991] , which is stable in the case of nearly incompressible materials. This discretization does not suffer from so-called locking effects as they are observed when using standard low(est) order conforming methods for the pure displacement formulation. In case of pure traction boundary conditions optimal order error estimates are available based on an appropriate discrete version of Korn's second inequality.

The focus of this work is on the construction of a uniform solver for the linear systems arising from this discretization scheme. We introduce a successive subspace correction method with two subspaces in order to solve the problem. The first subspace is the space of divergence-free functions. On that space, the original bilinear form is independent of the LamÃ© constant $\lambda$. The second sub-space is related to the space of Raviart-Thomas functions. With robust solvers on both subspaces we can efficiently solve the original problem.

We discuss the details of the construction, derive spectral equivalence results and present numerical experiments.