An algorithm for stabilizing hybridizable discontinuous Galerkin methods for nonlinear elasticity

It is now a well known fact that hybridizable discontinuous Galerkin for nonlinear elasticity may not converge to the exact solution if their inter-element jumps are not properly penalized or, equivalently, if the values of their stabilization function are not suitably chosen. For example, if their stabilization function is of order one, the method generates spurious oscillations in the region in which the elastic moduli tensor is indefinite. This phenomenon disappears when the values of the stabilization function are increased. Mixed methods display the same problem, as their stabilization function is identically zero. Here, we find explicit formulas for the lower bound of the values of the stabilization function which allow us to avoid this unpleasant phenomenon. Our numerical experiments show that, when we use polynomials of degree k>0 for the approximate gradient, first-order Piola-Kirchhoff and displacement, all the approximations converge with the optimal order, k+1. They also show that, if we increase the polynomial degree of the local spaces by one and update the values of the stabilization function only for the elements for which the elasticity tensor is indefinite, we obtain that all approximate solutions converge with order k+1 and that a local post processing of the displacement converges with order k+2.