General DG-Methods for Highly Indefinite Helmholtz Problems

Abstract

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in \(\mathbb R ^{d}\), \(d\in \{1,2,3\}\). The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the \(hp\)-version of the finite element method explicitly in terms of the mesh width \(h\), polynomial degree \(p\) and wavenumber \(k\). It is shown that the optimal convergence order estimate is obtained under the conditions that \(kh/\sqrt{p}\) is sufficiently small and the polynomial degree \(p\) is at least \(O(\log k)\). On regular meshes, the first condition is improved to the requirement that \(kh/p\) be sufficiently small.

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Notes

  1. 1.

    The DG method can also be formulated for geometries with curved boundaries.

  2. 2.

    To see this, e.g., for \(j=3\), we employ the interpolation inequality [36, (B.5)] to \(\nabla ^{3}u\) to obtain

    $$\begin{aligned} \left\| \nabla ^{3}u\right\| _{L^{\infty }( \widehat{K}) }\le C\left\| \nabla ^{3}u\right\| _{L^{2}( \widehat{K}) }^{1-d/\left( 2\left( s-3\right) \right) }\left\| \nabla ^{3} u\right\| _{H^{s-3}( \widehat{K}) }^{d/\left( 2\left( s-3\right) \right) }\quad \forall u\in H^{s}( \widehat{K}) \end{aligned}$$

    since \(s>3+d/2\). The combination with (7.6) yields the desired bound in (7.7).

  3. 3.

    For a face \(f\), the face normal \(n_{f}:\partial f\rightarrow \mathbb S _{2}\) is defined to have length \(1\), lies in the plane of \(f\), and points to the exterior of \(f\). The face normal derivative on \(\partial f\) is then given by \(\partial _{n_{f}}:=\left\langle n_{f},\nabla \cdot \right\rangle \).

  4. 4.

    The condition \(p \ge j\) can be dropped if \(E_{1,e} u\) vanishes to higher order at the vertex (0,1) due to appropriate assumptions on the function \(w\).

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Appendices

Appendix 1: Details for the Proof of Theorem 4.5

We start with an extension of [37, Lemma 4.6] for the modified Helmholtz equation.

Lemma 6.1

Let \(\varOmega \) be a bounded Lipschitz domain with a smooth boundary. Let \(S_{k}^{\varDelta }\) be the solution operator for the boundary value problem

$$\begin{aligned} -\varDelta u+k^{2}u=0\quad \text{ in } \varOmega ,\quad \partial _{n} u+\mathrm{i}ku=g\quad \text{ on } \partial \varOmega . \end{aligned}$$

Then, for every \(s\in \mathbb{N }_{0}\) there exists \(C>0\) independent of \(k\ge k_{0}\) such that

$$\begin{aligned}&\Vert S_{k}^{\varDelta }(g)\Vert _{H^{s+2}(\varOmega )} \le C\left[ \Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}+k^{s+1/2}\Vert g\Vert _{L^{2}(\partial \varOmega )}\right] ,\end{aligned}$$
(6.1)
$$\begin{aligned}&\Vert S_{k}^{\varDelta }(g)\Vert _{H^{1}(\varOmega )}+k\Vert S^{\varDelta }(g)\Vert _{L^{2}(\varOmega )} \le Ck^{-1/2}\Vert g\Vert _{L^{2}(\partial \varOmega )}. \end{aligned}$$
(6.2)

Proof

The case \(s=0\) in (6.1) as well as the estimate (6.2) is given in [37, Lemma 4.6]. For \(s\ge 1\), we employ induction and the standard shift theorem for the Laplacian: Since \(u\) solves

$$\begin{aligned} -\varDelta u=-k^{2}u\quad \text{ in } \varOmega ,\quad \partial _{n} u=g-\mathrm{i}ku\quad \text{ on } \partial \varOmega , \end{aligned}$$

we have

$$\begin{aligned} \Vert u\Vert _{H^{s+2}(\varOmega )}&\le C\left[ k^{2}\Vert u\Vert _{H^{s}(\varOmega )}+\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}+k\Vert u\Vert _{H^{s+1/2}(\partial \varOmega )}\right] \\&\le C\left[ k^{2}\Vert u\Vert _{H^{s}(\varOmega )}+\Vert g\Vert _{H^{s+1/2} (\partial \varOmega )}+k\Vert u\Vert _{H^{s+1}(\varOmega )}\right] , \end{aligned}$$

where we used a trace inequality. Using the induction hypothesis then leads to an estimate that involves norms of \(g\) other than \(\Vert g\Vert _{H^{s+3/2} (\partial \varOmega )}\) and \(\Vert g\Vert _{L^{2}(\partial \varOmega )}\). These can be removed by an interpolation inequality (see, e.g., [18, Thm. 1.4.3.3 ]) and an appropriate use of the Young inequality. \(\square \)

The analog of [37, Lemma 4.7] is the following (we use the operator \(H_{\partial \varOmega }^{N}\) defined in [37, (4.1c)]):

Lemma 6.2

Let \(\varOmega \) be a bounded Lipschitz domain with a smooth boundary. Fix \(q\in (0,1)\) and \(s\in \mathbb{N }_{0}\). Then, the operator \(H_{\partial \varOmega }^{N}\) can be selected such that the operator \(S_{k} ^{\varDelta }\circ H_{\partial \varOmega }^{H}\) satisfies for some \(C>0\) independent of \(k\)

$$\begin{aligned} k^{s+2}\Vert S_{k}^{\varDelta } (H_{\partial \varOmega }^{N}g)\Vert _{L^{2}(\varOmega )}+k^{2}\Vert S_{k}^{\varDelta }(H_{\partial \varOmega }^{N}g)\Vert _{H^{s}(\varOmega )}&\le q\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}, \end{aligned}$$
(6.3)
$$\begin{aligned} \Vert S_{k}^{\varDelta }(H_{\partial \varOmega }^{N}g) \Vert _{H^{s+2}(\varOmega )}&\le C\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}. \end{aligned}$$
(6.4)

Proof

Estimates (6.3) and (6.4) are shown in [37, Lemma 4.7] for the special case \(s=0\). For \(s\ge 1\), these estimates are derived as in [37, Lemma 4.7] by combining Lemma 6.1 with [37, Lemma 4.2]. We illustrate the procedure for the second term of the left-hand side of (6.3) for the case \(s\ge 2\): Lemma 6.1 yields

$$\begin{aligned} \Vert S_{k}^{\varDelta }(H_{\partial \varOmega }^{N})\Vert _{H^{s}(\varOmega )}&\le C\left[ \Vert H_{\partial \varOmega }^{N}g \Vert _{H^{s-3/2}(\partial \varOmega )}+k^{s-3/2}\Vert H_{\partial \varOmega }^{N}g\Vert _{L^{2}(\varOmega )}\right] \\&\le C\left[ (q/k)^{2}\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )} +k^{s-3/2}(q/k)^{s+1/2}\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}\right] , \end{aligned}$$

where we used [37, Lemma 4.2]. Rearranging terms yields the result. \(\square \)

We also need properties of the Newton potential \(N_{k}\), which generalize [37, Lemma 4.5]:

Lemma 6.3

Let \(\varOmega \) be a bounded Lipschitz domain. Fix \(s\in \mathbb{N }_{0}\) and \(q\in (0,1)\). Then the operator \(H_{\varOmega }\) of [37, (4.1b)] can be selected such that for \(0\le s^{\prime }\le s+2\)

$$\begin{aligned} \Vert N_{k}(H_{\varOmega }f)\Vert _{H^{s^{\prime }}(\varOmega )}\le C(q/k)^{s+2-s^{\prime }}\Vert f\Vert _{H^{s}(\varOmega )}. \end{aligned}$$
(6.5)

Proof

Follows from the procedure in [37]; see also [35, Lemma 4.2]. The essential point is that [36, (3.35)] can be generalized (by using the notation therein) to

$$\begin{aligned} \left\| \partial ^{\alpha }v_{\mu ,H^{2}}\right\| _{L^{2}\left( \mathbb R ^{d}\right) }=\left( 2\pi \right) ^{d/2}\left\| P_{\alpha -\beta }\widehat{G_{k}M}\left( 1-\chi _{\lambda k}\right) \widehat{\partial ^{\beta }f}\right\| _{L^{2}\left( \mathbb R ^{d}\right) } \end{aligned}$$

for all \(\alpha \in \mathbb N _{0}^{d}\) and \(\beta \in \mathbb N _{0}^{d}\). By selecting \(\vert \alpha \vert =s^{\prime }\) and \(\vert \beta \vert =s^{\prime }-2\), we see that \(\vert \alpha -\beta \vert =2\) and this case is considered in [36, (3.35)]. By performing the same estimates as in [36, after (3.35)], we derive for \(\vert \alpha -\beta \vert =2\) the estimate

$$\begin{aligned} \left\| \partial ^{\alpha }N_{k}(H_{\varOmega }f)\right\| _{L^{2}\left( \varOmega \right) }\le C\left\| \partial ^{\beta }H_{\varOmega }f\right\| _{L^{2}\left( \varOmega \right) } \end{aligned}$$

so that

$$\begin{aligned} \Vert N_{k}(H_{\varOmega }f)\Vert _{H^{s^{\prime }}(\varOmega )}\le C\Vert H_{\varOmega }f\Vert _{H^{s^{\prime }-2}(\varOmega )} \end{aligned}$$

follows. The combination with [37, Lemma 4.2] leads to the assertion (6.5). \(\square \)

The next lemma generalizes [37, Lemma 4.15] (note that the boundary condition (1.2) differs from that in [37] by a sign):

Lemma 6.4

Let \(\varOmega \) be a bounded Lipschitz domain with a smooth boundary. Fix \(s\in \mathbb{N }_{0}\). Assume that the solution operator \((f,g)\mapsto S_{k}(f,g)\) for (1.1), (1.2) satisfies (4.4). Then \(S_{k}\) admits the following decomposition: \(u=S_{k}(f,0)=u_{\fancyscript{A}}+u_{H^{s+2}}+\widetilde{u}\), where

$$\begin{aligned} \Vert u_{\fancyscript{A}}\Vert _{H^{1}(\varOmega )}+k\Vert u_{\fancyscript{A}}\Vert _{L^{2}(\varOmega )}&\le Ck^{\vartheta }\Vert f\Vert _{L^{2}(\varOmega )},\\ \Vert \nabla ^{n+2}u_{\fancyscript{A}}\Vert _{L^{2}(\varOmega )}&\le Ck^{\vartheta -1}\gamma ^{n}\max \{k,n\}^{n+2}\Vert f\Vert _{L^{2}(\varOmega )}\quad \forall n\in \mathbb{N }_{0},\\ k^{s+2}\Vert u_{H^{s+2}}\Vert _{L^{2}(\varOmega )}+\Vert u_{H^{s+2}}\Vert _{H^{s+2}(\varOmega )}&\le C\Vert f\Vert _{H^{s}(\varOmega )} \end{aligned}$$

for constants \(C\), \(\gamma >0\) independent of \(k\) and \(n\), and the remainder \(\widetilde{u}=S_{k}(\widetilde{f},0)\) satisfies the boundary value problem

$$\begin{aligned} -\varDelta \widetilde{u}-k^{2}\widetilde{u}=\widetilde{f}\quad \text{ in } \varOmega ,\quad \partial _{n}\widetilde{u}+\mathrm{i} k\widetilde{u}=0\quad \text{ on } \partial \varOmega \end{aligned}$$

for a right-hand side \(\widetilde{f}\in H^{s}(\varOmega )\) with

$$\begin{aligned} \Vert \widetilde{f}\Vert _{H^{s}(\varOmega )}\le q\Vert f\Vert _{H^{s}(\varOmega )},\quad \Vert \widetilde{f}\Vert _{L^{2}(\varOmega )}\le q\Vert f\Vert _{L^{2}(\varOmega )}. \end{aligned}$$

