In this article, the condition number of the stiffness matrix $\kappa \left(\mathbf{K}\right)$ 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 $\kappa \left(\mathbf{K}\right)$. 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.

本文中，刚度矩阵的条件数 $\kappa （\mathbf{ķ}）$比较了三种高阶有限元方法（FEM），即有限元方法的p版本，光谱元素方法（SEM）和基于NURBS的等几何分析（IGA）。注意，在分析中仅考虑线性弹性问题。众所周知，条件数是强烈影响直接求解器的有效位数或迭代求解方案所需的迭代计数的一个因素。因此，研究形状函数选择和元素变形对$\kappa （\mathbf{ķ}）$。基于大量的一维，二维和三维示例，对这些影响进行了全面研究，并将数值结果与从文献中提取的条件数估计值进行了比较。总体而言，对p -FEM，SEM和二维IGA观察到了很好的一致性，而对于三维等几何元素则存在差异。这些都是重要的发现，因为理论结果可能仅在非常受限的情况下可用，其中考虑了一个元素的几何形状，元素映射的常量Jacobi矩阵等。