Boundary concentrated finite elements for optimal control problems with distributed observation

Abstract

We consider the discretization of an optimal boundary control problem with distributed observation by the boundary concentrated finite element method. If the constraint is a $H^{1+\delta }(\Omega )$ regular elliptic PDE with smooth differential operator and source term, we prove for the two dimensional case that the discretization error in the $L_2$ norm decreases like $N^{-\delta }$, where $N$ is the number of unknowns. Our approach is suitable for solving a wide class of problems, among them piecewise defined data and tracking functionals acting only on a subdomain of $\Omega$. We present several numerical results.

Appendix

We will construct a BC-fem interpolation operator. Since we allow hanging nodes, these results generalize [19]. For the interpolation error we obtain approximation results comparable to those obtained in [19] for regular meshes.

1.1 Estimates of local element size and polynomial degrees

In the interpolation estimates below, it will be important to have comparable element size and element polynomial degree for neighboring elements. For meshes without hanging nodes, we have the following result from [23, Lemma 2.3], its extension to meshes with hanging nodes as used here is straightforward.

Lemma 4.1

Let $\tau$ be a $\gamma$-shape-regular mesh. Then there exists a constant $C(\gamma )$ such that for two neighboring elements $K,K'$ with $\bar{K} \cap \bar{K}'\ne \emptyset$ there holds

$$\begin{aligned} C(\gamma )^{-1}h_K \le h_{K'}\le C(\gamma ) h_K. \end{aligned}$$

(4.2)

Theorem 4.2

Let $\tau$ be a geometric mesh with a linear polynomial degree vector and slope $\alpha$. Then there is a constant $C(\alpha )$ depending on $\gamma$ such that for two neighboring elements $K,K'$ with $\bar{K} \cap \bar{K}'\ne \emptyset$ it holds

$$\begin{aligned} C(\alpha )^{-1}p_K\le p_{K'}\le C(\alpha )p_K. \end{aligned}$$

Moreover, $C(\alpha )\in {\mathcal {O}}(\alpha )$.

Proof

The constants $c_1, c_2$ defining the linear degree vector naturally satisfy $c_2>c_1$, cf. Definition 3.3. Using the properties of the linear degree vector and Lemma 4.1 we can estimate

$$\begin{aligned} p_{K'}&\le 1+\alpha c_2\log (h_{K'}/h)\\&\le 1+\alpha c_2\log (C(\gamma ) h_K/h)\\&\le 1+\alpha c_2 \log (h_K/h) + \alpha c_2 \log (C(\gamma ))\\&\le c_2c_1^{-1}(1+\alpha c_1 \log (h_K/h)+\alpha c_2\log (C(\gamma )))\\&\le c_2c_1^{-1}(p_K+p_K\alpha c_2\log (C(\gamma ))) \\&\le c_2c_1^{-1}(1+\alpha c_2\log (C(\gamma ))) p_K. \end{aligned}$$

The same computation yields a bound of $p_K$ from above. This proves the claim with $C(\alpha ):=\frac{c_2}{c_1}(1+\alpha c_2\log (C(\gamma )))$. $\square$

1.2 Extension and projection operators

The reference element we have in mind is the square $[-1,1]^2$, but we will keep the notation relatively neutral to make the results applicable to triangles as well. The index $i$ is taken from $\{1,2,3(,4)\}$. We take the reference element $\hat{K}$ and the space $Q_p(\hat{K}):=span\{x^i y^j\ |\ 0\le i,j\le p\}$. Triangles would require the space $span\{x^i y^j\ |\ 0\le i+j\le p\}$.

As our mesh will have hanging nodes, we assume that each edge $e_i$ of the reference element has an associated polynomial degree $p_i:=p_{e_i}$ [see (3.1)] with $p_i\le p$. The constructed approximant will lie in

$$\begin{aligned} P_{\mathbf {p}(K)}(\hat{K}):=\left\{ f\in Q_p(\hat{K})\ |\ \hbox {deg}(f|_{e_i})=p_i,\quad \hbox {deg}(f)\le p \right\} . \end{aligned}$$

(4.3)

We first need an extension operator acting from $\partial \hat{K}$ to $\hat{K}$ (see [22, Lemma 3.2.3]).

Lemma 4.3

Let $f\in C(\partial \hat{K})$ be a polynomial of degree $p_i$ on the $i$th edge of the reference element for all $i$. There exists a linear extension mapping $E:C(\partial \hat{K})\rightarrow P_{\mathbf {p}(K)}(\hat{K})$ with the following properties

$$\begin{aligned} (Ef)|_{e_i}&=f \end{aligned}$$

(4.4)

$$\begin{aligned} \Vert \, Ef \, \Vert _{L_\infty (\hat{K})}+p^{-2}\Vert \, \nabla Ef \, \Vert _{L_\infty (\hat{K})}&\le c \Vert \, f \, \Vert _{L_\infty (\partial \hat{K})} \end{aligned}$$

(4.5)

Proof

We prove this only in the case of $\hat{K}$ being the reference square. The extension to triangular $\hat{K}$ is straightforward, see e.g. [22, Lemma 3.2.3].

By subtracting a bilinear function from $f$ we can assume that it vanishes on the vertices of the reference element. For each $f_i:=f|_{e_i}$ we construct an extension $E_i(f_i)\in P_{\mathbf {p}(K)}(\hat{K})$ which is zero at all other edges $e_j$, $j\ne i$.

