Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Abstract

We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this, we first generalize the well-known Céa lemma to nonlinear function spaces. In a second step, we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an $H^1$-type Finsler norm and with the $H^1$-norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application, we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high-order scheme for this problem.

Appendix: Taylor Series Remainder Estimates

In this appendix, we prove two technical results about certain remainder terms in Taylor series expansions. They are used in the proof of the interpolation error bound in Sect. 5.1.

Lemma 8.1

For a function $H(x,y) : T_\text {ref} \times T_{\text {ref}}\rightarrow \mathbb {R}$ and a multi-index $\vec {k}$ with ${|\vec {k}|} >d/2$, we have the inequality

$$\begin{aligned} \left\| \int \limits _0^1 t^{|{\vec {k}}|-1}x^{\vec {k}} H(x,tx)\,dt\right\| _{L^2({T_\text {ref}})} \lesssim \Big \Vert \sup _{z\in {T_\text {ref}}}|H(z,x)|\Big \Vert _{L^2({T_\text {ref}})}, \end{aligned}$$

where the integration in the $L^2$-norms above occurs in the variable $x$. The implicit constant only depends on the diameter of $T_{\text {ref}}$.

Proof

We only treat the case $d=2$, the general case being similar. Also, for simplicity, we assume that $T_\text {ref}$ is contained in the unit ball. Using polar coordinates $(x_1,x_2) = rs_\varphi :=r(\cos (\varphi ),\sin (\varphi ))$ and the substitution $\tau = rt$, we can write

$$\begin{aligned} \int \limits _0^1 t^{|{\vec {k}}|-1}x^{\vec {k}} H(x,tx)\,dt = \int \limits _0^1 t^{|{\vec {k}}|-1}r^{|{\vec {k}}|} s_\varphi ^{\vec {k}} H(x,trs_\varphi )\,dt = \int \limits _0^r \tau ^{|{\vec {k}}|-1} s_\varphi ^{\vec {k}} H(x,\tau s_\varphi )\,d\tau . \end{aligned}$$

We need to estimate the $L^2$-norm of this expression. Since $T_\text {ref}$ is contained in the unit ball, we get

$$\begin{aligned} \left\| \int \limits _0^1 t^{|{\vec {k}}|-1}x^{\vec {k}} H(x,tx)\,dt\right\| _{L^2({T_\text {ref}})}^2 \le \int \limits _{0}^1\int \limits _{0}^{2\pi } \left( \int \limits _0^r \tau ^{|{\vec {k}}|-1} s_\varphi ^{\vec {k}} H(x,\tau s_\varphi )\,d\tau \right) ^2 r\,dr\,d\varphi . \end{aligned}$$

Using the Cauchy–Schwarz inequality, we can bound this expression by

$$\begin{aligned} \int \limits _0^1\int \limits _{0}^{2\pi } \int \limits _0^r H(x,\tau s_\varphi )^2 \tau \, d\tau \, d\varphi \int \limits _0^r \tau ^{2|{\vec {k}}|-4}\tau \,d\tau \, r\,dr \end{aligned}$$

which can in turn be bounded by

$$\begin{aligned}&\int \limits _{0}^1\int \limits _{0}^{2\pi } \int \limits _0^1 \sup _{x\in {T_\text {ref}}} H(x,\tau s_\varphi )^2 \tau \,d\tau \,d\varphi \int \limits _0^1 \tau ^{2|{\vec {k}}|-3} \, d\tau \, r \, dr\\&\quad \le \Big \Vert \sup _{x\in {T_\text {ref}}}|H(x,\cdot )|\Big \Vert _{L^2({T_\text {ref}})}^2 \int \limits _{0}^1\int \limits _0^1 \tau ^{2|{\vec {k}}|-3} \, d\tau \, r\, dr. \end{aligned}$$

Since the double integral on the right is no greater than 1, we get the desired expression. $\square$

Lemma 8.2

For a function $U$ defined on ${T_\text {ref}}$ and a multi-index $\vec {e}$ with $|\vec {e}| = 1$, we have

$$\begin{aligned} \sum _{|{\vec {k}} | = k}\frac{|{\vec {k}}|}{{\vec {k}} !} \partial ^{\vec {e}} \int \limits _0^1 t^{|{\vec {k}}| - 1}x^{\vec {k}} \partial ^{\vec {k}} U(tx)\,dt = \sum _{|{\vec {l}} | = k-1}\frac{(-1)^{k-1}}{{\vec {l}} !} x^{\vec {l}} \partial ^{{\vec {l}} + \vec {e}}U(x). \end{aligned}$$

(44)

Proof

The term on the left-hand side of (44) is the derivative of the residual term

$$\begin{aligned} R(x) :=U(0) - \sum _{|{\vec {l}}|<k}(-1)^{|{\vec {l}}|}\frac{x^{\vec {l}}}{{\vec {l}} !} \partial ^{\vec {l}} U(x) \end{aligned}$$

in the Taylor expansion of $U$ around $x$ and evaluated at zero. Using this interpretation, one can check the statement by direct computation. Indeed, applying the operator $\partial ^{\vec {e}}$ to $R$ and using the product rule, we get

$$\begin{aligned} \partial ^{\vec {e}} R(x)&=- \partial ^{\vec {e}} \sum _{|{\vec {l}}|<k}(-1)^{|{\vec {l}}|} \frac{x^{\vec {l}}}{{\vec {l}} !} \partial ^{\vec {l}} U(x)\\&=\sum _{|{\vec {l}}|<k}(-1)^{|{\vec {l}}|} \frac{x^{{\vec {l}}-\vec {e}}}{({\vec {l}} - \vec {e})!} \partial ^{\vec {l}} U(x) + (-1)^{|{\vec {l}}|}\frac{x^{\vec {l}}}{{\vec {l}} !} \partial ^{{\vec {l}} +\vec {e}} U(x) \\&=-\sum _{|{\vec {l}}|<k-1}(-1)^{|{\vec {l}}|} \frac{x^{{\vec {l}}}}{{\vec {l}}!} \partial ^{{\vec {l}}+\vec {e}} U(x) + \sum _{|{\vec {l}}|< k} (-1)^{|{\vec {l}}|}\frac{x^{\vec {l}}}{{\vec {l}} !} \partial ^{{\vec {l}} +\vec {e}} U(x)\\&= \sum _{|{\vec {l}}|=k-1} (-1)^{|{\vec {l}}|}\frac{x^{\vec {l}}}{{\vec {l}} !} \partial ^{{\vec {l}} +\vec {e}} U(x). \end{aligned}$$

In the second line, we have used the convention $x^{\vec {l}-\vec {e}} \equiv 0$ whenever $\vec {l} -\vec {e} \in \mathbb {Z}^d$ has a negative entry. $\square$