Finite element analysis involves the solution of linear systems described by large size sparse matrices. Iterative Krylov methods are well suited for such type of problems. These methods require linear algebra operations, including sparse matrix-vector multiplication which can be computationally expensive for large size matrices. In this paper, we present the best way to perform these operations, in double precision, on Graphics Processing Unit (GPU). Several linear algebra libraries are considered and compared to our proper implementation. These libraries and our proper implementation are then integrated within an iterative Krylov method on the GPU. Numerical experiments done on a set of finite element matrices are presented and illustrate the performance, robustness and accuracy of our proper implementation compared to the existing libraries and its suitability for finite element analysis. Dynamic tuning of the gridification, upon the GPU architecture and the finite element matrix characteristics, is finally applied to faster the sparse matrix-vector multiplication operation.

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Direct and iterative methods are often used to solve linear systems in engineering. The matrices involved can be large, which leads to heavy computations on the central processing unit. A graphics processing unit can be used to accelerate these computations. In this paper, we propose a new library, named Alinea, for advanced linear algebra. This library is implemented in C++, CUDA and OpenCL. It includes several linear algebra operations and numerous algorithms for solving linear systems. For both central processing unit and graphic processing unit devices, there are different matrix storage formats, and real and complex arithmetics in single- and double-precision. The CUDA version includes a self-tuning of the grid, i.e. threading distribution, depending upon the hardware configuration and the size of the problems. Numerical experiments and comparison with existing libraries illustrates the efficiency, accuracy and robustness of the proposed library.