A semi-implicit finite element method (FEM) is presented for the two-dimensional computer simulation of solid-liquid phase change controlled by natural convection and conduction. The algorithm is based on a combination of (1) a projection method to uncouple velocity calculations from pressure calculations for incompressible fluid flow, (2) the backward Euler and explicit Adams-Bashforth schemes to effectively integrate diffusion and advection in time, and (3) an enthalpy-porosity approach to account for the latent heat effect on a fixed finite element grid. Credibility of the obtained numerical predictions is investigated through computational model verification and validation procedures. Commonly used benchmark problems are employed to verify the algorithm accuracy and performance. The natural convection of freezing pure water is studied experimentally through the use of sophisticated full-field acquisition experimental techniques. The measured velocity and temperature fields are compared with the pertinent calculations. The range of congruity of the experimental and numerical results is thoroughly studied, and potential reasons of some disparity in a local structure of the natural convection flow and in the interface shape are discussed.