Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Analytical estimates

This is the second of two papers in which we motivate, introduce and analyze a new type of strategy for the stabilization of discontinuous Galerkin (DG) methods in nonlinear elasticity problems. The foremost goal behind it is to enhance the robustness of the method without deteriorating the accuracy of the resulting solutions. Its distinctive property is that for nonlinear elastic problems the stabilization term is solution dependent, and hence it is termed an adaptive stabilization mechanism. The key contribution of this paper is the construction of a stabilization strategy for which the method is perfectly stable, since the stabilization parameters can be explicitly computed as part of the numerical solution. This is accomplished through the main result of this paper, which consists of a theorem that provides lower bounds for the size of the stabilization parameters. Numerical examples confirm the guaranteed stability of the resulting method. However, they also show that the computed lower bounds overestimate the minimum amount of stabilization needed, negatively affecting the approximation properties of the method. These results underscore the fact that a better understanding of the stabilization mechanisms is needed in order to construct a method that is both robust and efficient.