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We consider heat conduction models with phase change in heterogeneous materials. We are motivated by important applications including heat conduction in permafrost, phase change materials (PCM), and human tissue. We focus on the mathematical and computational challenges associated with the nonlinear and discontinuous character of constitutive relationships related to the presence of free boundaries and material interfaces. We propose a monolithic discretization framework based on lowest order mixed finite elements on rectangular grids well known for its conservative properties. We implement this scheme which we call P0-P0 as cell centered finite differences, and combine with a fully implicit time stepping scheme. We show that our algorithm is robust and compares well to piecewise linear approaches. While various basic theoretical properties of the algorithms are well known, we prove several results for the new heterogeneous framework, and point out challenges and open questions; these include the approximability of fluxes by piecewise continuous linears, while the true flux features a jump. We simulate a variety of scenarios of interest.