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

TL;DR

This paper introduces a robust, uniform block-diagonal preconditioner for divergence-conforming HDG schemes applied to saddle point problems like the generalized Stokes and linear elasticity equations, improving solver efficiency.

Contribution

It develops an optimal, parameter-robust preconditioner using auxiliary space techniques and explicit inverses, enhancing the computational performance of HDG methods for saddle point problems.

Findings

Preconditioner is robust against model parameters.

Numerical results confirm effectiveness across mesh sizes.

Explicit inverse via Woodbury identity improves computational efficiency.

Abstract

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 global velocity/displacement space via the auxiliary space preconditioning (ASP) technique \cite{Xu96}. A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.