In the simulation of realistic solid mechanical problems, linear
equation systems with hundreds of million unknowns can arise. For the
efficient solution of such systems, parallel multilevel methods are
mandatory that are able to exploit the capabilities of modern hardware
technologies. The finite element and solution toolbox FEAST, which is
designed to solve scalar equations, pursues exactly this goal. It
combines hardware-oriented implementation techniques with a multilevel
domain decomposition method called ScaRC that achieves high numerical
and parallel efficiency. In this thesis a concept is developed to
solve multivariate elasticity problems based on the FEAST library. The
general strategy is to reduce the solution of multivariate problems to
the solution of a series of scalar problems. This approach facilitates
a strict separation of `low level` scalar kernel functionalities (in
the form of the FEAST library) and `high level` multivariate
application code (in the form of the elasticity problem), which is
very attractive from a software-engineering point of view: All efforts
to improve hardware-efficiency and adaptations to future technology
trends can be restricted to scalar operations, and the multivariate
application automatically benefits from these enhancements. In the
first part of the thesis, substantial improvements of the scalar ScaRC
solvers are developed, which are then used as essential building
blocks for the efficient solution of multivariate elasticity problems.
Extensive numerical studies demonstrate how the efficiency of the
scalar FEAST library transfers to the multivariate solution process.
The solver strategy is then applied to treat nonlinear problems of
finite deformation elasticity. A line-search method is used to
significantly increase the robustness of the Newton-Raphson method,
and different strategies are compared how to choose the accuracy of
the linear system solves within the nonlinear iteration. In order to
treat the important class of (nearly) incompressible material, a mixed
displacement/pressure formulation is used which is discretised with
stabilised bilinear finite elements (Q1/Q1). An enhanced version of
the classical `pressure Poisson` stabilisation is presented which is
suitable for highly irregular meshes. Advantages and disadvantages of
the Q1/Q1 discretisation are discussed, especially in the context of
transient computations. Two solver classes for the resulting saddle
point systems are described and compared: on the one hand various
kinds of (accelerated) segregated solvers (Uzawa, pressure Schur
complement methods, block preconditioners), and on the other hand
coupled multigrid solvers with Vanka-smoothers. Efficient Schur
complement preconditioners, which are required for the former class,
are discussed for the static and the transient case. The main strategy
to reduce the solution of multivariate systems to the solution of
scalar systems is only applicable in the case of segregated methods.
It is shown that for the class of elasticity problems considered in
this thesis, segregated solvers are clearly superior to Vanka-type
solvers in terms of numerical and parallel efficiency.