This thesis presents the formulation and partial analysis of a class of discontinuous Galerkin methods for quasi-static nonlinear elasticity problems. In contrast with most discontinuous Galerkin (DG) methods, we begin with the formulation of a DG approximation to derivatives. By doing so a class of DG methods deriving from a one-field minimization principle naturally ensues. Not only is the resulting method always symmetric, but the implementation of a computationally efficient approach may be constructed in which repeatedly looping over element faces is avoided.

In this thesis we perform extensive numerical studies for one particular DG method. For problems in which low-order conforming approximations are prone to locking, the DG methods remains nearly locking free. This benefit was observed for problems prone to both shear and volumetric locking in the nonlinear regime. It was also found that for some problems the approximations of low-order conforming methods may be detrimentally affected by kinematic constraints imposed by the mesh, whereas a low-order DG approximation does not suffer from this pathology. For example, a study of beam buckling revealed that a mesh based constraint allowed a conforming method to only identify one buckling mode, whereas a DG method captured both buckling modes with the same mesh. For problems suffering from a kinematic constraint or locking the DG method displays a clear benefit over a low-order conforming method. For problems where low order conforming methods are known to converge with optimal order it was expected that a DG method would offer no additional benefits and would have a substantial increase in computational cost to achieve the same accuracy as that of a low-order conforming method. In a selected three dimensional problem this was indeed found to be the case. However, a surprising characteristic found in a two-dimensional example shows a case where a DG method offers a lower computational cost to achieve the same level of accuracy obtained from a low-order conforming method.

As with other nonconforming methods such as enhanced strain (ES), DG may display numerical instabilities when a conforming method does not. In this thesis we motivate, introduce and demonstrate a novel approach to stabilizing DC methods in nonlinear elasticity problems. The stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities, with the goal of better approximating the solution. Consequently, it is called an adaptive stabilization strategy. The need for such a strategy is motivated through two- and three-dimensional examples in nonlinear elasticity. The proposed scheme is simple to implement and compute. This thesis provides a theorem that estimates lower bounds for the size of the stabilization parameters needed to guarantee stability. Although it was not possible to obtain sharp lower bounds, the theorem proves the stabilization parameters are bounded, and thus the adaptive stabilization technique can always be used to stabilize the problem, a fact that had been only previously assumed. Since the estimated lower bounds are not sharp, the stabilization parameters may excessively penalize discontinuities resulting in locking for nearly incompressible materials. However, the adaptive method is tailored in such a way that the stabilization scheme automatically adapts to the problem. Numerical experiments show that the actual stabilization parameter needed to stabilize the problem is generally of order one. The benefits mentioned in the last paragraph were all obtained with the adaptive method.

In addition to DG, the ES method is another nonconforming method that suffers from numerical instabilities for nonlinear elasticity problems. This thesis examines the striking similarities between DG and ES for nonlinear elasticity problems. Due to the similarities, a stabilization technique is presented which uses the insight gained from DG and current stabilization techniques for ES. The analysis of the adaptive stabilization method for ES parallels that of the one presented for DG. (Abstract shortened by UMI.)