This dissertation is dedicated to the selected mathematical and computational challenges for nonlinear partial differential equation models of heat conduction. In particular, we illustrate the mathematical structure and develop robust computational techniques for a class of problems with generalized constitutive laws.

The models we consider describe heat conduction with phase transitions in partially frozen to unfrozen soils for which the generalized constitutive laws may be expressed as multi-valued graphs. We show that these generalized constitutive laws can be framed in a single-valued monotone relationship in an appropriate product space.

The solutions to these models may feature limited regularity or even discontinuities across the phase change interface and must be posed in the sense of distributions; we develop these formulations in detail. This framework also leads naturally to a new family of analytical solutions. However, the approximation of these solutions present additional computational challenges due to this low regularity, motivating the need for the development of robust nonlinear iterative solvers. In this work, we propose and evaluate an accelerated iterative computational algorithm that improves the convergence of schemes compared to those from the mathematical literature. Finally, we test these solvers with uncertain data through non-intrusive uncertainty quantification techniques which aim to provide robust insight when modeling with realistic data.