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

## MSc Erwin Karer

**March 29, 2011, 4 p.m. HS 14**

In this talk we consider a nite 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

suer from so-called locking eects 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 li-

near systems arising from this discretization scheme. We introduce a successive

subspace correction method with two subspaces in order to solve the problem.

The rst subspace is the space of divergence-free functions. On that space, the

original bilinear form is independent of the Lame constant . The second sub-

space is related to the space of Raviart-Thomas functions. With robust solvers

on both subspaces we can eciently solve the original problem.

We discuss the details of the construction, derive spectral equivalence re-

sults and present numerical experiments.