Basis function selection and preconditioning high degree finite element and spectral methods. (English) Zbl 0708.65105

Carey, G. F.; Barragy, E.

BIT 29, No. 4, 794-804 (1989).

The article is devoted to the interesting question of the choice of the basis in a given finite element space with the purpose of diminishing the condition number. This choice for the finite element method based on square high degree \(p\geq 1\) finite elements is suggested to make on the element level. The space of each finite element is the space of polynomials of degree p in each variable \(x_ 1\), \(x_ 2\). Variants of bases are considered which are obtained as products of the next one- dimensional polynomials: 1) Chebyshev, 2) Lagrange interpolation polynomials using Lobatto points, 3) Legendre, 4) integrated Legendre, 5) Lagrange interpolation polynomials using equispaced points.
A comparison of the bases is performed numerically for the Dirichlet problem \(-\Delta u+cu=f\) in the unit square on uniform square discretizations with \(p=1,2,...,8\) and \(p/h=24\), where h is the size of an element. The systems of algebraic equations are solved by a standard conjugate gradient method with Jacobi (diagonal) preconditioning.
The number of iterations needed to achieve an accuracy corresponding to the accuracy of approximation, condition numbers for element matrices and condition number estimates of assembled matrices extracted from iterations are compared.
The conclusion of the authors is that the choice of the basis has a significant effect on the condition number. Using the Lobatto points produces a dramatic improvement for the Lagrange basis. Other choices of orthogonal polynomials are also effective.

Reviewer: R.Lezius

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10	Iterative numerical methods for linear systems
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65F35	Numerical computation of matrix norms, conditioning, scaling
35J25	Boundary value problems for second-order elliptic equations

Keywords:

basis function selection; spectral methods; choice of the basis; finite element space; condition number; finite element method; Dirichlet problem; conjugate gradient method; preconditioning

References:

[1]	Babuska, I., Szabo, B. A., and Katz, I. N.,The p-Version of the Finite Element Method, SIAM J. Numer. Anal., 18, 3, pp. 515–454, June 1981. · Zbl 0487.65059
[2]	Barragy, E. and Carey, G. F.,EBE Block Preconditioners, CNA Report, Univ. Texas 1989 (in press).
[3]	Fried. I.,Conditions of finite element matrices generated from nonuniform meshes, AIAA Journal, 10, pp. 219–221, 1972. · Zbl 0242.65046
[4]	Fried, I.,Bounds on the spectral and maximum norms of the finite element stiffness, flexibility and mass matrices, Int. J. Solids & Structures, 9, pp. 1013–1034, 1973. · Zbl 0263.73049
[5]	Ronquist, E. M. and Patera, A. T.,Spectral element multigrid: formulation and numerical results. J. Scient. Comp., 2, 4, p. 389, 1987. · Zbl 0666.65055
[6]	Szabo, B.,Mesh design for the p-version of the finite element method, Comp. Meth. Appl. Mech. & Eng., 55, 1, 2, pp. 181–197, 1986. · Zbl 0587.73106
[7]	Wathen,Spectral bounds and preconditioning methods using element-by-element analysis for Galerkin finite element equations, Report No. AM-87-04, Dept. of Mathematics, University of Bristol, Bristol England, 1987. · Zbl 0691.65021