Approximation in FEM, DG and IGA: a theoretical comparison

Abstract

In this paper we compare the approximation properties of degree p spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, $\mathcal {C}^{p-1}$ splines provide better a priori error bounds for the approximation of functions in $H^{p+1}(0,1)$. Our result holds for all practically interesting cases when comparing $\mathcal {C}^{p-1}$ splines with $\mathcal {C}^{-1}$ (discontinuous) splines. When comparing $\mathcal {C}^{p-1}$ splines with $\mathcal {C}^{0}$ splines our proof covers almost all cases for $p\ge 3$, but we can not conclude anything for $p=2$. The results are generalized to the approximation of functions in $H^{q+1}(0,1)$ for $q<p$, to broken Sobolev spaces and to tensor product spaces.

Appendix

Lemma 7

Let $p\ge 1$ be any odd number. Then the interpolation problem: find $g \in \mathcal {P}^{p}$ such that for all $s=0,2,\ldots ,p-1$,

$$\begin{aligned} \left\{ \begin{aligned} \partial ^s g(0)&=a_s\\ \partial ^s g(1)&=b_s \end{aligned}\right. \ \end{aligned}$$

(22)

admits a solution for all $\{a_s\}$, $\{b_s\}$.

Proof

We proceed by induction on p. If $p=1$ then the linear interpolant $g(x)=a_0 + x (b_0-a_0)$ satisfies $g(0)=a_0$ and $g(1)=b_0$ and is a solution. Now, let $p\ge 3$ be any odd number and assume the result is true for $p-2$. Let $f\in \mathcal {P}^{p-2}$ be the solution of

$$\begin{aligned} \partial ^s f(0)=a_{s+2},\quad \partial ^s f(1)=b_{s+2}\quad s=0,2,\ldots ,p-3, \end{aligned}$$

which we know exist by the induction hypothesis. We then define the function g by integrating f twice, i.e.,

$$\begin{aligned} g(x)=cx+d+\int _0^x\int _0^yf(z)dzdx. \end{aligned}$$

This function then satisfies the cases $s\ge 2$ of (22) for all $c,d\in \mathbb {R}$. We finish the proof by picking the constants c and d such that the case $s=0$, meaning $g(0)=a_0$ and $g(1)=b_0$, is also satisfied. $\square$

Lemma 8

Let $p\ge 0$ be any even number. Then the interpolation problem: find $g \in \mathcal {P}^{p}$ such that for all $s=1,3,\ldots ,p-1$,

$$\begin{aligned} \left\{ \begin{aligned} \partial ^s g(0)&=a_k\\ \partial ^s g(1)&=b_k \end{aligned}\right. \ \end{aligned}$$

(23)

admits a solution for all $\{a_s\}$, $\{b_s\}$.

Proof

For $p=0$ there is nothing to prove, and so we consider an even number $p\ge 2$. We then let $f\in \mathcal {P}^{p-1}$ be the solution of

$$\begin{aligned} \partial ^s f(0)=a_{s-1},\quad \partial ^s f(1)=b_{s-1}\quad s=0,2,\ldots ,p-2, \end{aligned}$$

which we know exist by Lemma 7. The function $g(x)=c+\int _0^xf(y)dy$ is then a solution of (23) for any $c\in \mathbb {R}$. $\square$