On condition numbers in hp -FEM with Gauss-Lobatto-based shape functions

By Jens Markus Melenk. 2002

64 citations recorded by the source corpus.

Sharp bounds on the condition number of stiffness matrices arising in hp/spectral discretizations for two-dimensional problems elliptic problems are given. Two types of shape functions that are based on Lagrange interpolation polynomials in the Gauss-Lobatto points are considered. These shape functions result in condition numbers O(p) and O(p ln p) for the condensed stiffness matrices, where p is the polynomial degree employed. Locally refined meshes are analyzed. For the discretization of Dirichlet problems on meshes that are refined geometrically toward singularities, the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/b37987be59440681670e3936698d415f4c255889

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

High order edge elements on simplicial meshes (2007, result): “Clearly the condition number shows a O(1/h(2)) behavior, in agreement with classical theoretical results [16].”
Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis (2014, methodology): “This has been shown for example in the context of the C-collocationGalerkin method [91–94], multidomain spectral or pseudospectral elements [44–46, 95, 96], and hp finite elements with Gauss-Lobatto basis functions [97, 98].”
Use of higher‐order shape functions in the scaled boundary finite element method (2006, methodology): “When higher-order shape functions are employed in the finite element method, it is common to examine the condition numbers of the coefficient matrices generated [15, 18].”
Spectral element methods on triangles and quadrilaterals: comparisons and applications (2004, result): “For the SEM, the OðN Þ behavior is in agreement with the theoretical result obtained in [19] (in the present study the mass matrix is the identity so that the result obtained in [19] for the stiffness matrix should be relevant).”
Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements (2007, unknown): “Such methods were studied in [Man90, BCM91, Pav94, Ain96a, Ain96b, Cas97, Bic97, GC98, SC01, Mel02 , EM04].”