The choice of high order coordinate functions and preconditioning in frames of the method of domain decomposition
The development of the \(p\)-method, as proposed by L. Babuska and others, has led to the consideration of finite element Lagrange interpolation with polynomials of higher degree than in the \(h\)-method approach. However, numerical results have suggested that the degree shouldn't be too high. The minimax property of Lagrange interpolation at the zeroes of Chebyshev polynomials has been at the heart of using their coordinates in some numerical calculations. [See for instance example 3 in `Towards automatic multigrid algorithms for SPD, nonsymmetric and indefinite problems', by \textit{Y. Shapira}, \textit{M. Israeli} and \textit{A. Sidi}, in Siam J. Sci. Comput., 17, No. 2, 439-453 (1996; Zbl 0851.65086).] In the paper under review, Lagrange interpolation in order to produce the Lagrange basis functions is proposed, not at the zeros of Chebyshev polynomials, but at their extrema's coordinates. The stiffness matrix associated to them can then easily and very satisfactorily be preconditioned, while the same easy preconditioning is bad for the stiffness matrix arising from Lagrange basis functions interpolation at uniformly distributed points. The striking results can be briefly described in the following way: while with the authors' approach, the final conditioning is bounded by the degree of the polynomials multiplied by an absolute constant, the final conditioning associated to uniform Lagrange interpolation grows exponentially with the degree. The results are generalized for two-dimensional elements associated to elliptic equations. Although the English version of the paper is not easy to read, its mathematical content is well worth the effort.
scientific article; zbMATH DE number 857504