Proof

The proof follows that of [37, Lemma 4.15]. We flag that the boundary condition (1.2) studied in the present paper differs from that in [37], which accounts for sign differences between the procedure here and in [37, Lemma 4.15]. We only need to show the additional bound \(\Vert u_{H^{s+2}}\Vert _{H^{s+2}(\varOmega )}\le C\Vert f\Vert _{H^{s}(\varOmega )}\). To that end, we have to consider, in the notation of [37, Lemma 4.15], the terms

$$\begin{aligned} u_{H^{2}}^{I}&= N_{k}(H_{\varOmega }f),\end{aligned}$$
(6.6)
$$\begin{aligned} u_{H^{2}}^{II}&= S_{k}^{\varDelta }\left( H_{\partial \varOmega }^{N}\big (-\mathrm{i} ku_{H^{2}}^{I}-\partial _{n}u_{H^{2}}^{I}\big )\right) . \end{aligned}$$
(6.7)

For (6.6), we use Lemma 6.3 to get

$$\begin{aligned}&k^{s+2}\Vert N_{k}(H_{\varOmega }f)\Vert _{L^{2}(\varOmega )}+k\Vert N_{k}(H_{\varOmega }f)\Vert _{H^{s+1}(\varOmega )}+\Vert N_{k}(H_{\varOmega }f)\Vert _{H^{s+2}(\varOmega )} \le C\Vert f\Vert _{H^{s}(\varOmega )},\\&\Vert N_{k}(H_{\varOmega }f)\Vert _{H^{s}(\varOmega )} \le C(q/k)^{2}\Vert f\Vert _{H^{s}(\varOmega )}. \end{aligned}$$

This implies in particular with a trace inequality that

$$\begin{aligned} \Vert -\mathrm{i}ku_{H^{2}}^{I}-\partial _{n}u_{H^{2}}^{I}\Vert _{H^{s+1/2}(\partial \varOmega )}\le Ck\Vert u_{H^{2}}^{I}\Vert _{H^{s+1}(\varOmega )}+C\Vert u_{H^{2}}^{I}\Vert _{H^{s+2}(\varOmega )}\le C\Vert f\Vert _{H^{s} (\varOmega )}, \end{aligned}$$

so that also for (6.7), we can obtain, with the aid of Lemma 6.2, the bounds

$$\begin{aligned}&\Vert S_{k}^{\varDelta }(H_{\partial \varOmega }^{N}(-\mathrm{i}ku_{H^{2}} ^{I}-\partial _{n}u_{H^{2}}^{I}))\Vert _{H^{s+2}(\varOmega )} \le C\Vert f\Vert _{H^{s}(\varOmega )},\\&\quad k^{s+2}\Vert S_{k}^{\varDelta }(H_{\partial \varOmega }^{N}(-\mathrm{i}ku_{H^{2} }^{I}-\partial _{n}u_{H^{2}}^{I})) \Vert _{L^{2}(\varOmega )}+k^{2}\Vert S_{k}^{\varDelta }(H_{\partial \varOmega }^{N}(-\mathrm{i}ku_{H^{2}}^{I} -\partial _{n}u_{H^{2}}^{I}))\Vert _{H^{s}(\varOmega )}\\&\qquad \le q\Vert f\Vert _{H^{s}(\varOmega )}. \end{aligned}$$

From the above estimates follows the bound for \(\Vert u_{H^{s+2}} \Vert _{H^{s+2}(\varOmega )}\). The estimate for \(\widetilde{f}\) follows also from the above observations by noting that we have to set \(\widetilde{f} :=2k^{2}u_{H^{2}}^{II}\) and then suitably adjust \(q\) as in the proof [37, Lemma 4.15]. \(\square \)

Finally, we formulate the analog of [37, Lemma 4.16]:

Lemma 6.5

Assume the hypotheses of Lemma 6.4. Fix \(q\in (0,1)\) and \(s\in \mathbb{N }_{0}\). Then the solution \(u=S_{k}(0,g)\) can be written as \(u=u_{\fancyscript{A}}+u_{H^{s+2}}+\widetilde{u}\), where

$$\begin{aligned} \Vert u_{\fancyscript{A}}\Vert _{H^{1}(\varOmega )}+k\Vert u_{\fancyscript{A}}\Vert _{L^{2}(\varOmega )}&\le Ck^{\vartheta }\Vert g\Vert _{H^{1/2}(\partial \varOmega )},\end{aligned}$$
(6.8)
$$\begin{aligned} \Vert \nabla ^{n+2}u_{\fancyscript{A}}\Vert _{L^{2}(\varOmega )}&\le Ck^{\vartheta -1}\gamma ^{n}\max \{n,k\}^{n+2}\Vert g\Vert _{H^{1/2}(\partial \varOmega )} \quad \forall n\in \mathbb{N }_{0},\nonumber \\\end{aligned}$$
(6.9)
$$\begin{aligned} k^{s+2}\Vert u_{H^{s+2}}\Vert _{L^{2}(\varOmega )}+\Vert u_{H^{s+2}}\Vert _{H^{s+2}(\varOmega )}&\le C\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}, \end{aligned}$$
(6.10)

where the constants \(C\), \(\gamma >0\) are independent of \(k\) and \(n\). The remainder \(\widetilde{u}\) satisfies the boundary value problem

$$\begin{aligned} -\varDelta \widetilde{u}- k^{2}\widetilde{u}= 0\quad \text{ in } \varOmega ,\quad \partial _{n} \widetilde{u}+ \mathrm{i} k\widetilde{u}= \widetilde{g} \quad \text{ on } \partial \varOmega \end{aligned}$$

for data \(\widetilde{g}\in H^{s+1/2}(\partial \varOmega )\) with

$$\begin{aligned} \Vert \widetilde{g}\Vert _{H^{s+1/2}(\partial \varOmega )}\le q\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}. \end{aligned}$$

Proof

The proof follows [37, Lemma 4.16], and we will only discuss (6.10). Again, we mention the sign difference between the boundary condition (1.2) and that studied in [37]. We have to consider, in the notation of [37, Lemma 4.16], the terms

$$\begin{aligned} u_{H^{2}}^{I}&= S_{k}^{\varDelta }(H_{\partial \varOmega }^{N} g),\end{aligned}$$
(6.11)
$$\begin{aligned} u_{H^{2}}^{II}&= N_{k}(H_{\varOmega }(2k^{2}u_{H^{2}}^{I})). \end{aligned}$$
(6.12)

For the term in (6.11), we use Lemma 6.2 to get

$$\begin{aligned} k^{s+2}\Vert u_{H^{2}}^{I}\Vert _{L^{2}(\varOmega )}+\Vert u_{H^{2}}^{I} \Vert _{H^{s+2}(\varOmega )}&\le C\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )},\\ k^{2}\Vert u_{H^{2}}^{I}\Vert _{H^{s}(\varOmega )}&\le q\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}. \end{aligned}$$

For the term in (6.12), we use Lemma 6.3 to arrive at

$$\begin{aligned} k\Vert u_{H^{2}}^{II}\Vert _{H^{s+1}(\varOmega )}+k^{s+2}\Vert u_{H^{2}}^{II} \Vert _{L^{2}(\varOmega )}+\Vert u_{H^{2}}^{II}\Vert _{H^{s+2}(\varOmega )}&\le Ck^{2}\Vert u_{H^{2}}^{I}\Vert _{H^{s}(\varOmega )}\\&\le Cq\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}. \end{aligned}$$

As in the proof of [37, Lemma 4.16], we then set \(\widetilde{g}:=-\mathrm{i}ku_{H^{2}}^{II}-\partial _{n}u_{H^{2}}^{II}\) and use the above estimates to get with the trace inequality

$$\begin{aligned} \Vert \widetilde{g}\Vert _{H^{s+1/2}(\partial \varOmega )}\le C\left[ k\Vert u_{H^{2}}^{II}\Vert _{H^{s+1}(\varOmega )}+\Vert u_{H^{2}}^{II}\Vert _{H^{s+2} (\varOmega )}\right] \le Cq\Vert g\Vert _{H^{s+1/2}(\partial \varOmega )}. \end{aligned}$$

Suitably adjusting the constant \(q\) yields the result. \(\square \)

Appendix 2: \(H^{1}\)-Conforming Approximation

In this appendix we construct an \(H^{1}\)-conforming approximation operator that features optimal rates of convergence not only in \(L^2\) and \(H^1\) but also for the trace and the normal derivative on the element boundaries. This operator can be constructed in an element-by-element fashion. That is, its value at the geometric entities (vertices, edge, faces, elements) is only determined by the function values at these entities. Our construction is closely related to the projection-based interpolation of [11] and the construction in [36, Appendix 2]. In contrast to [36, Appendix 2], where optimal rates in \(L^{2}\) and \(H^{1}\) were sought, we ensure that the optimal rate of convergence for the trace of the gradient is also achieved. We stress that our construction is done with a view to simplicity rather than minimal regularity assumptions.

Definition 7.1

(element-by-element construction in 2D) Let \(\widehat{K}\) be the reference triangle. Let \(s>5/2\). A polynomial \(\pi \) is said to permit an element-by-element construction of boundary polynomial degree \(p \ge 7\) for \(u\in H^{s}(\widehat{K})\) if

  1. (i)

    \(\pi (V)=u(V)\) for all \(d+1\) vertices \(V\) of \(\widehat{K}.\)

  2. (ii)

    For every edge \(e\) of \(\widehat{K}\), the restriction \(\pi |_{e}\in {\fancyscript{P}}_{p}\) is the unique minimizer of

    $$\begin{aligned} \pi \mapsto p^{2}\Vert u-\pi \Vert _{L^{2}(e)}+p \vert u - \pi \vert _{H^{1}(e)} + \vert u-\pi \vert _{H^{2}(e)} \end{aligned}$$
    (7.1)

    under two constraints: first, \(\pi \) satisfies (i) and second, the derivative (along \(e\)) of \(u-\pi \) vanishes in the endpoints of \(e\) (i.e., \((u - \pi )|_e \in H^2_0(e)\)).

Definition 7.2

(element-by-element construction in 3D) Let \(\widehat{K}\) be the reference tetrahedron. Let \(s>5\). A polynomial \(\pi \) is said to permit an element-by-element construction of edge polynomial degree \(p\ge 10\) and face polynomial degree \(2p\) for \(u\in H^{s}(\widehat{K})\) if

  1. (i)

    \(\pi (V)=u(V)\) for all \(d+1\) vertices \(V\) of \(\widehat{K}.\)

  2. (ii)

    For every edge \(e\) of \(\widehat{K}\), the restriction \(\pi |_{e}\in {\fancyscript{P}}_{p}\) is the unique minimizer of

    $$\begin{aligned} \pi \mapsto p^{4} \sum _{j=0}^{4} p^{-j} \vert u-\pi \vert _{H^{j}(e)} \end{aligned}$$
    (7.2)

    under two constraints: first, \(\pi \) satisfies (i) and second, the tangential derivatives (along \(e\)) up to order \(3\) vanish in the endpoints of \(e\) (i.e., \((u-\pi )|_e\in H_{0}^{4}(e)\)).

  3. (iii)

    For every face \(f\) of \(\widehat{K}\), the restriction \(\pi |_{f}\in {\fancyscript{P}}_{2p}\) is the unique minimizer of

    $$\begin{aligned} \pi \mapsto p^{4}\sum _{j=0}^{4}p^{-j}\left| u-\pi \right| _{H^{j}(f)} \end{aligned}$$
    (7.3)

    under two constraints: first, \(\pi \) satisfies (i), (ii) for all vertices and edges of \(f\) and second, the mixed derivatives of \(u-\pi \) vanish in the vertices, i.e., \(\partial _{e_{1}} \partial _{e_{2}}(u-\pi )(V)=0\) for each vertex \(V\) of \(f\), where \(e_{1}\), \(e_{2}\) are two tangential vectors associated with the edges \(e_{1}\), \(e_{2}\) of the face \(f\) that meet in \(V\).