Let us demonstrate the construction of $E_i(f_i)$ for $e_1$, $e_1:=\{(x,y)\in \mathbb {R}^2\ |\ x\in [-1,1], y=-1\}$. Here we define $E_1(f_1):=\frac{1-y}{2}f(x)$. Analogously we define the extension from the edges $e_i$, $i>1$. This way we get an extension $F:=E(f):=\sum _i E_i(f_i)$.

With the inverse estimate $\Vert \, \nabla F \, \Vert _{L_\infty (\hat{K})}\le c p^2 \Vert \, F \, \Vert _{L_\infty (\hat{K})}$ ([25, Theorem 4.76]) with $p\ge p_i$ we only need to show $\Vert \, F \, \Vert _{L_\infty (\hat{K})}\le c\Vert \, f \, \Vert _{L_\infty (\partial \hat{K})}$. This is a trivial estimate: $\Vert E_1(f_1)\Vert _{L_\infty (\hat{K})} \le \Vert f_1\Vert _{L^\infty (e_1)}$, as $\frac{1-y}{2}\le 1$ on $\hat{K}$.

In the case that $f$ does not vanish in the vertices let us denote by $F_0$ the bilinear interpolation of $f$ that is exact in the vertices. Then we set $Ef:= F_0 + \sum _i E_i(f_i-F_0)$. It is now easy to argue that the extension fulfills the claim. $\square$

1.3 Construction of the $bc$-fem interpolation operator on meshes with hanging nodes

In the following, $u$ will denote just a function and not the control variable as before. The aim of this section is to construct an interpolant on the reference element. It is desired to interpolate a function $u$ living on the physical domain $\Omega$ by pulling it back to the reference element for each element of the finite element discretization $\tau$.

The constructed interpolator will be needed for elements in the interior of $\Omega$. There, we need to distinguish between elements possessing a hanging node or not.

At first, we will construct the interpolator for elements without hanging nodes. The following theorem is similar to [19, Lemma 2.9]. We give a proof here in order to track the dependence of the constants on the parameter $\alpha$ of the linear degree vector.

In the sequel we will denote by $GL(q,f)$ the one-dimensional Gauss–Lobatto interpolation of degree $q$ for the function $f$ on an edge.

Theorem 4.4

Let $\hat{K}$ be the reference element. Let $u$ be a function on $\Omega$ whose pull back $\hat{u}=u\circ F_K$ is analytic on $\bar{\hat{K}}$ and satisfies

$$\begin{aligned} \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{L_\infty (\hat{K})}\le C_u\gamma _u^q q!,\quad q=0,1,2,\ldots \end{aligned}$$

Then there exists an interpolant $I(u)\in P_{\mathbf {p}(K)}(\hat{K})$ such that

• 1. $I(\hat{u})|_{e_i} = GL(p_i,\hat{u}|_{e_i})$,

• 2. $\Vert \, I(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}\le C_\alpha C_u e^{-b p_m}$,

where $b>0$ depends on $\gamma _u$, and $C_\alpha >0$ depends on $\gamma _u$ and $\alpha$ with $C_\alpha =\mathcal {O}(\alpha ^6)$ for $\alpha \rightarrow \infty$.

Here, $p_m$ denotes the minimal polynomial degree is defined by $p_m:={\min }_i \{ p_i\}$ and naturally $p_m\le p_i\le p$ with $p$ being the degree of the image of $I$, i.e. $P_{\mathbf {p}(K)}(\hat{K})$.

Proof

We restrict $\hat{u}\in C(\hat{K})$ to the boundary $\partial \hat{K}$ and define the piecewise Gauss–Lobatto interpolation operator

$$\begin{aligned} i:C(\partial \hat{K})&\rightarrow \{ f\in C(\partial \hat{K})\ |\ f|_{e_i}\ \text {is polynom with degree}\ p_i\},\\ i(\hat{u})(x)&= GL(p_i,\hat{u}|_{e_i})(x) \quad \forall x\in \partial \hat{K}. \end{aligned}$$

Let us define the finite-dimensional subspace

$$\begin{aligned} V:=\{u\in P_{\mathbf {p}(K)}(\hat{K})\ :\ \hat{u}|_{\partial \hat{K}} = 0\}. \end{aligned}$$

Since $V$ is finite-dimensional, there is a linear and bounded projection operator $\Pi :P_{\mathbf {p}(K)}(\hat{K}) \rightarrow V$ with $\Vert \Pi \Vert _{\mathcal L(C(\bar{\hat{K}}),C(\bar{\hat{K}}))}\le \sqrt{\hbox {dim}V}$, confer [22, Theorem A.4.1]. Since $V\subset P_{\mathbf {p}(K)} \subset Q_p(\hat{K})$, we have $\hbox {dim}(V)\le (p+1)^2$, which shows $\Vert \Pi \Vert _{\mathcal L(C(\bar{\hat{K}}),C(\bar{\hat{K}}))}\le p+1$.

