Efficient assembly for high order unstructured FEM meshes

The Finite Element Method (FEM) is a common numerical technique used for solving Partial Differential Equations (PDEs) on complex domain geometries. Large and unstructured FEM meshes are used to represent the computation domains which makes an efficient mapping of the Finite Element Method onto FPGAs particularly challenging. The focus of this paper is on assembly mapping, a key kernel of FEM, which induces the sparse and unstructured nature of the problem. We translate FEM vector assembly mapping into data access scheduling to perform vector assembly directly on the FPGA, as part of the hardware pipeline. We show how to efficiently partition the problem into dense and sparse sub-problems which map well onto FPGAs. The proposed approach, implemented on a single FPGA could outperform highly optimised FEM software running on two Xeon E5-2640 processors.