Theorem 7.3

Let \(\widehat{K}\) be the reference triangle or the reference tetrahedron. Set \(V_{p}:=\{v\in {\fancyscript{P}} _{2p}\,|\,v|_{e}\in {\fancyscript{P}}_{p} \text{ for } \text{ all } \text{ edges } e\}\) if \(d=2\) and \(V_{p}:=\{v\in {\fancyscript{P}}_{4p+1}\,|\,v|_{f}\in {\fancyscript{P}}_{2p} \text{ for } \text{ all } \text{ faces } f,v|_{e}\in {\fancyscript{P}}_{p} \text{ for } \text{ all } \text{ edges } e \}\) if \(d=3\). Assume \(s>5/2\) if \(d=2\) and \(s>5\) for \(d=3\). Then, for \(p\ge \max \{10,s-1\}\) for \(d=3\) and \(p\ge \max \{7,s-1\}\) for \(d=2\), there exists a linear operator \(\pi :H^{s}(\widehat{K})\rightarrow V_{p}\) that permits an element-by-element construction in the sense of Definition 7.1 (for \(d=2\)) or Definition 7.2 (for \(d=3\)) such that

$$\begin{aligned} p^{2}\Vert u-\pi (u)\Vert _{L^{2}(\widehat{K})}+p|u-\pi (u)|_{H^{1}(\widehat{K} )}+|u-\pi (u)|_{H^{2}(\widehat{K})}\le C p^{-(s-2)} |u|_{H^{s}(\widehat{K})}.\qquad \quad \end{aligned}$$
(7.4)

The constant \(C>0\) depends only on \(s\).

Proof

We will only present the arguments for the case \(d=3\). We construct \(\pi ( u ) \) directly—inspection of the proof shows that \(u\mapsto \pi \left( u\right) \) is a linear operator. To begin with, we mention that the condition \(p \ge 10\) ensures that an element-by-element construction in the form of Definition 7.2 is feasible: Taking in Lemma 7.13 \(i = 3\) (and the parameter \(p\) there as \(p = i+1 = 4\)) one can find a polynomial of degree \(p^\prime = 2i+p = 10\) that coincides with \(u\) and all its derivatives up to order \(i=3\) in all vertices.

Before actually embarking on the proof, we note a trace estimate that will be required frequently, namely, for any edge \(e\) of the tetrahedron \(\widehat{K} = \widehat{K}^{3D}\), we have for arbitrary but fixed \(t > 1\)

$$\begin{aligned} \Vert v\Vert _{L^2(e)} \le C_t \Vert v\Vert _{L^2(\widehat{K})}^{(t-1)/t} \Vert v\Vert _{H^t(\widehat{K})}^{1/t} \quad \forall v \in H^t(\widehat{K}); \end{aligned}$$
(7.5)

this embedding can be shown with appropriate trace estimates \(e \rightarrow f \rightarrow \widehat{K}\) or by combining the continuity assertion for the trace mapping of [43, Thm. 2.9.3] with interpolation inequalities (cf. also the proof of [36, Lemma B.3] where a similar argument is employed).

From [36, Lemma B.3] we have an approximation \(\pi ^{0}\in {\fancyscript{P}}_{p}\) with

$$\begin{aligned} |u-\pi ^{0}|_{H^{t}(\widehat{K})}\le Cp^{-(s-t)}\Vert u\Vert _{H^{s} (\widehat{K})},\quad t\in \left[ 0,s\right] . \end{aligned}$$
(7.6)

Also, [36, Lemma B.3] gives the following \(L^{\infty }\)-estimate and, by a similar reasoning, also an \(L^{\infty }\)-estimates for the derivatives up to order 3:Footnote 2

$$\begin{aligned} \sum _{j=0}^{3}p^{-j}\Vert \nabla ^{j}(u-\pi ^{0}) \Vert _{L^{\infty }(\widehat{K} )}\le Cp^{-(s-d/2)}\Vert u\Vert _{H^{s}(\widehat{K})}. \end{aligned}$$
(7.7)

Vertex Correction. With the vertex liftings of Lemma 7.13 we can construct a polynomial \(\pi ^{1} \in {\fancyscript{P}}_{p}\) with the following properties:

$$\begin{aligned} |u-\pi ^{1}|_{H^{t}(\widehat{K})}&\le Cp^{-(s-t)}\Vert u\Vert _{H^{s}(\widehat{K})},\quad t\in [0,s],\end{aligned}$$
(7.8)
$$\begin{aligned} D^{\beta }(u-\pi ^{1})(V)&= 0,\qquad \qquad \qquad \qquad \quad \quad 0\le |\beta |\le 3. \end{aligned}$$
(7.9)

To see this, we employ the vertex liftings \(E^{3D}_{V}\) of Remark 7.14. Specifically, we fix a vertex \(V\) and take in Remark 7.14 the parameter \(q = 3\) and the parameter \(p\) there as \(p-6\) to obtain the polynomial

$$\begin{aligned} \widetilde{\pi }_{1}:= \pi _{0} + E^{3D}_{V} (u - \pi _{0}) \in {\fancyscript{P}}_{p}. \end{aligned}$$

By construction in Remark 7.14, the polynomial \(\widetilde{\pi }_{1}\) satisfies \(D^{\beta }(u - \widetilde{\pi }_{1})(V) = 0\) for \(|\beta | \le 3\) and, by (7.37),

$$\begin{aligned}&\Vert \widetilde{\pi }_{1} - u\Vert _{H^{t}(\widehat{K})} \le C \sum _{|\alpha | \le 3} \Vert D^{\alpha }(u - \pi _{0})\Vert _{L^{\infty }(\widehat{K})} p^{-d/2+t} p^{-|\alpha |} + \Vert u-\pi _{0}\Vert _{H^{t}(\widehat{K})}\\&\overset{(7.6), (7.7)}{\le } C p^{-(s-t)}\Vert u\Vert _{H^{s}(\widehat{K})}. \end{aligned}$$

Proceeding in this way for all vertices yields a polynomial \(\pi ^{1} \in {\fancyscript{P}}_{p}\) with the properties (7.8), (7.9).

The condition (7.9) implies in particular that \(u-\pi ^{1}\in H_{0}^{4}(e)\) and \(\nabla (u-\pi ^{1})\in H_{0}^{3}(e)\) for each edge \(e\). With the trace estimates (7.5) we get from (7.8) the following estimates on edges:

$$\begin{aligned} p^{4}\Vert u-\pi ^{1}\Vert _{L^{2}(e)}+\sum _{j=0}^{3}p^{3-j}|\nabla (u-\pi ^{1})|_{H^{j}(e)}\le Cp^{-(s-5)}\Vert u\Vert _{H^{s}(\widehat{K})} \quad \forall \text{ edges } e \text{ of } \widehat{K}.\nonumber \\ \end{aligned}$$
(7.10)

Edge Correction I. Fix an edge \(e\). Since \(\pi ^{1}\) satisfies both side constraints in Definition 7.2.(ii), the minimizer \(\pi _{e}\) of (7.2) satisfies by (7.10)

$$\begin{aligned} p^{4}\sum _{j=0}^{4}p^{-j}|u-\pi _{e}|_{H^{j}(e)}\le p^{4}\sum _{j=0}^{4} p^{-j}|u-\pi ^{1}|_{H^{j}(e)}\le Cp^{-(s-5)}\Vert u\Vert _{H^{s}(\widehat{K} )}. \end{aligned}$$

We note that the difference \(\pi _{e}-\pi ^{1}|_{e}\) is a polynomial of degree \(p\) and \(\partial _{e}^{j}(\pi _{e}-\pi ^{1})\) vanishes at the endpoints of \(e\) for \(j\in \{0,1,2,3\}\), i.e., \(\pi _{e}-\pi ^{1}\in H_{0}^{4}(e)\cap {\fancyscript{P} }_{p}\). By writing \(\pi ^{1}-\pi _{e}=\left( \pi ^{1}-u\right) +\left( u-\pi _{e}\right) \) we obtain with the triangle inequality

$$\begin{aligned} p^{4}\sum _{j=0}^{4}p^{-j}|\pi ^{1}-\pi _{e}|_{H^{j}(e)}\le Cp^{-(s-5)}\Vert u\Vert _{H^{s}(\widehat{K})}. \end{aligned}$$
(7.11)

With the aid of Lemma 7.15 we can find an edge lifting \(L_{e}:=E_{1,e}^{3D}\left( \pi ^{1}-\pi _{e}\right) \in {\fancyscript{P} }_{2p}\) (take as the parameter \(p\) in the statement of Lemma 7.15 the value \(p-1\)) to correct the discrepancy \(\pi ^{1}-\pi _{e}\) with the following propertiesFootnote 3:

$$\begin{aligned} p^{4}\sum _{j=0}^{4}p^{-j}\Vert L_{e}\Vert _{H^{j}(\widehat{K})}&\overset{\text{ Lem. } \text{7.15.(vi), } \text{(vii) } \text{ and } \text{(7.11) }}{\le }Cp^{-\left( s-4\right) }\Vert u\Vert _{H^{s}(\widehat{K})},\\ p^{4}\sum _{j=0}^{4}p^{-j}\Vert L_{e}\Vert _{H^{j}(f)}&\ \ \overset{\text{ Lem. } \text{7.15.(viii) } \text{ and } \text{(7.11) }}{\le }Cp^{-\left( s-4-1/2\right) }\Vert u\Vert _{H^{s}(\widehat{K})} \quad \text{ for } \text{ all } \text{ faces } f,\\ L_{e}&=(\pi ^{1}-\pi _{e})\quad \text{ on } e,\\ L_{e}&=0\quad \text{ on } \text{ all } \text{ other } \text{ edges } \text{ of } \widehat{K},\\ (L_{e})|_{f}&=0\quad \text{ on } \text{ all } \text{ faces } f \text{ that } \text{ have } \text{ not } e \text{ as } \text{ an } \text{ edge, }\\ (\partial _{n_{f}}L_{e})|_{\partial f}&=0\quad \text{ for } \text{ each } \text{ face } f. \end{aligned}$$

With the aid of such a lifting for each edge \(e\), we can find a polynomial \(\pi ^{2}\in {\fancyscript{P}}_{2p}\) with

$$\begin{aligned} p^{4}\sum _{j=0}^{4}p^{-j}\Vert u-\pi ^{2}\Vert _{H^{j}(\widehat{K})}&\le Cp^{4-s}\Vert u\Vert _{H^{s}(\widehat{K})}, p^{4}\sum _{j=0}^{4}p^{-j}\Vert u-\pi ^{2}\Vert _{H^{j}(f)}\end{aligned}$$
(7.12)
$$\begin{aligned}&\le Cp^{1/2}p^{4-s}\Vert u\Vert _{H^{s}(\widehat{K})}\quad \text{ for } \text{ all } \text{ faces } f \end{aligned}$$
(7.13)

and the following two additional properties:

$$\begin{aligned} \pi ^{2}|_{e}=\pi _{e} \quad \text{ for } \text{ all } \text{ edges } e \quad \text{ and } \quad \partial _{n_{f}}\pi ^{2}|_{\partial f}=\partial _{n_{f} }\pi ^{1}|_{\partial f} \quad \text{ for } \text{ each } \text{ face } f. \end{aligned}$$
(7.14)

In other words, \(\pi ^{2}\) satisfies conditions (i), (ii) of Definition 7.2.

