Efficient high-order spectral element discretizations for building block operators of CFD

High-order methods gain more and more attention in computational fluid dynamics. Among these, spectral element methods and discontinuous Galerkin methods introduce element-wise approximations by means of a polynomial basis. This leads to a small number of operators consuming a large portion of the runtime of CFD applications. The present paper addresses tensor-product bases which are among the most frequent in these applications. Various implementations are developed and performance tests conducted for the interpolation operator, the Helmholtz operator, and the fast diagonalization operator. For each, up to 50% of the peak performance is attained, beating matrix-matrix multiplication for every polynomial degree relevant for simulations. This extremely high efficiency of the method developed is then demonstrated on a combustion problem with 1.72 · 109 mesh points.