A Least Squares Finite Element Method (LSFEM) formulation for the detection of unknown boundary conditions in steady heat conduction is presented. The method is capable of determining temperatures and heat fluxes in locations where such quantities are unknown provided such quantities are sufficiently overspecified in other locations. In several finite element and boundary element inverse implementations, the resulting system of equations becomes become rectangular if the number of overspecified conditions exceeds the number of unknown conditions. In the case of the finite element method, these rectangular matrices are sparse and can be difficult to solve efficiently. Often we must resort to the use of direct factorizations that require large amounts of core memory for realistic geometries. This difficulty has prevented the solution of large-scale inverse problems that require fine meshes to resolve complex 3-D geometries and material interfaces. In addition, the Galerkin finite element method (GFEM) does not provide the same level of accuracy for both temperature and heat flux. In this paper, an alternative finite element approach based on LSFEM will be shown. The LSFEM formulation always results in a symmetric positivedefinite matrix that can be readily treated with standard sparse matrix solvers. In this approach, the differential equation is cast in first-order form so equal order basis functions can be used for both temperature and heat flux. Enforcement of the overspecified boundary conditions is straightforward in the proposed formulation. The methods allows for direct treatment of complex geometries composed of heterogeneous materials.
