Spectral methods on triangles and other domains
Journal of Scientific Computing
From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex
SIAM Journal on Numerical Analysis
Basis Functions for Triangular and Quadrilateral High-Order Elements
SIAM Journal on Scientific Computing
Journal of Computational Physics
An Algorithm for Computing Fekete Points in the Triangle
SIAM Journal on Numerical Analysis
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Fourierization of the Legendre--Galerkin method and a new space--time spectral method
Applied Numerical Mathematics
A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems
Computers and Structures
A Triangular Spectral Element Method Using Fully Tensorial Rational Basis Functions
SIAM Journal on Numerical Analysis
An unconditionally stable rotational velocity-correction scheme for incompressible flows
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.46 |
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 superior, 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 identical 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.