Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form

Abstract: A block diagonal preconditioner with the minimal residual method and a block triangular preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods of linear elasticity. They are based on a new stability result of the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while the displacement it is spectral equivalent to Schur complement. A fast auxiliary space preconditioner based on the $H^1$ conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lamé constant and the mesh-size. Numerical examples are presented to support theoretical results. As byproducts, a new stabilized low order mixed finite element method is proposed and analyzed and superconvergence results of Hu-Zhang element are obtained.

• The paper introduces a fast auxiliary space preconditioner that leverages an H1-conforming subproblem to achieve parameter-robustness in mixed formulations of linear elasticity.
• It presents mesh-dependent norms and block preconditioners that ensure uniform convergence and stable discretization even in nearly incompressible regimes.
• Numerical experiments demonstrate constant iteration counts up to large DOFs, while the framework generalizes to other complex coupled PDE systems.

Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form

Problem Formulation and Motivation

The paper addresses iterative solution strategies for the Hu-Zhang family of mixed finite element methods (MFEMs) applied to linear elasticity, discretized via an $H(\text{div})$-conforming and symmetric-stress formulation. These lead to large-scale saddle point systems where the (1,1) block is a compliance-weighted mass matrix and the off-diagonal block encodes the discretized divergence operator. Robust solvers for these mixed systems are challenging to design, especially in the nearly incompressible limit (high Lamé parameter $\lambda$), and it is essential to achieve preconditioners whose performance is robust with respect to both mesh size $h$ and material parameters.

Mesh-Dependent Norms and Stability Theory

A primary theoretical contribution is the derivation of novel mesh-dependent norms under which the mixed linear elasticity MFEM system is stable and admits operator preconditioners whose performance is uniform in both $h$ and $\lambda$. For the stress space, the norm is equivalent to the standard mass-matrix weighted $L^2$ norm, and for the displacement, the norm is equivalent to the mass-based Schur complement norm, avoiding $H^2$ regularity assumptions. The authors rigorously establish discrete inf-sup conditions and robust coercivity for the MFEM system in these mesh-dependent norms, including detailed analysis for both high-order ($k \geq n+1$) and stabilized low-order ($1 \leq k \leq n$) MFEMs. Notably, the proofs apply without requiring full regularity of the domain or the discrete kernel structure, which removes typical technical barriers in analogous works.

Preconditioning Strategy

Leveraging the mesh-dependent stability, two classes of block preconditioners are constructed:

• Block Diagonal Preconditioner: Uses the compliance-mass inverse and Schur complement inverse blocks. This structure enables application of symmetric Krylov methods (e.g., MINRES).
• Block Triangular Preconditioner: Based on a block LU factorization with diagonally scaled mass and a modified Schur complement, suitable for nonsymmetric solvers (e.g., GMRES).

The main computational challenge is efficient application of the Schur complement inverse $S_h = B_h M_h^{-1} B_h^T + C_h$, especially since this subproblem is indefinite, large-scale, and does not in general admit a direct multilevel hierarchy.

Auxiliary Space Preconditioner

A central innovation is the development, analysis, and implementation of an auxiliary space preconditioning (ASP) framework for the Schur complement. The "auxiliary space," in the sense of Xu [Xu, Comput. 1996], is chosen to be the $H^1$-conforming, piecewise linear finite element space for the vector-valued Poisson (or equivalently, linear elasticity with $\lambda=0$) subproblem on the same mesh. This reduction allows for the ASP to leverage geometric or algebraic multigrid schemes that are highly optimized for $H^1$-problems, thereby translating the original inf-sup constrained, nonnested, and potentially ill-conditioned problem into one that is feasible for scalable multigrid solvers. Theoretical results guarantee spectral equivalence between the Schur complement and the ASP, yielding parameter-independent bounds on the condition number of the preconditioned system.

Strong Numerical Results

The empirical evaluation includes large-scale 2D elasticity problems, discretized via Hu-Zhang elements of order $k=1,2,3$. The ASP-enabled block preconditioners lead to uniformly bounded iteration numbers with respect to both mesh refinement and the Lamé parameter; e.g., for the triangular block preconditioner, the GMRES iteration count remains approximately constant (20–59 steps) up to over 400,000 DOFs and for $\lambda$ up to $+\infty$, consistently achieving residual reduction to $10^{-8}$. These results outperform previous approaches and demonstrate the uniformity of the preconditioner, in line with theoretical predictions.

Additional Contributions

The analysis results in several important byproducts:

• A rigorous superconvergent postprocessing scheme for displacement, leveraging the superconvergent auxiliary problem to yield higher accuracy with negligible additional cost.
• Development and analysis of a new stabilized low-order MFEM.
• The theoretical framework applies, with minimal adaptation, to a broad class of $H(\text{div})$-conforming symmetric stress elements beyond the Hu-Zhang family, including all cases where the stress-divergence operator is surjective onto the discrete displacement space.

Implications and Future Developments

This work provides a significant algorithmic advance for robust and efficient solvers for MFEM-discretized linear elasticity, particularly in settings where parameter-robustness and scalability are critical (e.g., high-contrast and nearly incompressible regimes). The auxiliary space reduction to $H^1$-type solvers opens the door for straightforward integration of highly optimized geometric and algebraic multigrid technologies. The theoretical framework and construction methodology are anticipated to generalize to mixed methods for other complex coupled PDE systems (e.g., poroelasticity, viscoelasticity) and to applications involving non-nested, hybridized, or nonconforming MFEMs.

Future work could investigate extensions to:

• Fully adaptive or anisotropic meshes
• Three-dimensional elasticity in complex geometries
• Nonlinear elasticity via Newton–Krylov–FASP methods

Conclusion

The paper introduces and comprehensively analyzes a fast auxiliary space preconditioner for linear elasticity problems discretized with mixed symmetric MFEMs, achieving robust performance independent of both discretization and physical parameters. The main contributions are the mesh-dependent stability analysis, the construction of block preconditioners, and the design of a scalable ASP exploiting $H^1$-conforming subproblems. Numerical results confirm the uniformity and efficiency of the approach, and the theoretical framework provides a foundation for future generalizations in computational PDEs.

Reference: "Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form" (1604.02568).