Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements

Abstract

We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial degree, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates. As hexahedral finite elements are special cases of NURBS, our results specialize to parametric hexahedral finite elements, and our analysis also generalizes to T-spline-based isogeometric analysis. We compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants, and we discuss application of our results to a Laplace problem. We finish this paper with a brief exploration of so-called patch-wise trace inequalities for isogeometric analysis.

Appendix A: An alternate Hölder inequality

Lemma A.1

Let $D \subset \mathbb R ^d$ denote an open domain for $d$ a positive integer. If $f \in L^{\infty }(D)$ and $g \in L^2(D)$, then

$$\begin{aligned} \Vert fg \Vert _{L^2(D)} \le \Vert f \Vert _{L^{\infty }(D)} \Vert g \Vert _{L^2(D)}. \end{aligned}$$

(86)

Proof

Let $f \in L^{\infty }(D)$ and $g \in L^2(D)$. By construction,

$$\begin{aligned} \Vert fg \Vert _{L^2(D)} = \Vert f^2g^2 \Vert ^{1/2}_{L^1(D)}. \end{aligned}$$

By the classical Hölder Inequality,

$$\begin{aligned} \Vert fg \Vert _{L^2(D)}&\le (\Vert f^2 \Vert _{L^{\infty }(D)} \Vert g^2 \Vert _{L^1(D)})^{1/2} \nonumber \\&= (\Vert f \Vert ^2_{L^{\infty }(D)} \Vert g \Vert ^2_{L^2(D)})^{1/2} \nonumber \\&= \Vert f \Vert _{L^{\infty }(D)} \Vert g \Vert _{L^2(D)}. \end{aligned}$$

(87)

$\square$