Uniform block diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes problem and linear elasticity

We propose a uniform block diagonal preconditioner for condensed H(div)-conforming HDG schemes for parameter-dependent saddle point problems including the generalized Stokes problem and the linear elasticity. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique [49]. A robust preconditioner spectrally equivalent to the Schur complement on the element-wise constant pressure space is also constructed. Finally, numerical results of the generalized Stokes and the steady linear elasticity equations verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.