A Reduced Local Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

Abstract EN

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 with C11=0.

However, RLDG method is actually not covered by LDG method, where C11 must 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 in L2-norm are O(hk) and O(hk+1), respectively, which are both uniform with respect to λ.

Moreover, we obtain a H(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 in H(div)-seminorm and L2-norm, which are also uniform with respect to λ.

Finally, some numerical results are shown to demonstrate the theoretical results.