The interpolation operator $I$ is now defined by

$$\begin{aligned} I(\hat{u}):= E(i(\hat{u})) + \Pi (\hat{u}-E(i(\hat{u}))) \end{aligned}$$

with the extension operator $E$ from Lemma 4.3. By construction, the first property is fulfilled. If $\hat{u}\in P_{\mathbf {p}(K)}(\hat{K})$ it follows that $i(\hat{u})=\hat{u}|_{\partial \hat{K}}$ and therefore $\hat{u}-E(i(\hat{u}))\in V$. Thus, $I$ interpolates functions of $P_{\mathbf {p}(K)}(\hat{K})$ exactly.

Let $\hat{u}\in C(\bar{\hat{K}})$ be given. Let us first estimate the norm of $I$ by

$$\begin{aligned} \Vert \, I(\hat{u}) \, \Vert _{L_\infty (\hat{K})}&\le c\Vert \, i(\hat{u}) \, \Vert _{L_\infty (\partial \hat{K})}+(p+1)\Vert \, \hat{u}-E(i(\hat{u})) \, \Vert _{L_\infty (\hat{K})} \\&\le c(1+\ln p)\Vert \, \hat{u} \, \Vert _{L_\infty (\partial \hat{K})}+(p+1)\Vert \, \hat{u} \, \Vert _{L_\infty (\hat{K})}\\&\quad +c(1+\ln p)(p+1)\Vert \, \hat{u} \, \Vert _{L_\infty (\partial \hat{K})}, \end{aligned}$$

where we used [22, Lemma 3.2.1] to bound the Gauss–Lobatto-interpolation operator $i$. Exploiting $\Vert \, \hat{u} \, \Vert _{L_\infty (\partial \hat{K})}\le \Vert \, \hat{u} \, \Vert _{L_\infty (\hat{K})}$ for $\hat{u}\in C(\bar{\hat{K}})$ yields the estimate

$$\begin{aligned} \Vert \, I(\hat{u}) \, \Vert _{L_\infty (\hat{K})}&\le C_I p(1+\ln p)\Vert \, \hat{u} \, \Vert _{L_\infty (\hat{K})}. \end{aligned}$$

Regarding approximation properties, it now follows with arbitrary $v\in P_{\mathbf {p}(K)}(\hat{K})$ and using $v=Iv$

$$\begin{aligned} \Vert \, \hat{u}-I(\hat{u}) \, \Vert _{L_\infty (\hat{K})}&=\Vert \, (\hat{u}-v)-I(\hat{u}-v) \, \Vert _{L_\infty (\hat{K})} \\&\le (1+C_I p(1+\ln p))\Vert \, \hat{u}-v \, \Vert _{L_\infty (\hat{K})}. \end{aligned}$$

In order to achieve an approximation property in $W^{1,\infty }(\hat{K})$, we need to estimate the first derivatives of $\hat{u}-I(\hat{u})$:

$$\begin{aligned} \Vert \, \nabla (\hat{u}&-I(\hat{u})) \, \Vert _{L_\infty (\hat{K})}\\ {}&=\Vert \, \nabla ((\hat{u}-v)- I(\hat{u}-v)) \, \Vert _{L_\infty (\hat{K})}\\&\le \Vert \, \nabla (\hat{u}-v) \, \Vert _{L_\infty (\hat{K})}+\Vert \, \nabla (I(\hat{u})-v) \, \Vert _{L_\infty (\hat{K})}\\&\le \Vert \, \nabla (\hat{u}-v) \, \Vert _{L_\infty (\hat{K})}+Cp^2\Vert \, (I(\hat{u})-v) \, \Vert _{L_\infty (\hat{K})}\\&\le \Vert \, \nabla (\hat{u}-v) \, \Vert _{L_\infty (\hat{K})}+C p^2(\Vert \, (I(\hat{u})-\hat{u}) \, \Vert _{L_\infty (\hat{K})}+\Vert \, \hat{u}-v \, \Vert _{L_\infty (\hat{K})})\\&\le \Vert \, \nabla (\hat{u}-v) \, \Vert _{L_\infty (\hat{K})}+ C p^2(2+C_I p(1+\ln p))\Vert \, \hat{u}-v \, \Vert _{L_\infty (\hat{K})}. \end{aligned}$$

In the last two estimates, we can pass to the infimum because $v$ was arbitrary, which shows

$$\begin{aligned} \Vert \, \hat{u}-I(\hat{u}) \, \Vert _{L_\infty (\hat{K})}&\le \hat{C}_1 p(1+\ln p) \inf _{v\in P_{\mathbf {p}(K)}(\hat{K})}\Vert \, \hat{u}-v \, \Vert _{L_\infty (\hat{K})},\\ \Vert \, \nabla (\hat{u}-I(\hat{u})) \, \Vert _{L_\infty (\hat{K})}&\le \inf _{v\in P_{\mathbf {p}(K)}(\hat{K})}\left\{ \Vert \, \nabla (\hat{u}-v) \, \Vert _{L_\infty (\hat{K})} \right. \\&\quad \left. +\, \hat{C}_2 p^3(1+\ln p)\Vert \, \hat{u}-v \, \Vert _{L_\infty (\hat{K})}\right\} \!. \end{aligned}$$

Relying on best approximation results in the space $P_{\mathbf {p}(K)}(\hat{K})$, we have [22, Theorem 3.2.19]

$$\begin{aligned} \inf _{v\in P_{p}(\hat{K})}\Vert \, \hat{u}-v \, \Vert _{L_\infty (\hat{K})}&\le CC_u e^{-b' p_m},\\ \inf _{v\in P_{p}(\hat{K})}\Vert \, \nabla (\hat{u}-v) \, \Vert _{L_\infty (\hat{K})}&\le CC_u e^{-b' p_m} \end{aligned}$$

with constants $C,b'$ depending both on $\gamma _u$.

