Subspace Correction Methods for Linear Elasticity

MSc Erwin Karer

Dec. 19, 2011, 9 a.m. S2 416

The framework of subspace correction methods describes approaches to solve finite element discretizations of elliptic partial differential equations. Examples of efficient solution techniques are multigrid methods, domain decomposition methods, and auxiliary space methods.

As a first result, we derive the error norm of the method of successive subspace corrections in case of two sub-spaces using strengthened Cauchy-Bunyakovsky-Schwarz inequalities to estimate energy minimizing restrictions of the operator on subspaces.

Next, we focus on the system of elliptic partial differential equations modeling the stresses and displacements in linear elastic materials in primal variables and consider the standard discretization of the linear elasticity equations by means of continuous piecewise linear finite elements. It is well known that the classical algebraic multigrid (AMG) methods do not perform well on this problem without modifications.

We study one competitive AMG method for solving the symmetric positive definite system resulting from the discretization. In this method, the coarsening is based on so-called edge matrices, which allows to generalize the concept of strong and weak connections, as used in classical AMG, to “algebraic vertices” that accumulate the nodal degrees of freedom in case of vector-field problems. We devise a measure for the nodal dependence which guides the generation of the edge matrices, which are the basic building blocks of this method. Further, we present a two-level convergence analysis of the method and numerical results underlining the expected behavior.

In a second part, we investigate the equations of elasticity in primal variables for nearly incompressible materials, like rubber. For such materials the problem becomes ill-posed and the resulting discrete problem is nearly singular.

Alternatively, we consider a stable nonconforming finite element discretization based on reduced integration. One main question which we address here is how to construct a robust (uniform in the problem parameters, such as Lamé's first parameter) iterative solution method for the resulting system of linear algebraic equations. We intro-duce a specific space decomposition into two overlapping subspaces that serves as a basis for devising a uniformly convergent subspace correction algorithm. The first subspace consists of weakly divergence-free functions. The second subspace is the complementary space which we augment with a suitably chosen overlap by adding certain weakly divergence free components. We solve the two subproblems exactly. This subspace correction method gives rise to a preconditioner for the system operator. We additionally address how to solve the subproblems by efficient solution methods. A numerical test confirms the uniform convergence.