Relation to face minimizer \(\pi _{f}\). For a face \(f\), we denote by \(\pi _{f}\) the polynomial that is obtained by the minimizing procedure (7.3). We claim that

$$\begin{aligned} p^{4}\sum _{j=0}^{4}p^{-j}\Vert u-\pi _{f}\Vert _{H^{j}(f)}\le Cp^{-(s-9/2)} \Vert u\Vert _{H^{s}(\widehat{K})}. \end{aligned}$$
(7.15)

To see this, we estimate the error of a modification of \(\pi ^{2}\). An interpolation inequality and estimate (7.12) imply

$$\begin{aligned} \Vert \nabla ^{2}(u-\pi ^{2})\Vert _{L^{\infty }(f)}&\le \Vert \nabla ^{2} (u-\pi ^{2})\Vert _{L^{\infty }(\widehat{K})}\le C\Vert u-\pi ^{2}\Vert _{H^{2}(\widehat{K})}^{1-3/4}\Vert u-\pi ^{2}\Vert _{H^{4}(\widehat{K})} ^{3/4}\nonumber \\&\le Cp^{-1/2+4-s}\Vert u\Vert _{H^{s}(\widehat{K})}. \end{aligned}$$
(7.16)

We note that the polynomial \(\pi ^{2}\) coincides with \(\pi _{e}\) for each edge \(e\) of \(\partial f\). The second order mixed derivatives of \((u-\pi ^{2})|_{f}\) may not vanish at the vertices. This can be corrected with a lifting of Lemma 7.13. Specifically, for each vertex \(V\) Lemma 7.13 provides a lifting \(L_{V}\in {\fancyscript{P} }_{p}\) (take the parameter \(p\) in Lemma 7.13 as \(p-4\)) that vanishes on \(\partial f\) such that the mixed derivative at \(V\) equals 1 and

$$\begin{aligned} p^{4}\sum _{j=0}^{4}p^{-j}\Vert L_{V}\Vert _{H^{j}(f)}\le Cp^{-1+4-2}, \end{aligned}$$

where we used appropriate trace theorems again. Combining this with (7.16), we can construct a function \(\widetilde{\pi }_{f}\) that satisfies all the desired constraints on \(\partial f\) and at the vertices of \(f\) and additionally the estimate

$$\begin{aligned} p^{4}\sum _{j=0}^{4}p^{-j}\Vert u-\widetilde{\pi }_{f}\Vert _{H^{j}(f)}&\le Cp^{4}\sum _{j=0}^{4}p^{-j}\Vert u-\pi ^{2}\Vert _{H^{j}(f)}\!+\!Cp^{-1/2+4-s} p^{-1+4-2}\Vert u\Vert _{H^{s}(\widehat{K})} \\&\le Cp^{1/2+4-s}\Vert u\Vert _{H^{s}(\widehat{K})}, \end{aligned}$$

where we used (7.12) to control \(u-\pi ^{2}\). We conclude for the minimizer \(\pi _{f}\) that (7.15) holds.

Edge Correction II The minimizer \(\pi _{f}\) satisfies

$$\begin{aligned} D_{f}^{\beta }\left( \pi _{f}-u\right) \left( V\right) =0\quad \text{ for } \text{ all } 0\le \left| \beta \right| \le 2 \text{ at } \text{ all } \text{ vertices } V, \end{aligned}$$
(7.17)

where the subscript \(f\) in \(D_{f}^{\beta }\) indicates that differentiation is taken in the plane given by \(f\). The observations (7.14), (7.17), and (7.9) ensure that for each face \(f\) and each edge \(e\) of \(f\), we have \(\partial _{n_{f}}(\pi _{f}-\pi ^{2})|_{e} =\partial _{n_{f}}(\pi _{f}-\pi ^{1})|_{e}=\partial _{n_{f}}(\pi _{f} -u)|_{e}+\partial _{n_{f}}(u-\pi ^{1})|_{e}\in H_{0}^{2}(e)\). With the trace estimate (7.5) we get from (7.15) and (7.12)

$$\begin{aligned}&p^{2}\sum _{j=0}^{2}p^{-j}|\partial _{n_{f}}(\pi ^{2}-\pi _{f})|_{H^{j}(e)}\nonumber \\&\quad \le p^{2}\sum _{j=0}^{2}p^{-j}|\partial _{n_{f}}(\pi ^{2}-u)|_{H^{j}(e)} +p^{2}\sum _{j=0}^{2}p^{-j}|\partial _{n_{f}}(u-\pi _{f})|_{H^{j}(e)}\nonumber \\&\quad \le Cp^{4-s}\Vert u\Vert _{H^{s}(\widehat{K})} . \end{aligned}$$
(7.18)

We are now in position to construct for each face a lifting \(L_{f} \in {\fancyscript{P}}_{3p+1}\) (which is composed of liftings \(E_{2,e}^{3D}\left( \left. \partial _{n_{f}}(\pi ^{2}- \pi _{f})\right| _{e}\right) \) with the lifting operator \(E_{2,e}^{3D}\) of Lemma 7.16 ) with the following properties:

$$\begin{aligned} p^{2}\sum _{j=0}^{2}p^{-j}\Vert L_{f}\Vert _{H^{j}(\widehat{K})}&\le Cp^{2-s}\Vert u\Vert _{H^{s}(\widehat{K})},\end{aligned}$$
(7.19)
$$\begin{aligned} L_{f}&= 0\quad \text{ on } \text{ all } \text{ faces } \text{ except } f,\end{aligned}$$
(7.20)
$$\begin{aligned} \partial _{n_{f}}L_{f}|_{\partial f}&= \partial _{n_{f}}(\pi ^{2}-\pi _{f})|_{\partial f}. \end{aligned}$$
(7.21)

With these liftings, we may adjust \(\pi ^{2}\) to produce a polynomial \(\pi ^{3}\in {\fancyscript{P}}_{3p+1}\) with

$$\begin{aligned} p^{2}\sum _{j=0}^{2}p^{-j}\Vert u-\pi ^{3}\Vert _{H^{j}(\widehat{K})}&\le Cp^{2-s}\Vert u\Vert _{H^{s}(\widehat{K})},\end{aligned}$$
(7.22)
$$\begin{aligned} \pi ^{3}-\pi _{f}&\in H_{0}^{2}(f)\quad \text{ on } \text{ all } \text{ faces } f. \end{aligned}$$
(7.23)

Face Correction. In view of \(\pi ^{3}-\pi _{f}\in H_{0}^{2}(f)\) we may use the final face lifting of Lemma 7.17 to produce a polynomial \(\pi ^{4}\in {\fancyscript{P}}_{4p+1}\) to enforce the desired behavior on the faces. Since \(\left. \pi ^{4}\right| _{f}=\pi _{f}\) for all faces, it satisfies the conditions of Definition 7.2 and additionally

$$\begin{aligned} p^{2}\sum _{j=0}^{2}p^{-j}\Vert u-\pi ^{4}\Vert _{H^{j}(\widehat{K})}\le Cp^{2-s}\Vert u\Vert _{H^{s}(\widehat{K})}. \end{aligned}$$
(7.24)

Volume correction. As a final step, we replace \(\Vert u\Vert _{H^{s}(\widehat{K})}\) on the right-hand side of (7.24) by the seminorm \(|u|_{H^{s} (\widehat{K})}\) with the classical compactness argument due to Deny-Lions. Specifically, we take \(\pi \left( u\right) \in {\fancyscript{P}}_{4p+1}\) as the minimizer of

$$\begin{aligned} v\mapsto p^{2}\sum _{j=0}^{2}p^{-j}\Vert u-v\Vert _{H^{j}(\widehat{K})} \end{aligned}$$

under the constraint that \(v|_{\partial \widehat{K}}=\pi ^{4}|_{\partial \widehat{K}}\). Then \(u\mapsto \pi \left( u\right) \) is a projection on the space \(V_{p}\) (as defined in the theorem) and the full norm \(\Vert u\Vert _{H^{s}(\widehat{K})}\) can be replaced with \(|u|_{H^{s}(\widehat{K})}\) for \(p\ge s-1\). \(\square \)

Corollary 7.4

Let \({\fancyscript{T}}\) be an \(H^1\)-regular mesh in the sense of the beginning of Sect. 4.2.2 and \(S = S^{p,1}({\fancyscript{T}})\) be the space of piecewise mapped polynomials of degree \(p\) on \({\fancyscript{T}}\). Let \(s > 5/2\) for \(d = 2\) and \(s > 5\) for \(d = 3\). Then, for every \(p\ge s-1\) there exists a linear operator \(I:H^{s}(\varOmega )\rightarrow S\cap H^{1}(\varOmega )\) such that for all \(K\in {\fancyscript{T}}\)

$$\begin{aligned}&\left( \frac{h_{K}}{p}\right) ^{2}\Vert \nabla ^2( u-Iu)\Vert _{L^{2}(K)}+\left( \frac{h_{K}}{p}\right) \Vert \nabla (u-Iu)\Vert _{L^{2}(K)}+\Vert u-Iu \Vert _{L^{2}(K)}\\&\quad \le C\left( \frac{h_{K}}{p}\right) ^{s}\Vert u\Vert _{H^{s}(K)}. \end{aligned}$$

Proof

For large \(p\), we use the operator constructed in Theorem 7.3. For example, for \(d = 3\) and \(p^\prime \ge \max \{10,s-1\}\) with \(p^\prime := \lfloor (p-1)/4\rfloor \), we can define \(Iu\) on the reference element \(\widehat{K}\) by taking the operator constructed in Theorem 7.3 (with \(p^\prime \) taking the role of \(p\) there); this yields the desired estimates in \(p\) and the appropriate powers of \(h_K\) arise from scaling arguments (cf. Lemma 4.7). If \(p^\prime < \max \{10,s-1\}\), this corresponds to finitely many possible values of \(p\) and the \(p\)-dependence in the desired estimate is irrelevant. We take \(Iu\) as any standard Lagrange interpolation operator and obtain the required \(h_K\)-dependence again by the scaling arguments of Lemma 4.7. \(\square \)

Lifting Operators

Preliminaries

We start with a convenient definition of the reference triangle \(\widehat{K}^{2D}\) and the reference tetrahedron \(\widehat{K}^{3D}\):

$$\begin{aligned} \widehat{K}^{2D}&:= \{(x,y)\,|\, -1 < x < 1, \quad 0 < y < 1 - |x|\},\end{aligned}$$
(7.25)
$$\begin{aligned} \widehat{K}^{3D}&:= \{(x,y,z)\,|\, -1 < x < 1, \quad 0 < y, \quad 0 < z,\quad 0 < y +z < 1 - |x|\}.\qquad \quad \end{aligned}$$
(7.26)

Below, we will frequently require the following asymptotics of the Beta function \(B\) for \(\alpha > -1\) and \(p \ge 0\) (cf., e.g., [39, Secs. 1.6, 5.1]):

$$\begin{aligned} \int \limits _{0}^{1} x^{\alpha }(1-x)^{p}\,dx = B(\alpha +1,p+1) = \frac{\varGamma (\alpha +1)\varGamma (p+1)}{\varGamma (\alpha +p+2)} \le C_{\alpha }(p+1)^{-1-\alpha }.\qquad \end{aligned}$$
(7.27)

We need a preliminary result that will prove useful for the construction of various vertex liftings:

Lemma 7.5

For \(q\in \mathbb{N }\) define on (0,1) the function \(L_{q}(r):=(1-r)^{q}\). Fix \(i\in \mathbb{N }_{0}\). Then there exists a polynomial \(\pi _{i}\in {\fancyscript{P}}_{i}\) of the form

$$\begin{aligned} \pi _{i}(r)=\sum _{j=0}^{i}\alpha _{j}(qr)^{j} \end{aligned}$$

and a constant \(C_{i}\) (which depends solely on \(i\)) with the following properties:

$$\begin{aligned} |\alpha _{j}|&\le C_{i},\quad j=0,\ldots ,i,\\ (\pi _{i}L_{q})^{(j)}(0)&= {\left\{ \begin{array}{ll} 1 &{} \text{ if } j=0\\ 0 &{} \text{ if } 0 < j\le i. \end{array}\right. } \end{aligned}$$

Furthermore, the polynomial \(\pi _{i}L_{q}\) satisfies, for every \(a\in [0,1]\), \(\alpha \ge 0\), and every \(s\in \mathbb{N }_{0}\)

$$\begin{aligned} \left| \int \limits _{0}^{1-a}|r^{\alpha }(\pi _{i}L_{q})^{(s)}(a+r)|^{2} \,dr\right| \le C_{s,i,\alpha }(1-a)^{2(q-s+\alpha )+1}q^{-1+2s-2\alpha } \sum _{j=0}^{i}(qa)^{2j}.\qquad \quad \end{aligned}$$
(7.28)

The constant \(C_{s,i,\alpha }\) depends only on \(s\), \(\alpha \), and \(i\).

