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The strong dependence of the rheology of a fluid on temperature has a great impact on the style of thermally driven convection. When the viscosity contrast is sufficiently large, the viscosity of the coldest fluid at the top of a bottom-heated box is so high that this fluid layer becomes very stiff and a so-called cold “stagnant lid” develops on top of a hot convecting layer. Studying this style of convection is relevant for planetary mantles since the rheology of mantle material is likely to be very strongly temperature dependent. In this paper, the Rayleigh number dependence of stagnant-lid convection with a viscosity contrast of 106 is studied numerically in two and three dimensions in wide Cartesian domains. Like in constant-viscosity cases, the convection in the layer underneath the stagnant lid undergoes the typical transition from steady to time-dependent with the thinning of plumes and with the appearance of boundary layer instabilities as the Rayleigh number increases. A stagnant-lid style of convection was obtained in 2D and 3D for all supercritical Rayleigh numbers considered and the interior temperature appeared not to depend on the Rayleigh number. We have compared our results with other theoretical and numerical results and we found a close agreement.

The subject of this paper belongs to the field of numerical solution of time-dependent partial differential equations. Attention is focussed on parabolic problems in two space dimensions the solutions of which possess sharp moving transitions in space and time, such as steep moving fronts and emerging and disappearing layers. For such problems, a space grid held fixed throughout the entire calculation can be computationally inefficient, since, to afford an accurate approximation, such a grid would easily have to contain a very large number of nodes. We consider a so-called static-regridding method that adapts the space grid using nested, locally uniform grids. The notion of locally uniform grid refinement is an example of "domain decomposition", the general idea of which is to decompose the original physical or computational domain into smaller subdomains and to solve the original problem on these subdomains. Hence, our computational subdomains are nested, locally uniform space grids with nonphysical boundaries which are generated up to a level of refinement good enough to resolve the anticipated fine scale structures. This way a fine grid covering the entire physical domain can be avoided. We discuss several aspects occurring in a static-regridding algorithm like ours, such as the data structure, the regridding strategy and the determination of initial and boundary conditions at coarse-fine grid interfaces. We also present two numerical examples to demonstrate the performance of the method. The first example is hypothetical and is used to illustrate the convergence behaviour of the method. The second example originates from practice and describes a model combustion process.