On the Efficiency of the Spectral Element Method and Possible Speed-Up in Comparison to the Finite Element Method

  • Dornisch, Wolfgang (RPTU University Kaiserslautern-Landau)
  • Affan, Shadi (RPTU University Kaiserslautern-Landau)

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The term spectral element method commonly refers to a Bubnov–Galerkin finite element method based on Lagrange basis functions with unevenly spaced nodes defined by the roots of Legendre polynomials [1] in combination with Gauss—Lobatto-–Legendre quadrature. In this case, the location of nodes and integration points coincides. Thus, the evaluation of the basis functions can be replaced by the Kronecker-Delta symbol. This leads to a block-diagonal mass matrix, which is frequently made use of in the literature for transient analysis [2]. In contrast to that, the global stiffness matrix retains the same sparsity pattern as in the standard finite element method since the spatial derivatives of the basis functions do not possess the Kronecker property. However, the situation for the computation of the element stiffness matrix is different. As elaborated in [3], only the derivatives in the same row and column of the parameter space as the considered integration point are non-zero. By making use of this cross-pattern of nodes, a significant reduction of computational costs, which rises strongly with the order, has been reported in [3]. To the authors’ knowledge, the exploitation of this effect has neither been reported nor quantified in the literature, except in [3]. In this contribution we assess the speed-up of the cross-scheme in comparison to a standard assembly as required in the finite element method. We use the Julia-based finite element framework Ferrite.jl [4] and implement static condensation on element level, as proposed e.g. in [5]. Two-dimensional elasticity examples with known analytical solutions allow a proper evaluation of accuracy, computational costs and efficiency. We also compare the efficiency to standard first and second order elements in Ferrite.jl. The gain in efficiency will be discussed.