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We propose a uniform block-diagonal preconditioner for condensed$H$(div)-conforming HDG schemes for parameter-dependent saddle point problems,including the generalized Stokes equations and the linear elasticity equations.An optimal preconditioner is obtained for the stiffness matrix on the globalvelocity/displacement space via the auxiliary space preconditioning (ASP)technique \cite{Xu96}. A spectrally equivalent approximation to the Schurcomplement on the element-wise constant pressure space is also constructed, andan explicit computable exact inverse is obtained via the Woodbury matrixidentity. Finally, the numerical results verify the robustness of our proposedpreconditioner with respect to model parameters and mesh size.

ima journal of numerical analysis

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