Symmetry-Free, p-Robust Equilibrated Error Indication for the hp-Version of the FEM in Nearly Incompressible Linear Elasticity

We consider the extension of the p-robust equilibrated error estimator due to Braess, Pillwein and Schöberl to linear elasticity. We derive a formulation where the local mixed auxiliary problems do not require symmetry of the stresses. The resulting error estimator is p-robust, and the reliability estimate is also robust in the incompressible limit if quadratics are included in the approximation space. Extensions to other systems of linear second-order partial differential equations are discussed. Numerical simulations show only moderate deterioration of the effectivity index for a Poisson ratio close to .

A Reduced Local Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

A reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity is proposed in this paper, which is locking-free. RLDG method can be formally regarded as a special case of LDG method withC11=0. However, RLDG method is actually not covered by LDG method, whereC11must be chosen to be positive to ensure the stability of LDG method. RLDG method can also be considered as the localization of some symmetric nonconforming mixed finite element method. The implementation of RLDG method is discussed. By introducing a lifting operator as LDG method, RLDG method can be rewritten as primal formulation with unknown displacement only. Next, we obtain that the convergence rates of the approximation to stress tensor in energy norm and displacement inL2-norm areO(hk)andO(hk+1), respectively, which are both uniform with respect toλ. Moreover, we obtain aH(div)-conforming displacement by projecting the displacement and corresponding numerical trace of RLDG method into the Raviart-Thomas element space. And then we analyze the error estimates of this postprocessed displacement inH(div)-seminorm andL2-norm, which are also uniform with respect toλ. Finally, some numerical results are shown to demonstrate the theoretical results.