Collecting the estimates above, we obtain

$$\begin{aligned} \Vert \, I(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}\le & {} \hat{C}_1p(1+\ln p) C C_u e^{-b'p_m} \\&+\, CC_ue^{-b'p_m}+\hat{C}_2 p^3(1+\ln p)\hat{C}_1p(1+\ln p) C C_u e^{-b'p_m}. \end{aligned}$$

We have from Theorem 4.2 that $C(\alpha )^{-1}p_{K'}\le p_K\le C(\alpha ) p_{K'}$ for two neighboring elements $K, K'$. Hence, we can bound $p\le C(\alpha ) p_m$ because the minimal polynomial degree is determined by at least one neighbor. This way we get

$$\begin{aligned} \Vert \, I(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}&\le \hat{C}_3 C(\alpha )^6 p_m^6 CC_u e^{-b'p_m} \end{aligned}$$

(4.6)

Absorbing $p_m^6$ by decreasing the constant $b'$ yields

$$\begin{aligned} \Vert \, I(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}\le \ C_\alpha C_u e^{-b p_m} \end{aligned}$$

with $C_\alpha$ depending on $\alpha ,\gamma _u$ and $b$ on $\gamma _u$, and $C_\alpha = {\mathcal {O}}(\alpha ^6)$ for $\alpha \rightarrow \infty$. $\square$

Remark 4.5

We cannot avoid the constant $p(1+\ln p)$ in the estimates of $\Vert \, \hat{u}-I(\hat{u}) \, \Vert _{L_\infty (\hat{K})}$ and $\Vert \, \nabla (\hat{u}-I(\hat{u})) \, \Vert _{L_\infty (\hat{K})}$ as we allow different polynomial degrees in the interior and on the edges of elements.

In the second step, we will construct an interpolation operator that can deal with hanging nodes. To begin with, we cite an one-dimensional interpolation result of [22, Lemma 3.2.6].

Lemma 4.6

Let $u$ be analytic on the interval $[-1,1]$ and satisfy for some $C_u,\gamma _u$

$$\begin{aligned} \Vert \, \nabla ^{q+2} u \, \Vert _{L_\infty (I)}\le C_u\gamma _u^q q!\quad q=0,1,2,\ldots \end{aligned}$$

There are constants $C,b>0$ depending on $\gamma _u$ such that $GL(q,u)$ satisfies for $p=1,2,\ldots$

$$\begin{aligned} \Vert \, u-GL(p,u) \, \Vert _{W^{1,\infty }(I)}\le C C_u e^{-b p}. \end{aligned}$$

Proof

In [22, Lemma 3.2.6], the estimate $\Vert \, u-GL(p,u) \, \Vert _{W^{1,\infty }(I)}\le \kappa C_u\left( \frac{1}{1+\sigma }\right) ^{p+1}$ is proven with $\kappa ,\sigma >0$ depending on $\gamma _u$. With $C=\kappa (1+\sigma )^{-1}$ and $b=\ln (1+\sigma )$ we obtain the desired estimate. $\square$

Let us describe now the construction of an interpolator on elements with hanging nodes. Depending on the position of the hanging nodes, we prolong the local edge $e_j$ to the full coarse edge $\tilde{e}_j$, with $j$ from $\{1,2,3(,4)\}$. An exemplary situation is depicted in Fig. 5.

Fig. 5

Reference element with hanging node and its neighbor (possibly distorted)

Theorem 4.7

(hanging nodes) Let $\hat{K}$ be the reference element. Let $u$ be a function on $\Omega$ whose pull back $\hat{u}$ is analytic on $\bar{\hat{K}}$ and satisfies

$$\begin{aligned} \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{L_\infty (\hat{K})}\le C_u\gamma _u^q q!,\quad \quad q=0,1,2,\ldots \end{aligned}$$

(4.7)

Let the indices $i$ represent the free edges, whereas $j$ denotes constrained edges due to the existence of hanging nodes. If additionally it holds

$$\begin{aligned} \Vert \, \nabla ^{q+2} \hat{u} \, \Vert _{L_\infty (\tilde{e}_j)}\le C_u \gamma _u^q q!,\qquad q=0,1,2,\ldots \end{aligned}$$

(4.8)

with $C_2,\gamma _2>0$ then there exists an interpolant $I(\hat{u})\in P_{\mathbf {p}(K)}(\hat{K})$ such that

• 1. $\tilde{I}(\hat{u})|_{e_i} = GL(p_i, \hat{u}|_{e_i})$,

• 2. $\tilde{I}(\hat{u})|_{e_j}= GL(p_j, \hat{u}|_{\tilde{e}_j})|_{e_j}$.

• 3. $\Vert \, \tilde{I}(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}\le \tilde{C}(\alpha ) C_u e^{-b p_m}$

where $b$ depends on $\gamma _u,\gamma _2$. The constant $\tilde{C}(\alpha )$ is at most ${\mathcal {O}}(\alpha ^6)$ for $\alpha \rightarrow \infty$.

