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Solid state convection can take place in the rocky or icy mantles ofplanetary objects and these mantles can be surrounded above or below or both bymolten layers of similar composition. A flow toward the interface can proceedthrough it by changing phase. This behaviour is modeled by a boundary conditiontaking into account the competition between viscous stress in the solid, thatbuilds topography of the interface with a timescale $\tau_\eta$, and convectivetransfer of the latent heat in the liquid from places of the boundary wherefreezing occurs to places of melting, which acts to erase topography, with atimescale $\tau_\phi$. The ratio $\Phi=\tau_\phi/\tau_\eta$ controls whetherthe boundary condition is the classical non-penetrative one ($\Phi\rightarrow\infty$) or allows for a finite flow through the boundary (small $\Phi$). Westudy Rayleigh-B\'enard convection in a plane layer subject to this boundarycondition at either or both its boundaries using linear and weakly non-linearanalyses. When both boundaries are phase change interfaces with equal values of$\Phi$, a non-deforming translation mode is possible with a critical Rayleighnumber equal to $24\Phi$. At small values of $\Phi$, this mode competes with aweakly deforming mode having a slightly lower critical Rayleigh number and avery long wavelength, $\lambda_c\sim 8\sqrt{2}\pi/ 3\sqrt{\Phi}$. Both modeslead to very efficient heat transfer, as expressed by the relationship betweenthe Nusselt and Rayleigh numbers. When only one boundary is subject to a phasechange condition, the critical Rayleigh number is $\Ray_c=153$ and the criticalwavelength is $\lambda_c=5$. The Nusselt number increases about twice fasterwith Rayleigh number than in the classical case with non-penetrative conditionswhen the bottom boundary is a phase change interface.

Rayleigh-B\'enard convection in a creeping solid with melting and freezing at either or both its horizontal boundaries