Subject: Subspace Correction Methods for Linear Elasticity
Author: Dipl.-Ing. Erwin Karer
Supervision: Prov.-Doz. Dr. Johannes Kraus
External referee: Prof. Dr. Ludmil Zikatanov (The Pennsilvania State University)

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 subspaces 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. There are two basic approaches to set up a variational framework for such models. On one hand there is a mixed formulation resulting in indefinite linear systems of algebraic equations for the discrete solution. On the other hand, there is a formulation in primal variables, which gives rise to a symmetric positive (semi)-definite discrete problem.

First, we consider the standard discretization of the linear elasticity equations by means of continuous piecewise linear finite elements. This discretization suffers from volume locking as the material becomes nearly incompressible. We first consider the case in which such low order conforming methods provide sufficiently accurate approximation to the displacement field. 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 of the elasticity problem. 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. The major contribution is devising a measure for the nodal dependence which guides the generation of the edge matrices, which are the basic building blocks of this method. A natural measure is the abstract angle between the two subspaces spanned by the basis functions corresponding to the vertices forming an edge in our finite element partition. Another original contribution of this work is a two-level convergence analysis of the method. The presented numerical results cover also problems with jumps in the Young's modulus of elasticity and orthotropic materials, like wood or cancellous bone.

In a second part, we investigate the equations of elasticity in primal variables for nearly incompressible materials, like rubber. For such materials, i.e., when the first Lamé parameter tends to infinity this problem becomes ill-posed and the resulting discrete problem is nearly singular.

Due to the locking of approximations using conforming, low order polynomial spaces, to obtain any meaningful approximation to the displacement field, one has to use finite element spaces of at least order four (or even higher). Alternatively, and this is what we are studying here, one can consider stable nonconforming finite element discretizations based on reduced integration. One main question which then arises, and 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 introduce 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 which is a convex combination of a multiplicative preconditioner (based on the subspace splitting we mentioned above) plus a solution of a system equivalent to vector Laplace equation (for which efficient methods exist). We present a pool of numerical tests confirming the uniform convergence.

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