A Multigrid Preconditioner for the Mixed Formulation of Linear Plane Elasticity

In this paper, we develop a multigrid preconditioner for the discrete system of linear equations that results from the mixed formulation of the linear plane elasticity problem using the Arnold-Winther elements. This, in turn, can be reduced to the problem of finding a multigrid preconditioner for the form $(\cdot,\cdot)+(\vdiv\cdot,\vdiv\cdot)$ in the symmetric matrix space resulting from Arnold-Winther elements. Since the form is not uniformly elliptic, a Helmholtz-type decomposition is essential. The Arnold-Winther finite element space gives rise to nonnested multilevel spaces adding difficulty to the analysis. We prove that for the variable V-cycle multigrid preconditioner, the condition number of the preconditioned system is independent of the number of levels. The results of numerical experiments are also presented.

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