Bounds on spectral condition numbers of matrices arising in the \(p\)- version of the finite element method
Condition numbers for \(p\)-version finite elements can be quite large and substantial attention has therefore been directed towards finding basis sets for which the growth of the condition number is relatively modest as a function of \(p\). This issue is related to domain decomposition methods; for one many iterative substructuring methods finding well-conditioned basis for the polynomial space is the main practical and theoretical issue. The authors work exclusively with methods based on tensor product spaces, which greatly simplifies the analysis. The reference element is then a square or a cube. Two bases are considered in detail. The first, based on the Lagrangian nodal basis on uniform meshes is shown to lead to a spectacular, exponential growth of the condition number. The second consists of linear functions and the products of \((1 - x^2)\) and Chebyshev polynomials. Upper and lower bounds for the condition number, which are algebraic, and which invites pessimism, are provided. The degree of the growth function is shown to be at least \(5d/2\), where \(d\) is the dimension of the underlying Euclidean space. It is also conjectured that any basis which is hierarchical and also satisfies a local support condition will have condition numbers that grow as fast as the \(4d\)-th power of \(p\). Maybe somewhat surprising the experimental work does not include a study of the basis derived from the integrated Legendre polynomials. For this basis, \textit{Olivier Pourquier} of Lyons, France, in his 1994 thesis, has shown that the condition number grows as the \(4(d - 1)\)th power of \(p\). This appears to disprove the main conjecture of this paper. Pourquier also has good results on the Lagrangian basis using the Gauss-Lobatto- Legendre nodes which are favored by many practitioners.
scientific article; zbMATH DE number 739415