Robust multigrid methods for nearly incompressible elasticity using macro elements

Abstract

We present a mesh-independent and parameter-robust multigrid solver for the Scott-Vogelius discretisation of the nearly incompressible linear elasticity equations on meshes with a macro element structure. The discretisation achieves exact representation of the limiting divergence constraint at moderate polynomial degree. Both the relaxation and multigrid transfer operators exploit the macro structure for robustness and efficiency. For the relaxation, we use the existence of local Fortin operators on each macro cell to construct a local space decomposition with parameter-robust convergence. For the transfer, we construct a robust prolongation operator by performing small local solves over each coarse macro cell. The necessity of both components of the algorithm is confirmed by numerical experiments.

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Cited by in corpus (3)

• A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations
• An augmented Lagrangian preconditioner for implicitly-constituted non-Newtonian incompressible flow
• Finite element appoximation and augmented Lagrangian preconditioning for anisothermal implicitly-constituted non-Newtonian flow