Proof

The polynomials \(\pi _{i}\) can be defined inductively. We take \(\pi _{0}\equiv 1\). For \(\pi _{i+1}\) we make the ansatz \(\pi _{i+1}(r)=\pi _{i}(r)+\alpha _{i+1}r^{i+1}\). This implies for \(0\le m\le i\) that \(\left( \pi _{i+1}L_{q}\right) ^{\left( m\right) } \left( 0\right) =\left( \pi _{i}L_{q}\right) ^{\left( m\right) }\left( 0\right) \). The unknown coefficient \(\alpha _{i+1}\) is then determined by the condition

$$\begin{aligned} 0\overset{!}{=}(\pi _{i+1}L_{q})^{(i+1)}(0)= \sum _{j=0}^{i+1}\left( {\begin{array}{c}i+1\\ j\end{array}}\right) \pi _{i}^{(j)}(0)L_{q}^{(i+1-j)}(0)+\alpha _{i+1}q^{i+1}(i+1)!L_{q}(0). \end{aligned}$$

Since \(L_{q}^{(j)}(0)=(-1)^{j}\left( {\begin{array}{c}q\\ j\end{array}}\right) j!\) we get \(|L_{q}^{(j)}(0)|\le Cq^{j}\) for a constant \(C>0\) independent of \(q\in \mathbb{N }\). In view of \(\pi _{i}^{(j)}(0)=\alpha _{j}q^{j}j!\), the claimed estimate follows for \(\alpha _{i+1}\) by induction and (7.27). We finally show (7.28 ). For simplicity of notation, let \(q\ge s+1\). Since \(r^{\alpha }(\pi _{i}L_{q})^{(s)}\) consists of terms of the form \(\left( \left( 1-r\right) ^{q}(qr)^{j}\right) ^{\left( s\right) }r^{\alpha }\), the product rule shows that it consists of terms of the form \(\left( \left( 1-r\right) ^{q}\right) ^{\left( s-k\right) }\left( (qr)^{j}\right) ^{\left( k\right) }r^{\alpha }\) which can be estimated from above by

$$\begin{aligned} r^{\alpha }\,q^{s-k}q^{j}r^{j-k}(1-r)^{q-(s-k)},\quad 0\le j\le i,\quad 0\le k\le \min \{s,j\}. \end{aligned}$$

With these constraints on \(j\) and \(k\), we estimate with the change of variables \(r=(1-a)\rho \)

$$\begin{aligned} I_{j,k}&:=q^{2s}\int \limits _{r=0}^{1-a}r^{2\alpha }(q(a+r))^{2(j-k)} (1-(a+r))^{2(q-(s-k))}\,dr\\&=q^{2s+2(j-k)}(1\!-\!a)^{2(q-(s-k))+1+2\alpha } \int \limits _{\rho =0}^{1}\!\!\rho ^{2\alpha }(a\!+\!(1\!-\!a)\rho )^{2(j-k)}(1-\rho )^{2(q-(s-k))}\,d\rho \\&\lesssim q^{2s+2(j-k)}(1-a)^{2(q-(s-k))+1+2\alpha }\int \limits _{\rho =0}^{1} \rho ^{2\alpha }(a^{2(j-k)}+\rho ^{2(j-k)})(1\!-\!\rho )^{2(q-(s-k))}\,d\rho \\&\!\!\! \overset{(7.27)}{\lesssim }q^{2s+2(j-k)-1} (1-a)^{2(q-(s-k))+1+2\alpha }\left( q^{-2\alpha }a^{2(j-k)}+q^{-2(j-k)-2\alpha }\right) \\&\lesssim q^{2s-1-2\alpha }(1-a)^{2(q-(s-k))+1+2\alpha }\left( (qa)^{2(j-k)} +1\right) . \end{aligned}$$

Summation over all relevant \(j\), \(k\) gives the stated estimate. \(\square \)

We need a working lemma for the edge liftings in 2D and 3D:

Lemma 7.6

Consider \(\widehat{K}^{2D}\) and its edge \(e=(-1,1)\times \{0\}\). Let \(j\in \{0,1,2,3,4\}\). Let \({\fancyscript{V}}\) be the set of vertices of \(\widehat{K}^{2D}\) and \(d_{\fancyscript{V}}:=\mathrm{dist}(\cdot ,{\fancyscript{V}})\) be the distance from the vertices. Let \(w\in C^{\infty }(\mathbb{R }^{4})\). Let \(\alpha \in \mathbb{N }_{0}\). Then there is \(C>0\) such that for every \(p\ge 0\) the map \(E_{1,e}:H_{0}^{j}(e)\rightarrow H^{j}(\widehat{K}^{2D})\) given by

$$\begin{aligned} (E_{1,e}u)(x,y):=y^{\alpha }w\left( x,y,\frac{y}{1-x},\frac{y}{1+x}\right) (1-y)^{p}u(x) \end{aligned}$$

satisfies:

$$\begin{aligned} |E_{1,e}u|_{H^{j}(\widehat{K}^{2D})}\le C(p+1)^{-\alpha -1/2}\left[ p^{j}\Vert u\Vert _{L^{2}(e)}+p^{j-1}|u|_{H^{1} (e)}+\cdots +p^{0}|u|_{H^{j}(e)}\right] .\nonumber \\ \end{aligned}$$
(7.29)

Furthermore, if \(0\le \alpha \le j\) and \(0\le i\le j\) and additionallyFootnote 4 \(p \ge j\)

$$\begin{aligned} \Vert d_{\fancyscript{V}}^{-(j-i)}\nabla ^{i}E_{1,e}u\Vert _{L^{2}(e^{\prime })}\le C(p+1)^{-\alpha }\left[ |u|_{H^{j}(e)}+p|u|_{H^{j-1} (e)}+\cdots +p^{j}\Vert u\Vert _{L^{2}(e)}\right] \nonumber \\ \end{aligned}$$
(7.30)

for any simplex edge \(e^{\prime }\). In particular, therefore, \(E_{1,e}u\in H_{0}^{j}(e^{\prime })\) for every edge if \(p \ge j\).

Proof

We start with the proof of (7.29). Without explicitly stating it below, we will assume that \(p\) is sufficiently large (specifically, \(p\ge 2\)). For the case \(j=0\), (7.29) follows from the observation that \(w\) is a bounded function on \(\widehat{K}^{2D}\) since \(y/(1-|x|)\le 1\) on \(\widehat{K}^{2D}\) and the estimate (7.27). For the cases \(j\ge 1\), we have to control the derivatives. We use \(0<y<1-|x|\) and the smoothness of \(w\) to estimate

$$\begin{aligned} |D^{\beta }w|&\le C(1-|x|)^{-|\beta |},\quad (x,y)\in \widehat{K} ^{2D},\end{aligned}$$
(7.31)
$$\begin{aligned} |D^{\beta }(y^{\alpha }(1-y)^{p})|&\le Cp^{|\beta |-\alpha }(1-y)^{p-|\beta |}(1+(yp)^{\alpha }), \quad (x,y)\in \widehat{K}^{2D}, \end{aligned}$$
(7.32)

for arbitrary multiindices \(\beta \in \mathbb{N }_{0}^{2}\) and \(p\ge |\beta |\). Recall that \(i\mapsto a^{i}\) is convex for \(i\in \mathbb{N }_{0}\) and \(a>0\). From the product rule, we therefore infer for fixed \(\beta \in \mathbb{N } _{0}^{2}\) and \(p\ge |\beta |\)

$$\begin{aligned} |D^{\beta }(y^{\alpha }w\,(1-y)^{p})|&\le C\left[ ((1-|x|)^{-|\beta |} (1-y)^{|\beta |} +p^{\beta }) p^{-\alpha } (1 + (p y)^\alpha )\right] (1-y)^{p-|\beta |}\nonumber \\&\le C\left[ ((1-|x|)^{-|\beta |} +p^{\beta }) (p^{-\alpha } + y^\alpha )\right] (1-y)^{p-|\beta |}. \end{aligned}$$
(7.33)

(7.33) thus allows us to control the derivatives of the function \(W\) defined as

$$\begin{aligned} W(x,y):=y^{\alpha }w(1-y)^{p}. \end{aligned}$$

We now consider the case \(j=1\) and \(|\beta |=1\). Then Lemma 7.8 gives

$$\begin{aligned}&\int \limits _{x=-1}^{1}\int \limits _{y=0}^{1-|x|}\left( u(x)D^{\beta }W\right) ^{2}+\left( \partial _{x}u(x)W\right) ^{2}\,dy\,dx\\&\quad \le C\left( p^{-2\alpha -1}\Vert \frac{1}{1-x}u\Vert _{L^{2}(e)}^{2} +p^{2}p^{-2\alpha -1}\Vert u\Vert _{L^{2}(e)}^{2}+p^{-2\alpha -1}\Vert \partial _{x}u\Vert _{L^{2}(e)}^{2}\right) \\&\quad \le Cp^{-2\alpha -1}\Vert \partial _{x}u\Vert _{L^{2}(e)}^{2}+p^{2} p^{-2\alpha -1}\Vert u\Vert _{L^{2}(e)}^{2}, \end{aligned}$$

where, in the last step, we employed the Hardy inequality of Lemma 7.7 (with \(\beta =-2\) there). We now consider \(j=2\). Then, we have to bound \(\Vert uD^{\beta }W\Vert _{L^{2}(\widehat{K}^{2D})}\) for \(|\beta |=2\), \(\Vert \partial _{x}uD^{\beta }W\Vert _{L^{2}(\widehat{K}^{2D})}\) for \(|\beta |=1\) and \(\Vert \partial _{x}^{2}uW\Vert _{L^{2}(\widehat{K}^{2D})}\). Writing \(D^{2}W\) and \(D^{1}W\) for the sum of all derivatives of order \(2\) and \(1\), respectively, we estimate

$$\begin{aligned} \Vert \partial _{x}^{2}uW\Vert _{L^{2}(\widehat{K}^{2D})}^{2}&\le Cp^{-2\alpha -1}|u|_{H^{2}(e)}^{2},\\ \Vert \partial _{x}uD^{1}W\Vert _{L^{2}(\widehat{K}^{2D})}^{2}&\le C\left( p^{-2\alpha -1}\Vert \frac{1}{1-x}\partial _{x}u\Vert _{L^{2}(e)}^{2} +p^{2}p^{-2\alpha -1}\Vert \partial _{x}u\Vert _{L^{2}(e)}^{2}\right) \\&\le C\left( p^{-2\alpha -1}\Vert \partial _{x}^{2}u\Vert _{L^{2}(e)} ^{2}+p^{2}p^{-2\alpha -1}\Vert \partial _{x}u\Vert _{L^{2}(e)}^{2}\right) , \end{aligned}$$

where, in the last step, we used again the Hardy inequality of Lemma 7.7 with the assumption \(\partial _{x}u(1)=0\). Estimating \(uD^{2}W\) requires us to control

$$\begin{aligned}&\int \limits _{x=-1}^{1}\int \limits _{y=0}^{1-\left| x\right| }|u(x)|^{2} (1-|x|)^{-4}\left( p^{-\alpha }+y^{\alpha }\right) ^{2}(1-y)^{2p-4} \,dy\,dx\quad \text{ and } \\&\int \limits _{x=-1}^{1}\int \limits _{y=0}^{1-\left| x\right| }|u(x)|^{2} p^{4}\left( p^{-\alpha }+y^{\alpha }\right) ^{2}(1-y)^{2p-4}\,dy\,dx. \end{aligned}$$

The second term is readily bounded by \(p^{4-2\alpha -1}\Vert u\Vert _{L^{2} (e)}^{2}\). For the first term, an application of Lemma 7.8 yields

$$\begin{aligned} p^{-2\alpha -1}\Vert \frac{1}{(1-|x|)^{2}}u\Vert _{L^{2}(e)}^{2}. \end{aligned}$$

A two-fold application of the Hardy inequality Lemma 7.7 yields \(\Vert 1/(1-|x|)^{2}u\Vert _{L^{2}(e)}\le C\Vert \partial _{x}^{2}u\Vert _{L^{2}(e)}^{2}\), which is the desired estimate. The cases \(j=3\), 4 are shown with similar arguments.

