Natural heat convection, mixed heat convection and heat transfer by conduction and convection with solidification of a ternary alloy are described by the finite volume method using a geometric multigrid approach. The objective of this paper is to analyze the effects of the multigrid technique on the accuracy and efficiency in describing convective heat transfer in closed and open cavities with and without liquid-solid phase changes of Newtonian and shear-thinning non-Newtonian fluids. It is found that the multigrid scheme reduces the computation time in natural convection in a square cavity from two times for Rayleigh number Ra = 10(5) up to seven times for Ra = 10(3), between 50% and 2.7 times for mixed convection with an inner solid when the Richardson number decreases from RI = 10 to 0.1 and the Reynolds number Re = 100, and 20% for natural convection/heat conduction in solidification of a ternary aluminum alloy (Ra = 10(4)) with a shear-thinning rheology and a power index equal to 0.5.