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The simulation of heat flow through heterogeneous material is important forthe design of structural and electronic components. Classical analyticalsolutions to the heat equation PDE are not known for many such domains, eventhose having simple geometries. The finite element method can provideapproximations to a weak form continuum solution, with increasing accuracy asthe number of degrees of freedom in the model increases. This comes at a costof increased memory usage and computation time; even when taking advantage ofsparse matrix techniques for the finite element system matrix. We summarizerecent approaches in solving problems in structural mechanics and steady stateheat conduction which do not require the explicit assembly of any systemmatrices, and adapt them to a method for solving the time-depended flow ofheat. These approaches are highly parallelizable, and can be performed ongraphical processing units (GPUs). Furthermore, they lend themselves to thesimulation of heterogeneous material, with a minimum of added complexity. Wepresent the mathematical framework of assembly-free FEM approaches, throughwhich we summarize the benefits of GPU computation. We discuss ourimplementation using the OpenCL computing framework, and show how it is furtheradapted for use on multiple GPUs. We compare the performance of single and dualGPUs implementations of our method with previous GPU computing strategies fromthe literature and a CPU sparse matrix approach. The utility of the novelmethod is demonstrated through the solution of a real-world coefficient inverseproblem that requires thousands of transient heat flow simulations, each ofwhich involves solving a 1 million degree of freedom linear system overhundreds of time steps.

Analysis of heterogeneous computing approaches to simulating heat transfer in heterogeneous material