For the estimate (7.30), we argue in a similar way. We focus on the case \(e^{\prime }\ne e\), the case \(e^{\prime }=e\) being slightly simpler. The assumption \(p \ge j\) implies that \(E_{1,e} u\) vanishes to higher order at the vertex (0,1). We may therefore concentrate on the behavior of \(E_{1,e} u\) at the vertices (\(-\)1,0) and (1,0). For example, for \(i=0\) we have to estimate terms of the following form (the contribution \((1-y)^{p}\) is generously estimated by 1) in view of (7.33):

$$\begin{aligned} \int \limits _{x=-1}^{1}u^{2}(x)\left( 1-|x|\right) ^{-2j} \left( p^{-\alpha }+(1-|x|)^{\alpha }\right) ^{2}\,dx, \end{aligned}$$
(7.34)

where we observed that the factor \(y^{\alpha }\) arising in (7.33) is changed into \((1-|x|)\) due to the parametrization of \(e^{\prime }\). The integral (7.34) can then be treated with the Hardy inequality of Lemma 7.7. \(\square \)

From [43, Rem. 1, Sec. 3.2.6] we have the following variant of Hardy’s inequality:

Lemma 7.7

(Hardy inequality) For \(\beta <-1\) and \(\varphi \in C_{0}^{\infty }(0,1)\)

$$\begin{aligned} \int \limits _{x=0}^{1}|\varphi (x)|^{2}x^{\beta }\,dx\le \left( \frac{2}{|\beta +1|}\right) ^{2}\int \limits _{x=0}^{1}x^{\beta +2}\left| \varphi ^{\prime }(x)\right| ^{2}\,dx. \end{aligned}$$

Lemma 7.8

Fix \(\alpha \ge 0\) and \(\beta \in \mathbb{R }\) with \(\alpha +\beta \ge 0\). There is some \(C>0\) independent of \(x\in (0,1)\) and \(p\ge 0\) such that

$$\begin{aligned} \int \limits _{y=0}^{1-x}\left( \left( \frac{y}{1-x}\right) ^{\alpha }y^{\beta }(1-y)^{p}\right) ^{2}\,dy&\le C\left( \min \{1-x,p^{-1}\}\right) ^{1+2\beta },\end{aligned}$$
(7.35)
$$\begin{aligned} \int \limits _{y=0}^{1-x}\left( \frac{y^{\alpha }}{(1-x)^{\alpha +1/2}}(1-y)^{p}\right) ^{2}\,dy&\le C. \end{aligned}$$
(7.36)

Proof

We may assume \(p\ge 2\). Both estimates follow by distinguishing between the cases \(x<1-1/p\) and \(1-1/p<x<1\); in the latter case, we use additionally (7.27). \(\square \)

Lemma 7.9

Let \(f \in L^{1}(\widehat{K}^{2D})\). Then

$$\begin{aligned} \int \limits _{\widehat{K}^{3D}} f(x,y+z)\,dx\,dy\,dz&= \int \limits _{\widehat{K}^{2D}} y f(x,y)\,dy\,dx,\\ \int \limits _{\widehat{K}^{3D}} y f(x,y+z)\,dx\,dy\,dz&= \int \limits _{\widehat{K}^{2D}} \frac{1}{2} y^{2} f(x,y)\,dy\,dx. \end{aligned}$$

Proof

Follows from an appropriate application of Fubini’s theorem. \(\square \)

Liftings for the 2D Case

We start with vertex liftings that allow us to match the Taylor expansion in the vertices to any desired order.

Lemma 7.10

(vertex liftings in 2D) Fix \(i \in \mathbb{N }_{0}\) and a vertex \(V\) of \(\widehat{K}^{2D}\). Denote by \(e_{1}\), \(e_{2}\) the two edges meeting at \(V\) and by \(\partial _{e_{1}}\), \(\partial _{e_{2}}\) differentiation along \(e_{1}\), \(e_{2}\). Fix \((i_{1},i_{2}) \in \mathbb{N } _{0}^{2}\) with \(i_{1} + i_{2} \le i\). Then for \(p \ge i+1\) one can find polynomials \(L_{V,(i_{1},i_{2}),p} \in {\fancyscript{P}}_{p+2i}\) with

$$\begin{aligned} \partial _{e_{1}}^{j_{1}} \partial _{e_{2}}^{j_{2}} L_{V,(i_{1},i_{2}),p}(V)&= \delta _{i_{1},j_{1}} \delta _{i_{2},j_{2}} \quad \forall (j_{1},j_{2} )\in \mathbb{N }_{0}^{2} \text{ with } j_{1} + j_{2} \le i,\\ \nabla ^{j} L_{V,(i_{1},i_{2}),p}(V^{\prime })&= 0 \quad \forall 0 \le j \le i, \quad \forall \text{ vertices } V^\prime \ne V. \end{aligned}$$

Furthermore \(L_{V,(i_{1},i_{2}),p}\) vanishes on the edge opposite \(V\) and for every \(s \ge 0\), one has for a constant \(C_{s} > 0\) independent of \(p\) (but depending on \(s\) and \(i\))

$$\begin{aligned} \Vert L_{V,(i_{1},i_{2}),p}\Vert _{H^{s}(\widehat{K}^{2D})} \le C_{s} p^{-1+s - (i_{1}+i_{2})}. \end{aligned}$$

Proof

It is convenient to work with the reference triangle

$$\begin{aligned} \widetilde{K}^{2D}:=\{(x,y)\,|\,0<x<1,0<y<1-x\}. \end{aligned}$$

Let \(L_{1,p}\in {\fancyscript{P}}_{p+i}\) be the univariate polynomial given by Lemma 7.5 with the property \(L_{1,p} ^{(j)}(0)=\delta _{j,0}\) for \(j=0,\ldots ,i\) and \(L_{1,p}^{(j)}(1) = 0\) for \(j=0,\ldots ,p-1\). Set

$$\begin{aligned} L_{V,(i_{1},i_{2}),p}(x,y):= \frac{1}{i_{1}!}\frac{1}{i_{2}!}x^{i_{1}}y^{i_{2} }L_{1,p}(x+y)\in {\fancyscript{P}}_{p+i_{1}+i_{2}+i}. \end{aligned}$$

Since \(L_{1,p}(0)=1\) and \(L_{1,p}^{(j)}(0)=0\) for \(j=1,\ldots ,i\) and \(L_{1,p}^{(j)}(1)=0\) for \(j=0,\ldots ,p-1\ge i\), we see that \(L_{V,p}\) has the desired properties in the vertices of \(\widetilde{K}^{2D}\). To see the norm bounds, we consider \((s_{1},s_{2})\in \mathbb{N }_{0}^{2}\) with \(s_{1} +s_{2}=s\). Then, by the product rule, \(D^{(s_{1},s_{2})}L_{V,(i_{1},i_{2}),p}\) consist of terms of the form

$$\begin{aligned} x^{i_{1}-k_{1}}y^{i_{2}-k_{2}}L_{1,p}^{(s_{1}+s_{2}-k_{1}-k_{2})} (x+y),\quad 0\le k_{1}\le \min \{i_{1},s_{1}\},\quad 0\le k_{2}\le \min \{i_{2},s_{2}\}. \end{aligned}$$

Hence, we have to bound

$$\begin{aligned} I_{k_{1},k_{2}}:=\int \limits _{x=0}^{1}x^{2(i_{1}-k_{1})}\int \limits _{y=0}^{1-x} y^{2(i_{2}-k_{2})}|L_{1,p}^{(s_{1}+s_{2}-k_{1}-k_{2})}(x+y)|^{2}\, dy\,dx. \end{aligned}$$

With the aid of Lemma 7.5 in the first step and (7.27) in the second one, we get

$$\begin{aligned} I_{k_{1},k_{2}}&\lesssim \sum _{j=0}^{i}\int \limits _{x=0}^{1}\!\! x^{2(i_{1}-k_{1} )}p^{-1-2(i_{2}-k_{2})+ 2(s_{1}+s_{2}-k_{1}-k_{2})}(1\!-\!x)^{2(p-(s_{1} +s_{2}-k_{1}-k_{2}+i_{2}-k_{2}))+1}\left( xp\right) ^{2j}\\&\lesssim \sum _{j=0}^{i}p^{-2(i_{1}-k_{1})-1}p^{-1-2i_{2}+2s_{1} +2s_{2}-2k_{1}}\lesssim p^{2(s_{1}-i_{1}+s_{2}-i_{2})-2}=p^{2(s-i_{1} -i_{2})-2}, \end{aligned}$$

which implies the desired estimate. \(\square \)

Lemma 7.11

(edge liftings in 2D) For every edge of \(\widehat{K}^{2D}\) and \(j \ge 1\) and \(p \in \mathbb{N }\) there is a bounded linear operator \(E_{1,e}^{2D}:L^{2}(e)\rightarrow L^{2}(\widehat{K}^{2D})\) with the following properties with a \(C>0\) independent of \(p\) and \(u\):

  1. (i)

    \(\Vert E_{1,e}^{2D}u\Vert _{L^{2}(\widehat{K}^{2D})}\le Cp^{-1/2}\Vert u\Vert _{L^{2}(e)}\).

  2. (ii)

    \(|E_{1,e}^{2D}u|_{H^{k}(\widehat{K}^{2D} )}\!\le \! Cp^{{-}1/2}\left[ \! p^{k}\Vert u\Vert _{L^{2}(e)}{+}p^{k{-}1}\!\Vert \nabla _{e}u\Vert _{L^{2}(e)}{+}\cdots {+}|u|_{H^{k}(e)}\!\right] \! \text{ if }\,\, \text{ additionally } \) \(u \in H^k_0(e)\).

Additionally, \(E_{1,e}^{2D} u\) has a trace on \(\partial \widehat{K}^{2D}\) and

  1. (iii)

    \((E_{1,e}^{2D}u)|_{e}=u.\)

  2. (iv)

    \((E_{1,e}^{2D}u)|_{\partial \widehat{K} ^{2D}\setminus e}=0\).

Furthermore, if \(u \in H^j_0(e)\), then

  1. (v)

    \(\forall u\in {\fancyscript{P}}_{q}\cap H_{0} ^{j}(e):E_{1,e}^{2D}u\in {\fancyscript{P}}_{p+q}\).

  2. (vi)

    \((\nabla ^{k} E_{1,e}^{2D}u)|_{\partial \widehat{K}^{2D}\setminus e}=0\),    \(k=0,\ldots ,j-1\).

Proof

We consider the edge \(e=\{(x,y)\,|\,y=0\}\). The edge lifting for \(e\) is taken to be

$$\begin{aligned} (E_{1,e}^{2D}u)(x,y)&:= u(x)\frac{1}{(1-x^{2})^{j}}(1-x-y)^{j} (1+x-y)^{j}(1-y)^{p} \\&= u(x)\left( 1-\frac{y}{1-x}\right) ^{j}\left( 1-\frac{y}{1+x}\right) ^{j}(1-y)^{p}. \end{aligned}$$

Lemma 7.6 implies the norm bounds stated in (i), (ii), since \(E_{1,e}^{2D}u\) has the form studied there. The properties concerning the traces and derivatives on \(\partial \widehat{K}^{2D}\) given in (iii)—(vi) follow by inspection (and \(j > 0\)). \(\square \)

The following result is a variation of Lemma 7.11 and will be required for the 3D situation.

Lemma 7.12

Let \({\fancyscript{V}}\) be the vertices of \(\widehat{K}^{2D}\) and \(d_{\fancyscript{V}}:= \mathrm{dist}(\cdot ,{\fancyscript{V}})\). Then, for every edge \(e\) of \(\widehat{K}^{2D}\) and \(p \in \mathbb{N }\) there is a bounded linear operator \(E_{1,e}:L^{2}(e)\rightarrow L^{2}(\widehat{K}^{2D})\) with the following properties:

  1. (i)

    \(\Vert E_{1,e}u\Vert _{L^{2}(\widehat{K}^{2D})}\le Cp^{-1/2}\Vert u\Vert _{L^{2}(e)}\).