Let us comment on the impact of Theorem 4.7. Due to property 1. and 2., it is possible to construct a complete interpolant in an element by element fashion. Together with Theorem 4.4 it is guaranteed, that the resulting interpolant is continuous across each edge and therefore the global interpolant lies in the conforming finite element space $\mathbb {V}^\mathbf {p}_{l,\Gamma _\mathcal {D}}$. This is possible as the definition of the finite element space enforces that the polynomial degree on a constrained edge coincides with the polynomial degree on the corresponding coarse edge.

Proof

We define the piecewise Gauss–Lobatto interpolation operator as

$$\begin{aligned}&\tilde{i}:C(\partial \hat{K}\cup \bigcup _j\tilde{e}_j)\rightarrow \{ f\in C(\partial \hat{K}):\ f_i|_{e_i}\ \text {is polynomial of degree}\ p_i\},\\&\tilde{i}(\hat{u})(x)= GL(p_i,\hat{u}|_{e_i})(x), \quad x\in e_i,\\&\tilde{i}(\hat{u})(x)= GL(p_j,\hat{u}|_{\tilde{e}_j})|_{e_j}, \quad x\in e_j. \end{aligned}$$

The function $\hat{u}=u\circ F_K$ can also be evaluated at points outside of $\hat{K}$ since the mapping $F_K$ is analytic. Thus, the Gauss–Lobatto interpolation on $\tilde{e}$ is well defined.

With the operators defined in the proof of Theorem 4.4 we define the interpolation operator as

$$\begin{aligned} \tilde{I}= E(\tilde{i}(\hat{u}))-\Pi (\hat{u}-E(i(\hat{u}))). \end{aligned}$$

We compute

$$\begin{aligned} \Vert \, \tilde{I}(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}&\le \Vert \, I(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}+\Vert \, I(\hat{u})-\tilde{I}(\hat{u}) \, \Vert _{W^{1,\infty }(\hat{K})}, \end{aligned}$$

where $I$ is given by Theorem 4.4. The first addend is bounded by $C_\alpha C_u e^{-b p_m}$ due to Theorem 4.4. So we only need to estimate the second one. Using Lemma 4.3 we find

$$\begin{aligned} \Vert \, I(\hat{u})&-\tilde{I}(\hat{u}) \, \Vert _{W^{1,\infty }(\hat{K})}\\ {}&=\Vert \, E(i(u))-E(\tilde{i}(u)) \, \Vert _{W^{1,\infty }(\hat{K})}\\&\le c p^2 \Vert \, i(\hat{u})-\tilde{i}(\hat{u}) \, \Vert _{L_\infty (\partial \hat{K})}\\&=c p^2 \Vert \, \sum _j GL(p_j,\hat{u}|_{e_j})- GL(p_j,\hat{u}|_{\tilde{e}_j})|_{e_j} \, \Vert _{L_\infty (e_j)}\\&\le c p^2\sum _j\left( \Vert \, GL(p_j, \hat{u}|_{e_j}) - \hat{u} \, \Vert _{L_\infty (e_j)} + \Vert \, GL(p_j,\hat{u}|_{\tilde{e}_j})|_{e_j} -\hat{u} \, \Vert _{L_\infty (e_j)}\right) . \end{aligned}$$

The first addends are bounded due to (4.7) and Lemma 4.6.

$$\begin{aligned} \sum _j \Vert \, GL(p_j,\hat{u}|_{e_j})-\hat{u} \, \Vert _{L_\infty (e_j)} \le \sum _j CC_u e^{-b_1 p_j} \le 4CC_u e^{-b_1p_m}. \end{aligned}$$

(4.9)

If we use an affine mapping from $\tilde{e}_j$ to $[-1,1]$, the prerequisite (4.8) transforms into

$$\begin{aligned} \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{L_\infty (-1,1)}\le C_u (2\gamma _u)^q q!,\quad q=0,1,2,\ldots . \end{aligned}$$

Using again Lemma 4.6 we find

$$\begin{aligned} \sum _j \Vert \, GL(p_j,\hat{u}|_{\tilde{e}_j})|_{e_j} -\hat{u} \, \Vert _{L_\infty (e_j)}&\le \sum _j\Vert \, GL(p_j,\hat{u}|_{\tilde{e}_j}) -\hat{u} \, \Vert _{L_\infty (\tilde{e}_j)}\\&\le \sum _j CC_u e^{-b_2 p_j}\le 4CC_ue^{-b_2 p_m}. \end{aligned}$$

With $cp^2\le cC(\alpha )^2$, the final estimate reads

$$\begin{aligned} \Vert \, \tilde{I}(\hat{u})-\hat{u} \, \Vert _{W^{1,\infty }(\hat{K})}&\le C_\alpha C_ue^{-bp_m}+ cC(\alpha )^2(C C_ue^{-b_1 p_m}+ CC_ue^{-b_2p_m})\\&\le \tilde{C}_\alpha C_ue^{-\tilde{b}p_m}. \end{aligned}$$

with $\tilde{C}_\alpha$ depending on $\alpha ,\gamma _u$ and $\tilde{b}$ on $\gamma _u$. As $C_\alpha \in {\mathcal {O}}(\alpha ^6)$, it follows $\tilde{C}_\alpha \in {\mathcal {O}}(\alpha ^6)$. $\square$

Remark 4.8

Note that the interpolation operator projects $\hat{u}-E( i(\hat{u}))$ instead of $\hat{u}-E(\tilde{i}(\hat{u}))$ onto the subspace $V$ of polynomials vanishing at the boundary of the element. This simplifies the interpolation error estimates.

