Condition number estimates for matrices arising in NURBS based isogeometric discretizations of elliptic partial differential equations

Abstract

We derive bounds for the minimum and maximum eigenvalues and the spectral condition number of matrices for isogeometric discretizations of elliptic partial differential equations in an open, bounded, simply connected Lipschitz domain $Ω\subset \mathbb{R}^d$, $d\in\{2,3\}$. We consider refinements based on mesh size $h$ and polynomial degree $p$ with maximum regularity of spline basis functions. For the $h$-refinement, the condition number of the stiffness matrix is bounded above by a constant times $ h^{-2}$ and the condition number of the mass matrix is uniformly bounded. For the $p$-refinement, the condition number grows exponentially and is bounded above by $p^{2d+2}4^{pd}$ and $p^{2d}4^{pd}$ for the stiffness and mass matrices, respectively. Rigorous theoretical proofs of these estimates and supporting numerical results are provided.