  2. (ii)

    \(|E_{1,e} u|_{H^{j}(\widehat{K}^{2D})}\le Cp^{-1/2}p^{j}\sum _{\ell =0}^{j}p^{-\ell }\Vert u\Vert _{H^{\ell }(e)}\)    if \(u \in H^j_0(e)\), \(j \ge 0\).

  3. (iii)

    If \(u \in H^j_0(e)\) for a \(j \ge 1\), then \(E_{1,e}u|_{e^{\prime }}\in H_{0} ^{j}(e^{\prime })\) for every edge \(e^{\prime }\) of \(\widehat{K}^{2D}\) and in fact, for \(0\le i\le j \le p\),

    $$\begin{aligned} \Vert d_{\fancyscript{V}}^{-(j-i)}\nabla ^{i}E_{1,e} u\Vert _{L^{2}(e^{\prime })}\le C p^{j} \sum _{k=0}^{j} p^{-k} |u|_{H^{k}(e)}. \end{aligned}$$

In the above estimates, the constant \(C>0\) is independent of \(u\) and \(p\). Additionally, if \(u \in H^3_0(e)\), then

  1. (iv)

    \(\forall u\in {\fancyscript{P}}_{q}\cap H_{0}^{3}(e):E_{1,e}u\in {\fancyscript{P}}_{p+q+1}\).

  2. (v)

    \((E_{1,e}u)|_{\partial \widehat{K} ^{2D}\setminus e}=0\).

  3. (vi)

    \((\nabla E_{1,e}u)|_{\partial \widehat{K}^{2D}\setminus e}=0\).

  4. (vii)

    \((E_{1,e}u)|_{e}=u\).

  5. (viii)

    \((\partial _{n}E_{1,e}u)|_{e}=0\).

Proof

We modify the operator \(E_{1,e}^{2D}\) of Lemma 7.11 slightly and set

$$\begin{aligned} (E_{1,e}u)(x,y)&:= u(x)\frac{1}{(1-x^{2})^{2}}(1-x-y)^{2}(1+x-y)^{2}\left( 1+py+y\frac{4}{1-x^{2}}\right) (1-y)^{p}\\&= u(x)\left( 1-\frac{y}{1-x}\right) ^{2}\left( 1-\frac{y}{1+x}\right) ^{2}\left( 1+py+y\frac{4}{1-x^{2}}\right) (1-y)^{p}. \end{aligned}$$

The control of \(|E_{1,e}u|_{H^{j}(\widehat{K}^{2D})}\) stated in (i), (ii) follows from Lemma 7.6 by observing that \(2y/(1-x^2) = y/(1-x) + y/(1+x)\) so that \(E_{1,e}u=W_{1}u+pyW_{2}u\) with functions \(W_{1}\), \(W_{2}\) of the form studied in Lemma 7.6. Likewise, the bounds given in (iii) on edges \(e^{\prime }\) follow from Lemma 7.6 and the special form \(E_{1,e}u=W_{1}u+pyW_{2}u\). (In fact, the condition \(p\ge j\) on the degree \(p\) is not completely sharp.) The properties (iv)–(vii) result from the factor \((1-y/(1-x))^2 (1-y/(1+x))^2\). The property \((\partial _{n}E_{1,e}u)|_{e}=0\) is a consequence of the factor \(1+py+4y/(1-x^{2})\). \(\square \)

Liftings for the 3D Case

We start with the vertex liftings:

Lemma 7.13

(vertex liftings in 3D) Fix \(i\in \mathbb{N }_{0}\) and a vertex \(V\) of \(\widehat{K}^{3D}\). Denote by \(e_{1}\), \(e_{2}\), \(e_{3}\) the three edges meeting at \(V\) and by \(\partial _{e_{k}}\), differentiation along \(e_{k}\). Fix \((i_{1},i_{2},i_{3})\in \mathbb{N }_{0} ^{3}\) with \(i_{1}+i_{2}+i_{3}\le i\). Then one can find, for every \(p\ge i+1\), a polynomial \(L_{V,(i_{1},i_{2},i_{3}),p}\in {\fancyscript{P}}_{p+2i}\) with

$$\begin{aligned} \partial _{e_{1}}^{j_{1}}\partial _{e_{2}}^{j_{2}}\partial _{e_{3}}^{j_{3} }L_{V,(i_{1},i_{2},i_{3}),p}(V)&= \delta _{i_{1},j_{1}} \delta _{i_{2},j_{2}}\delta _{i_{3},j_{3}}\quad \forall (j_{1},j_{2},j_{3})\in \mathbb{N }_{0} ^{3} \text{ with } j_{1}+j_{2}+j_{3}\le i,\\ \nabla ^{j}L_{V,(i_{1},i_{2},i_{3}),p}(V^{\prime })&= 0\quad \forall 0\le j\le i,\quad \forall \text{ vertices } V^\prime \ne V. \end{aligned}$$

Furthermore, \(L_{V,(i_{1},i_{2},i_{3}),p}\) vanishes on the face opposite \(V\). Additionally, for every \(s\ge 0\), one has for a constant \(C_{s}>0\) independent of \(p\) (but depending on \(s\) and \(i\))

$$\begin{aligned} \Vert L_{V,(i_{1},i_{2},i_{3}),p}\Vert _{H^{s}(\widehat{K}^{3D})}\le C_{s}p^{-3/2+s-(i_{1}+i_{2}+i_{3})}. \end{aligned}$$

Proof

The proof parallels that of the 2D-version detailed in Lemma 7.10. It is convenient to work with the reference tetrahedron

$$\begin{aligned} \widetilde{K}^{3D}:=\{(x,y,z)\,|\,0<x<1,0<y<1-x,0<z<1-x-y\}. \end{aligned}$$

Let \(L_{1,p}\in {\fancyscript{P}}_{p+i}\) be the univariate polynomial given by Lemma 7.5 with \(L_{1,p}^{(j)}(0)=\delta _{j,0}\), \(j=0,\ldots ,i\) and \(L_{1,p}^{(j)}(1) = 0\) for \(j=0,\ldots ,p-1\). Set

$$\begin{aligned} L_{V,(i_{1},i_{2},i_{3}),p}(x,y,z):= \frac{1}{i_{1}!}\frac{1}{i_{2}!}\frac{1}{i_{3}!}x^{i_{1}}y^{i_{2}}z^{i_{3}}L_{1,p}(x+y+z)\in {\fancyscript{P}} _{p+i_{1}+i_{2}+i_{3}}. \end{aligned}$$

Since \(L_{1,p}(0)=1\) and \(L_{1,p}^{(j)}(0)=0\) for \(j=1,\ldots ,i\) and \(L_{1,p}^{(j)}(1)=0\) for \(j=0,\ldots ,p-1\ge i\), we see that \(L_{V,p}\) has the desired properties in the vertices of \(\widetilde{K}^{3D}\). To see the norm bounds, we consider a \((s_{1},s_{2},s_{3})\in \mathbb{N }_{0}^{3}\) with \(s_{1}+s_{2}+s_{3}=s\). Then, by the product rule, \(D^{(s_{1},s_{2},s_{3} )}L_{V,(i_{1},i_{2},i_{3}),p}\) consist of terms of the form

$$\begin{aligned} x^{i_{1}-k_{1}}y^{i_{2}-k_{2}}z^{i_{3}-k_{3}}L_{1,p}^{(s_{1}+ s_{2}+s_{3} -k_{1}-k_{2}-k_{3})}(x+y+z) \end{aligned}$$

where \((k_{1},k_{2},k_{3})\in \mathbb{N }_{0}^{3}\) is constrained to satisfy \(0\le k_{1}\le \min \{i_{1},s_{1}\}\), \(0\le k_{2}\le \min \{i_{2},s_{2}\}\), \(0\le k_{3}\le \min \{i_{3},s_{3}\}\). Hence, we have to bound

$$\begin{aligned}&I_{k_{1},k_{2},k_{3}}\\&\quad :=\int \limits _{x=0}^{1}\!x^{2(i_{1}-k_{1})}\int \limits _{y=0} ^{1-x}\!y^{2(i_{2}-k_{2})} \int \limits _{z=0}^{1-x-y}\!z^{2(i_{3}-k_{3})}|L_{1,p} ^{(s_{1}+s_{2}+s_{3}-k_{1}-k_{2}-k_{3})}(x+y+z)|^{2}\,dz\,dy\,dx. \end{aligned}$$

Abbreviating \(s=s_{1}+s_{2}+s_{3}\) and \(k=k_{1}+k_{2}+k_{3}\) we get with the aid of Lemma 7.5

$$\begin{aligned}&I_{k_{1},k_{2},k_{3}}\lesssim \sum _{j=0}^{i}p^{-1-2(i_{3}-k_{3})+2(s-k)} \\&\quad \int \limits _{x=0}^{1}x^{2(i_{1}-k_{1})}\int \limits _{y=0}^{1-x}y^{2(i_{2}-k_{2} )}(1-(x+y))^{2(p-(s-k+i_{3}-k_{3}))+1}\left( (x+y)p\right) ^{2j}. \end{aligned}$$

For the innermost integral, we use the change of variables \(y=(1-x)\eta \) and get in view of (7.27)

$$\begin{aligned} \int \limits _{y=0}^{1-x}&= (1-x)^{2+2(i_{2}-k_{2})+2(p-(s-k+i_{3}-k_{3}))}\\&\quad \int \limits _{\eta =0}^{1}\eta ^{2(i_{2}-k_{2})}(1-\eta )^{2(p-(s-k+i_{3}-k_{3} ))+1}((x+(1-x)\eta )p)^{2j}\\&\lesssim (1-x)^{2+2(i_{2}-k_{2})+2(p-(s-k+i_{3}-k_{3}))}p^{-2(i_{2} -k_{2})-1}p^{2j}\left[ x^{2j}+p^{-2j}\right] . \end{aligned}$$

Thus, we get

$$\begin{aligned} I_{k_{1},k_{2},k_{3}}&\lesssim \sum _{j=0}^{i}p^{-1-2(i_{3}-k_{3} )+2(s-k)}p^{-1-2(i_{2}-k_{2})}\\&\quad \int \limits _{x=0}^{1}x^{2(i_{1}-k_{1})}(1-x)^{2+2(i_{2} -k_{2})+2(p-(s-k+i_{3}-k_{3}))}(px)^{2j}\\&\lesssim \sum _{j=0}^{i}p^{-1-2(i_{3}-k_{3})+ 2(s-k)}p^{-1-2(i_{2}-k_{2} )}p^{-1-2(i_{1}-k_{1})}\lesssim p^{-3-2(i_{1}+i_{2}+i_{3})+2s}, \end{aligned}$$

which is the claimed estimate. \(\square \)

Remark 7.14

(vertex liftings matching to finite order) Let \(s>0\) and \(q\in \mathbb{N }_{0}\) such that the embedding theorem \(H^{s}(\widehat{K}^{3D})\subset C^{q}(\overline{\widehat{K}})\) is valid. Define, for \(p\ge q+1\), with the aid of the functions of \(L_{V,(i_{1},i_{2},i_{3}),p}\) of Lemma 7.13 the operator

$$\begin{aligned} E_{V}^{3D}u:=\sum _{\alpha \in \mathbb{N }_{0}^{3}:|\alpha |\le q}\frac{1}{\alpha !}D^{\alpha }u(V)L_{V,\alpha ,p}. \end{aligned}$$

Then \(E_{V}^{3D}u\in {\fancyscript{P}}_{p+2q}\). Furthermore \(D^{\beta }(u-E_{V}^{3D}u)(V)=0\) for all \(|\beta |\le q\) and \((D^{\beta }E_{V} ^{3D}u)(V^{\prime })=0\) for all \(|\beta |\le q\) and vertices \(V^{\prime }\ne V\), and \(E_{V}^{3D}u\) vanishes on the face opposite \(V\). Additionally, for \(t\ge 0\), we have

$$\begin{aligned} \Vert E_{V}^{3D}u\Vert _{H^{t}(\widehat{K}^{3D})}\le C_{t}\sum _{|\alpha |\le q}|D^{\alpha }u(V)|p^{-|\alpha |}p^{-3/2+t}. \end{aligned}$$
(7.37)

For the following lemmas, we recall our notion of face normal derivative operator \(\partial _{n_f}\): For a face \(f\) of \(\widehat{K}^{3D}\) with boundary \(\partial f\), we denote by \(\partial _{n_f} v = n_f \cdot \nabla v\), where \(n_f\) is the vector of length \(1\) normal to \(\partial f\) in the plane spanned by \(f\).

