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

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


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
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