An eigen-based high-order expansion basis for structured spectral elements X. Zheng, S. Dong ⇑ Center for Computational & Applied Mathematics, Department of Mathematics, Purdue University, United States article info Article history: Received 9 November 2010 Received in revised form 3 August 2011 Accepted 11 August 2011 Available online 26 August 2011 Keywords: Expansion basis High order methods Spectral element Simultaneous diagonalization abstract We present an eigen-based high-order expansion basis for the spectral element approach with structured elements. The new basis exhibits a numerical efficiency significantly supe- rior, in terms of the conditioning of coefficient matrices and the number of iterations to convergence for the conjugate gradient solver, to the commonly-used Jacobi polynomial- based expansion basis. This basis results in extremely sparse mass matrices, and it is very amenable to the diagonal preconditioning. Ample numerical experiments demonstrate that with the new basis and a simple diagonal preconditioner the number of conjugate gradient iterations to convergence has essentially no dependence or only a very weak dependence on the element order. The expansion bases are constructed by a tensor product of a set of special one-dimensional (1D) basis functions. The 1D interior modes are constructed such that the interior mass and stiffness matrices are simultaneously diagonal and have identi- cal condition numbers. The 1D vertex modes are constructed to be orthogonal to all the interior modes. The performance of the new basis has been investigated and compared with other expansion bases. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction In high-order approaches with spectral elements or p-finite elements, the computational domain is first partitioned using a number of elements, much like in low-order finite element methods. Within each element, however, an expansion of the field variables will further be performed, usually in terms of a set of high-order polynomial bases [21,28,17] or rational poly- nomial bases [24]. The selection of the high-order expansion basis functions within the elements directly influences the con- ditioning and sparsity of the resultant system matrix after discretization [4,14], which in turn influences the number of iterations to convergence in iterative solvers. The expansion basis therefore intimately influences the numerical efficiency and performance of the high-order methods. An ideal expansion basis would be expected to yield a linear algebraic system with a well-conditioned sparse matrix that can be solved using iterative solvers as efficiently as possible. One of the earliest set of high-order basis functions aiming to improve the matrix sparsity and conditioning utilizes the integrals of Legendre polynomials [28]. This basis produces a diagonal block in the elemental stiffness matrix involving the interior modes. However, for unstructured elements it leads to an exponential increase in the condition number of the sys- tem matrix with respect to the element order [1]. Carnevali et al. [7] employed an orthogonalization process in the basis con- struction for triangles and tetrahedra, which resulted in local matrices better-conditioned than those of [28]. The growth of the condition number with respect to the element order is however still exponential. 0021-9991/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2011.08.009 ⇑ Corresponding author. E-mail address: sdong@math.purdue.edu (S. Dong). Journal of Computational Physics 230 (2011) 8573–8602 Contents lists available at SciVerse ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp