Parallel finite element technique using Gaussian belief propagation. The computational efficiency of Finite Element Methods (FEMs) on parallel architectures is severely limited by conventional sparse iterative solvers. Conventional solvers are based on a sequence of global algebraic operations that limits their parallel efficiency. Traditionally, sophisticated programming techniques tailored to specific CPU architectures are used to improve the poor performance of sparse algebraic kernels. The introduced FEM Multigrid Gaussian Belief Propagation (FMGaBP) algorithm is a novel technique that eliminates all global algebraic operations and sparse data-structures. The algorithm is based on reformulating the FEM into a distributed variational inference problem on graphical models. We present new formulations for FMGaBP, which enhance its computation and communication complexities. A Helmholtz problem is used to validate the FMGaBP formulation for 2D, 3D and higher FEM degrees. Implementation techniques for multicore architectures that exploit the parallel features of FMGaBP are presented showing speedups compared to open-source libraries, specifically deal.II and Trilinos. FMGaBP is also implemented on manycore architectures in this work; Speedups of 4.8X, 2.3X and 1.5X are achieved on an NVIDIA Tesla C2075 compared to the parallel CPU implementation of FMGaBP on dual-core, quad-core and 12-core CPUs respectively.

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