1.4 Best-approximation and discretization error estimates

First we establish an easy lemma to conveniently check the prerequisites of Theorems 4.4 and 4.7.

Lemma 4.9

Let $u$ be a function on $\Omega$ that satisfies

$$\begin{aligned} \Vert \, \nabla ^q u \, \Vert _{L_2(\Omega )}\le C_u\gamma _u^q q!, \quad q=0,1,2,... \end{aligned}$$

(4.10)

Then $u$ is analytic on $\bar{\Omega }$ and scaling constants $C_s,c_s>0$ exist such that

$$\begin{aligned} \Vert \, \nabla ^q u \, \Vert _{C(\bar{\Omega })}\le C_sC_u (c_s\gamma _u)^q q!, \quad q=0,1,2,... \end{aligned}$$

Proof

For an arbitrary but fixed $q$, we have $\nabla ^q u\in H^2(\Omega )$. A Sobolev embedding implies

$$\begin{aligned} \Vert \, \nabla ^q u \, \Vert _{C(\bar{\Omega })}\le C \Vert \, \nabla ^q u \, \Vert _{H^2(\Omega )}. \end{aligned}$$

Estimating each derivative of $u$ appearing in the $H^2(\Omega )$ norm separately with (4.10) yields

$$\begin{aligned} \Vert \, \nabla ^q u \, \Vert _{C(\bar{\Omega })}\le C(1+\gamma _u+\gamma _u^2)C_u\gamma _u^q(q+2)!. \end{aligned}$$

Choosing $C_s:=2C(1+\gamma _u+\gamma _u^2)$ and $c_s=6$, which implies $c_s^q\ge (q+2)(q+1)$ for $q\ge 1$, proves the estimate, which in turn gives analyticity of $u$ on $\bar{\Omega }$. $\square$

The proof of the following theorem is inspired by [19, Proposition 2.10].

Theorem 4.10

Let $\tau$ be a $\gamma$-shape-regular geometric mesh with the properties of Sect. 1. Let $u\in B^2_{1-\delta }(C_u,\gamma _u)$ for some $\delta \in (0,1]$ and $\mathbf p$ the linear degree vector with slope $\alpha$. Then for sufficiently large $\alpha$ it holds

$$\begin{aligned} \hbox {inf } \left\{ \Vert \, u-v \, \Vert _{H^1(\Omega )}\ |\ v\in \mathbb {V}^\mathbf {p}_{l,\Gamma _\mathcal {D}}\right\} \le C \, C_u \, h^{\delta }. \end{aligned}$$

Here, $C$ depends on $\Omega ,\gamma _u,\alpha$ and the shape regularity constant $\gamma$, but not on $C_u$. The choice of $\alpha$ depends on all these constants as well but not on $C_u$.

We want to construct the interpolant element by element. On elements abutting the boundary we will use the linear interpolant because the linear degree vector does not allow larger polynomial degrees on elements of size $h$.

For elements not abutting the boundary we want to take advantage of the increased polynomial degree to achieve good approximation quality. The previous error estimates of the interpolants, however, depend on the minimal polynomial degree $p_m$ which is determined by at least one neighbor element. To guarantee that the neighbor’s polynomial degree (and thus $p_m$) can be increased sufficiently, we introduce a second layer of elements near the boundary.

Proof

Overall we distinguish the following cases:

• 1. Elements $K$ collected in $\tau _b$ abutting the boundary, i.e. $\bar{K}\cap \partial \Omega \ne \emptyset$.

• 2. Elements in the ’second’ layer near the boundary, i.e. $K\in \tau$ such that $\bar{K}\cap \partial \Omega =\emptyset$ and $\exists K'\in \tau$ with $\bar{K}\cap \bar{K}'\ne \emptyset ,\ \bar{K}'\cap \partial \Omega \ne \emptyset$. These elements are collected in $\tau _s$.

• 3. Elements without hanging nodes which do not belong to $\tau _b\cup \tau _s$. They are collected in $\tau _f$.

• 4. Elements that do not fall into the previous categories, i.e. elements with hanging nodes which do not belong to $\tau _b\cup \tau _s$. They form the set $\tau _c$.

Let $u\in B^2_{1-\delta }(C_u,\gamma _u)$. For an element $K$ we define the constant $C_K$ by

$$\begin{aligned} C_K^2=\sum _{q=0}^\infty \frac{1}{(2\gamma _u)^{2q}(q!)^2}\Vert \, r^{q+1-\delta }\nabla ^{q+2} u \, \Vert _{L_2(K)}^2. \end{aligned}$$

It holds

$$\begin{aligned} \Vert \, r^{q+1-\delta }\nabla ^{q+2}u \, \Vert _{L_2(K)}&\le C_K(2\gamma _u)^q q!, \end{aligned}$$

(4.11)

$$\begin{aligned} \sum _{K\in \tau }C_K^2&\le \frac{4}{3}C_u^2. \end{aligned}$$

(4.12)

Additionally, we define

$$\begin{aligned} \tilde{C}_K^2:= C_K^2 + \sum _{K':\bar{K}\cap \bar{K}'\ne \emptyset }C_{K'}^2, \end{aligned}$$

which implies $\sum _{K\in \tau } \tilde{C}_K^2 \le \frac{16}{3} C_u^2$.

