On the condition number of high order finite element methods: Influence of p-refinement and mesh distortion. (1st December 2020) - Eisenträger, S.; Atroshchenko, E.; Makvandi, R.
Abstract: In this article, the condition number of the stiffness matrix κ ( K ) is compared for three high order finite element methods (FEMs), i.e., the p -version of the FEM, the spectral element method (SEM), and the NURBS-based isogeometric analysis (IGA). Note that only problems in linear elasticity are considered in the analysis. It is well-known that the condition number is one factor strongly influencing the number of significant digits for direct solvers or the required iteration count for iterative solution schemes. Therefore, it is important to investigate the effect of the choice of the shape functions and the element distortion on κ ( K ) . Based on numerous one- and two-, and three-dimensional examples, these influences are comprehensively studied, and the numerical results are compared with condition number estimates extracted from the literature. Overall, a good agreement is observed for p -FEM, SEM and two-dimensional IGA, while discrepancies are noted for three-dimensional isogeometric elements. These are important findings as theoretical results may be only available for very restricted scenarios, where one-element geometries, constant Jacobi matrices of the element maps, etc. are considered. Highlights: Systematic comparison of κ ( K ) for p -FEM, SEM, and IGA of highest and lowest continuity w.r.t.: 1. p -refinement, 2. mesh distortion. The lowest κ ( K ) in 1D/2D applications is observed for the p -FEM. The lowest κ ( K ) in 3D applications is observed for the SEM defined on GLL-nodes. Comparison of the numerical growth rates with the theoretical estimates for κ ( K ) in p -refinement.