The direct measurement of temperatures and heat fluxes may be difficult or impossible on boundaries that are obstructed, such as internal cavities, or exposed to harsh environmental conditions that would destroy the thermal sensors. In such circumstances, one may inversely determine the temperature and heat fluxes on these unknown boundaries by using over-specified conditions on boundaries where such information can be readily collected. This assumes the geometry and material properties of the domain are known. Algorithms for solving these problems, such those based on finite difference, finite element, and boundary element, are well known for the case where measured boundary conditions are not a function of time. In this work, I demonstrate an inverse finite element method that effectively solves this inverse heat conduction problem using over-specified temperatures and heat fluxes that are time varying. The material properties may highly heterogeneous and non-linear. A boundary regularization method in space and time is used to stabilize the method for cases involving errors in temperature and heat flux measurements. Several three dimensional examples are given using simulated measurements with and without measurement errors, to demonstrate the accuracy of the method.

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