We construct an interpolant $u_h \in \mathbb {V}^\mathbf {p}_{l,\Gamma _\mathcal {D}}$ of $u$ for each element $K$ falling into one of the four categories above. In the following, the index $q$ will always be from $\mathbb {N}\cup \{0\}$.

• 1. $K\in \tau _b$. Let $I_{lin}$ denote the linear or bilinear interpolation. We set $u_h|_K:= I_{lin}u|_K$. We use [19, Appendix B.4] and the property $1.$ of Definition 3.2 to obtain

  $$\begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(K)} \le \Vert \, u-I_{lin}(u) \, \Vert _{H^1(K)}\le Ch_K^{\delta }\Vert \, r^{1-\delta }\nabla ^2u \, \Vert _{L_2(K)}\le Ch^{\delta }C_K. \end{aligned}$$

• 3. $K\in \tau _f$. The pullback $\hat{u}$ of $u$ on $\hat{K}$ satisfies

  $$\begin{aligned} \begin{aligned} \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{L_2(\hat{K})}&\le C h_K^{q+1}\Vert \, \nabla ^{q+2}u \, \Vert _{L_2(K)} \\&\le C h_K^{q+1}\Vert \, r^{q+1-\delta }\nabla ^{q+2}u \, \Vert _{L_2(K)} \frac{1}{\inf _{x\in K} r(x)^{q+1-\delta }} \end{aligned} \end{aligned}$$

  (4.13)

  As $r(x)$ for $x\in K$ is bounded from below by the diameter of the largest inscribed circle of a neighboring element, $\gamma$-shape-regularity yields

  $$\begin{aligned} \inf _{x\in K}r(x)\ge \tilde{c}(\gamma ) h_K \end{aligned}$$

  for a $\tilde{c}(\gamma )>0$. Consequently,

  $$\begin{aligned} \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{L_2(\hat{K})}\le C C_K h_K^{\delta } (2 \tilde{c}\gamma _u)^q q!. \end{aligned}$$

  where $C$ is possibly rescaled by $\tilde{c}(\gamma )$. We set $u_h|_K:= I(\hat{u}) \circ F_K^{-1}$, where $I$ is given by Theorem 4.4. Due to Lemma 4.9 we can apply Theorem 4.4 and get

  $$\begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(K)}\le C_\alpha C C_K h_K^{\delta } e^{-b p_{m,K}} \end{aligned}$$

  with $b$, $C_\alpha$ given by Theorem 4.4 depending on $\gamma _u$ but not on $C_u$ and $K$. Using

  $$\begin{aligned} p_m=p_{K'}\ge c\alpha \ln (h_{K'}/h) \end{aligned}$$

  for a neighbor $K'$ of element $K$, we arrive at

  $$\begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(K)}\le C_\alpha CC_K h_{K'}^{\delta -\alpha b}h^{\alpha b}. \end{aligned}$$

  Using $h_{K'}\ge c h$ yields

  $$\begin{aligned} h_{K'}^{\delta -\alpha b}h^{\alpha b}\le h^{\mathrm{min}}\{\delta , \alpha b\}. \end{aligned}$$

• 4. $K\in \tau _c$. We set $\hat{K}:= F_K^{-1}(K)$ and denote the edges of $\hat{K}$ that possess a hanging node by $e_j$, $j\in \{1,\dots , 4\}$. The coarse edge that contains $e_j$ is denoted by $\tilde{e}_j$ in reference coordinates. Let $K_j$ denote the neighboring element of $K$ that contains the same hanging node, i.e. $\bar{K}_j \cap F_K(\tilde{e}_j) \ne \emptyset$, and set $\hat{K}_j := F_K^{-1}(K_j)$. For an illustration see Fig. 6. In order to apply Theorem 4.7, we have to estimate $L^\infty$-norms of the pullback on the extended edge $\tilde{e}_j$. With the properties of the elements in $\tau _c$ we obtain

  $$\begin{aligned} \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{L_\infty (\tilde{e}_j)}&\le \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{C(\tilde{e}_j)} \le C \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{C(\hat{E}_j)} \end{aligned}$$

  with $\hat{E}_j = {\text {int}}{\text {conv}}( \hat{K} \cup \hat{e}_j) \subset \hat{K} \cup \hat{K}_j$. Let us emphasize that the constant $C$ depends on $\tilde{E}_j$ but not on $\tilde{K}_j$. Hence, $C$ is independent of $K_j$, and thus it is independent of the mesh. Since $h_{K_j'}$ and $h_K$ are comparable due to Lemma 4.1, we obtain analogously to (4.13)

  $$\begin{aligned} \Vert \, \nabla ^{q+2}\hat{u} \, \Vert _{L_2(\hat{K}_j)} \le C C_{K_j} h_K^{\delta } (2\gamma _u)^q q! \end{aligned}$$

