A matrix-dependent transfer multigrid method for strongly variable viscosity infinite Prandtl number thermal convection

We apply a two-dimensional Cartesian finite element treatment to investigat e infinite Prandtl number thermal convection with temperature, strain rate and yield stress dependent rheology using parameters in the range estimated for the mantles of the terrestrial planets. To handle the strong viscosity variations that arise from such nonlinear rheology in solving the momentum equation, we exploit a multigrid method based on matrix-dependent intergri d transfer and the Galerkin coarse grid approximation. We observe that the matrix-dependent transfer algorithm provides an exceptionally robust and ef ficient means for solving convection problems with extreme viscosity gradie nts. Our algorithm displays a convergence rate per multigrid cycle about fi ve times better than what other published methods (e.g., CITCOM of Moresi a nd Solomatov, 1995) offer for cases with similar extreme viscosity variatio n. The algorithm is explained in detail in this paper. When this method is applied to problems with temperature and strain rate de pendent rheologies, we obtain strongly time dependent solutions characteriz ed by episodic avalanching of cold material from the upper boundary layer t o the bottom of the convecting domain for a significantly broad range of pa rameter values. In particular, we observe this behavior for the relatively simple case of temperature dependent Newtonian rheology with a plastic yiel d stress. The intensity and temporal character of the episodic behavior dep ends sensitively on the yield stress value. The regions most strongly affec ted by the yield stress are thickened portions of the cord upper boundary l ayer which can suddenly become unstable and form downgoing diapirs. These c omputational results suggest that the finite yield properties of silicate r ocks must play a vitally important role in planetary mantle dynamics. Altho ugh our example calculations were selected mainly to illustrate the power o f our multigrid method, they suggest that many possible exotic behaviors in planetary mantles have yet to be discovered.