  (4.14)

  with a possibly larger constant $C$ independent of $K,K_j$. The two estimates (4.13) and (4.14) yield

  $$\begin{aligned} \Vert \, \nabla ^{q+2}u \, \Vert _{L_2(\hat{E}_j)}\le C (C_{K_j}+C_K) h^{\delta }_K (2\gamma _u)^q q! \end{aligned}$$

  and Lemma 4.9 shows that the prerequisites for Theorem 4.7 are fulfilled. So we set $u_h|_K:= \tilde{I}(\hat{u}) \circ F_K^{-1}$, with $\tilde{I}$ given by Theorem 4.7. The result of this theorem yields

  $$\begin{aligned} \begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(K)}&\le \tilde{C}_\alpha C\left( C_K + \sum _{K':\bar{K}\cap \bar{K}'\ne \emptyset }C_{K'}\right) h_K^{\delta } e^{-b p_{m,K}}\\&= \tilde{C}_\alpha C \tilde{C}_K h_K^{\delta } e^{-b p_{m,K}}. \end{aligned} \end{aligned}$$

  Arguing as in the case $K\in \tau _f$, we find

  $$\begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(K)}\le \tilde{C}_\alpha C\tilde{C}_K h^{\mathrm{min}}\{\delta , \alpha b\}. \end{aligned}$$

• 2. $K\in \tau _s$. Here, we set $u_h|_K:= I(\hat{u}) \circ F_K^{-1}$ if $K$ has no hanging nodes or $u_h|_K:= \tilde{I}(\hat{u}) \circ F_K^{-1}$ otherwise. Analogously as in the cases $K\in \tau _f$, $K\in \tau _c$, we obtain

  $$\begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(K)}\le \tilde{C}_\alpha C\tilde{C}_K h_K^{\delta } e^{-b p_{m,K}}. \end{aligned}$$

  However, we cannot apply $p_m\ge \alpha \ln (h_K/h)$ because $p_m=1$, and thus $p_m$ is fixed and cannot be increased. In geometric meshes, the element size $h_K$ is proportional to the size of a neighboring element. In the second layer, there is a neighbor abutting the boundary, so we find $C(\gamma )^{-1} h\le h_K\le C(\gamma ) c_2 h$. Thus, we obtain for a possibly adapted $C$

  $$\begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(K)}\le \tilde{C}_\alpha C\tilde{C}_K h^{\delta }. \end{aligned}$$

Overall we now estimate

$$\begin{aligned} \sum _{K\in \tau }\Vert \, u-u_h \, \Vert _{H^1(K)}^2\le & {} C^2\left( \sum _{K\in \tau _b}C_K^2 h^{2\delta } +\tilde{C}_\alpha ^2 \sum _{K\in \tau _s} \tilde{C}_K^2 h^{2\delta } \right. \\&\left. +\,C_\alpha ^2 \sum _{K\in \tau _f} C_K^2 h^{2\min \{\delta , \alpha b\}} +\tilde{C}_\alpha ^2\sum _{K\in \tau _c} \tilde{C}_K^2 h^{2\min \{\delta , \alpha b\}} \right) . \end{aligned}$$

Since $b$ is independent of $\alpha$, we can choose $\alpha$ large enough to obtain

$$\begin{aligned} \sum _{K\in \tau }\Vert \, u-u_h \, \Vert _{H^1(K)}^2 \le C^2 \ C_u^2 \ h^{2\delta }. \end{aligned}$$

By construction $u_h$ is a continuous function on $\bar{\Omega }$. Thus, it holds $u_h\in H^1(\Omega )$ and

$$\begin{aligned} \Vert \, u-u_h \, \Vert _{H^1(\Omega )}\le C \ C_u \ h^{\delta }. \end{aligned}$$

$\square$

Fig. 6

reference element $\hat{K}$ enlarged to $\hat{E}_j$ to handle a hanging node

Remark 4.11

The proof only works for affine linear or bilinear mappings $F_K$. The reason is that prolonged edges of the reference element have to be straight lines under $F_K$, so that in global coordinates they coincide with the coarse edges. Together with the property that hanging nodes are in the middle of a coarse edge, the described procedure and usage of interpolation operators works.

Theorem 4.12

(Lemma 3.6) Let $\tau$ be a geometric mesh on $\Omega$ with mesh size $h$, $\mathbf {p}$ a linear degree vector with slope $\alpha$. Suppose Assumptions 1 and 2 are satisfied. Let $y\in H^{1+\delta }(\Omega )$ for some $\delta \in (0,1]$ be a solution to the state equation (3.4) with data $u\in L_2(\Gamma _\mathcal {N})$ and $f\in B_{1-\delta }^0(C_f,\gamma _f)$, $C_f,\gamma _f>0$. Then for sufficiently large $\alpha$ there is $C>0$ independent of $h$ such that

$$\begin{aligned} \Vert \, y-y_h \, \Vert _{H^1(\Omega )}\le Ch^\delta \end{aligned}$$

holds.

Proof

According to Theorem 2.3 the solution also lies in $B^2_{1-\delta }(C_y,\gamma _y)$ for some constants $C_y,\gamma _y>0$. In view of the best approximation properties of the FE solution (Cea’s lemma) and the approximation quality from Theorem 4.10 the proof is complete. $\square$