Lemma 7.15

(edge trace lifting) For each edge \(e\) of \(\widehat{K}^{3D}\) denote by \(f_{1,e}\), \(f_{2,e}\) the two faces sharing \(e\). There is a lifting operator \(E_{1,e}^{3D}:H_{0}^{3}(e)\rightarrow H^{3}(\widehat{K}^{3D})\) with the following lifting properties:

  1. (i)

    \((E_{1,e}^{3D}u)|_{e}=u\).

  2. (ii)

    \(E_{1,e}^{3D}u\) vanishes on all faces that do not have \(e\) as an edge.

  3. (iii)

    \(E_{1,e}^{3D}u\) as well as \(\nabla E_{1,e}^{3D}u\) vanish on all edges except \(e\).

  4. (iv)

    For each of the two faces \(f_{1,e}\), \(f_{2,e}\), the face normal derivative of \(E_{1,e}^{3D}u\) vanish on \(e\), i.e.,

    $$\begin{aligned} (\partial _{n_{f_{i,e}}} E_{1,e}^{3D} u)|_e = 0\; { for}\; i=1, 2. \end{aligned}$$
  5. (v)

    If \(u\in {\fancyscript{P}}_{q}\cap H_{0}^{3}(e)\), then \(E_{1,e} ^{3D}u\in {\fancyscript{P}}_{q+p+1}\).

For each fixed \(j \ge 0\), the following stability bounds are valid:

  1. (vi)

    \(\Vert E_{1,e}^{3D}u\Vert _{L^{2}(\widehat{K}^{3D})}\le Cp^{-1}\Vert u\Vert _{L^{2}(e)}\).

  2. (vii)

    If \(u \in H^j_0(e)\), then \(|E_{1,e}^{3D}u|_{H^{j}(\widehat{K}^{3D})}\le Cp^{-1}[ p^{j}\Vert u\Vert _{L^{2}(e)}+p^{j-1}|u|_{H^{1}(e)}+\cdots +|u|_{H^{j} (e)}] \).

  3. (viii)

    If \(u \in H^j_0(e)\), then for the faces \(f_{i,e}\), \(i\in \{1,2\}\),

    $$\begin{aligned} |E_{1,e}^{3D}u|_{L^{2}(f_{i,e})}&\le Cp^{-1/2}\Vert u\Vert _{L^{2}(e)},\\ |E_{1,e}^{3D}u|_{H^{j}(f_{i,e})}&\le Cp^{-1/2}\left[ p^{j}\Vert u\Vert _{L^{2}(e)}+p^{j-1}|u|_{H^{1}(e)}+\cdots +|u|_{H^{j}(e)}\right] . \end{aligned}$$

Proof

Let \(e=(-1,1)\times \{0\}\times \{0\}\). With the operator \(E_{1,e}\) of Lemma 7.12 define \(E_{1,e}^{3D}\) by the formula

$$\begin{aligned} (E_{1,e}^{3D}u)(x,y,z):=(E_{1,e}u)(x,y+z). \end{aligned}$$

The statements (i)–(iv), about where \(E_{1,e}^{3D}u\) vanishes follows from the definition. The estimates (vii), follow from Lemma 7.12.(ii) and the simple observation that \(y=0\) or \(z=0\) for the faces \(f_{1,e}\), \(f_{2,e}\). For the volume bounds (v), (vi), we employ Lemma 7.9 and arguments similar to those of the 2D case in Lemma 7.6. \(\square \)

Lemma 7.16

(edge normal derivative lifting) For each edge \(e\) of \(\widehat{K}^{3D}\) denote by \(f_{1,e}\) and \(f_{2,e}\) the two faces that share the edge \(e\). There is a lifting operator \(E_{2,e}^{3D}:H_{0}^{2} (e)\rightarrow H^{2}(\widehat{K}^{3D})\) with the following properties:

  1. (i)

    \(E_{2,e}^{3D}u\) vanishes on \(\partial \widehat{K}^{3D}\setminus f_{1,e}.\)

  2. (ii)

    The face normal derivative \(\partial _{n_{f_{1,e}}} E_{2,e}^{3D} u\) satisfies

    $$\begin{aligned} \partial _{n_{f_{1,e}}} (E_{2,e}^{3D}u)|_{e}=u \ \mathrm{and}\ \partial _{n_{f_{1,e}}}(E_{2,e}^{3D}u)|_{\partial f_{1,e}\setminus e}=0. \end{aligned}$$
  3. (iii)

    \(\Vert E_{2,e}^{3D}u\Vert _{L^{2}(\widehat{K}^{3D})}\le Cp^{-2}\Vert u\Vert _{L^{2}(e)}.\)

  4. (iv)

    \(|E_{2,e}^{3D}u|_{H^{2}(\widehat{K}^{3D})}\le Cp^{-2}\left[ p^{2}\Vert u\Vert _{L^{2}(e)}+p|u|_{H^{1}(e)}+|u|_{H^{2}(e)}\right] .\)

  5. (v)

    For the face \(f_{1,e}\), we have

    $$\begin{aligned} |E_{2,e}^{3D}u|_{L^{2}(f_{1,e})}&\le Cp^{-2+1/2}\Vert u\Vert _{L^{2} (e)},\\ |E_{2,e}^{3D}u|_{H^{2}(f_{1,e})}&\le Cp^{-2+1/2}\left[ p^{2}\Vert u\Vert _{L^{2}(e)}+p|u|_{H^{1}(e)}+|u|_{H^{2}(e)}\right] . \end{aligned}$$
  6. (vi)

    If \(u\in {\fancyscript{P}}_{q} \cap H^{2}_{0}(e)\), then \(E_{2,e}u \in {\fancyscript{P}}_{q+p+1}\).

Proof

Let \(e=(-1,1)\times \{0\}\times \{0\}\) and let \(f_{1,e}=\{(x,y,z)\,|\,\partial \widehat{K}^{3D}\cap \{y=0\}\}\). With the operator \(E_{1,e}\) of Lemma 7.11 define \(E_{2,e}^{3D}\) by the formula

$$\begin{aligned} (E_{2,e}^{3D}u)(x,y,z):=y(E_{1,e}u)(x,y+z). \end{aligned}$$

The statements (i), (ii) about where \(E_{2,e}^{3D}u\) vanishes follows from the definition. The estimates (v) follow by reasoning as in the proof of Lemma 7.11. In view of Lemma 7.9, we see that we can proceed with analogous arguments as in the 2D case to get the volume bounds of (iii), (iv). \(\square \)

We finally need a lifting from faces.

Lemma 7.17

(face lifting) For each face \(f\) of \(\widehat{K}^{3D}\) there is a lifting operator \(E_{f}^{3D}:H_{0}^{2}(f)\rightarrow H^{2}(\widehat{K}^{3D})\) with the following properties:

  1. (i)

    \((E_{f}^{3D}u)|_{\partial \widehat{K}^{3D}\setminus f}=0\).

  2. (ii)

    \((E_{f}^{3D}u)|_{f}=u\).

  3. (iii)

    \(\Vert E_{f}^{3D}u\Vert _{L^{2}(\widehat{K}^{3D})}\le Cp^{-1/2}\Vert u\Vert _{L^{2}(f)}\).

  4. (iv)

    \(|E_{f}^{3D}u|_{H^{2}(\widehat{K}^{3D})}\le Cp^{-1/2}\left[ p^{2}\Vert u\Vert _{L^{2}(f)}+p|u|_{H^{1}(f)}+|u|_{H^{2}(f)}\right] \).

  5. (v)

    If \(u\in {\fancyscript{P}}_{q}\cap H_{0}^{2}(f)\), then \(E_{f}^{3D} u\in {\fancyscript{P}}_{p+q}\).

Proof

Let \(f=\widehat{K}^{2D}\times \{0\}\). Define \(E_{f}^{3D}\) by

$$\begin{aligned} (E_{f}^{3D}u)(x,y,z)&:= \frac{u(x,y)}{(1-x-y)(1+x-y)} (1-x-y-z)(1+x-y-z)(1-z)^{p}\\&= u(x,y)\left( 1-\frac{z}{1-x-y}\right) \left( 1-\frac{z}{1+x-y}\right) (1-z)^{p}. \end{aligned}$$

We focus on the bounds for the second derivatives of \(E_{f}^{3D}u\). We note that \(E_{f}^{3D}\) has the form

$$\begin{aligned} (E_{f}^{3D}u)(x,y,z)=u(x,y)w(x,y,z/(1-x-y),z/(1+x-y),z)(1-z)^{p} \end{aligned}$$
(7.38)

for a smooth function \(w\). Arguing as in the proof of Lemma 7.6 , we see that for multiindices \(\beta \in \mathbb{N }_{0}^{3}\), \(|\beta |\le 2\) we have by the smoothness of \(w\) and that fact that \(|z/(1-x-y)|\le 1\) as well as \(|z/(1+x-y)|\le 1\) on \(\widehat{K}^{3D}\)

$$\begin{aligned} |D^{\beta }w(x,y,z/(1-x-y),z/(1+x-y),z)|\le C\left[ \left( \frac{1}{1-x-y}\right) ^{|\beta |}+\left( \frac{1}{1+x-y}\right) ^{|\beta |}\right] .\nonumber \\ \end{aligned}$$
(7.39)

With the product rule we get with the abbreviation \(d(x,y):=\mathrm{dist}((x,y),\partial \widehat{K}^{2D})\)

$$\begin{aligned} |D^{\beta }(w(1-z)^{p})|\le C\left( \frac{1}{d}+p\right) ^{|\beta |}(1-z)^{p-|\beta |}. \end{aligned}$$
(7.40)

As in the proof of Lemma 7.6, we abbreviate \(D^{1}u\) and \(D^{2}u\) for the sum of all derivatives of order 1 and 2. From (7.38) we obtain with the product rule for differentiation and (7.40)

$$\begin{aligned}&\left| E_{f}^{3D}u\right| _{H^{2}\left( \widehat{K}^{3D}\right) } \le C\sum _{\ell =0}^{2}\left\| \left( D^{2-\ell }u\right) \left( \frac{1}{d}+p\right) ^{\ell }(1-z)^{p-\ell }\right\| _{L^{2}\left( \widehat{K}^{3D}\right) }\\&\quad \le C\sum _{\ell =0}^{2}\left( \left\| d^{-\ell }\left( D^{2-\ell }u\right) \left( 1-z\right) ^{p-\ell }\right\| _{L^{2}\left( \widehat{K}^{3D}\right) }+p^{\ell }\left\| \left( D^{2-\ell }u\right) \left( 1-z\right) ^{p-\ell }\right\| _{L^{2}\left( \widehat{K}^{3D}\right) }\right) \\&\quad \le Cp^{-1/2}\sum _{\ell =0}^{2}\left( \left\| d^{-\ell }D^{2-\ell }u\right\| _{L^{2}\left( \widehat{K}^{2D}\right) }+p^{\ell }\left| u\right| _{H^{2-\ell }\left( \widehat{K}^{2D}\right) }\right) \\&\quad \overset{\text{([20, } \text{ Thm. } \text{1.4.4.4]) }}{\le }Cp^{-1/2} \sum _{\ell =0}^{2}p^{2-\ell }\left| u\right| _{H^{\ell }\left( \widehat{K}^{2D}\right) }, \end{aligned}$$

which concludes the proof. \(\square \)

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Melenk, J.M., Parsania, A. & Sauter, S. General DG-Methods for Highly Indefinite Helmholtz Problems. J Sci Comput 57, 536–581 (2013). https://doi.org/10.1007/s10915-013-9726-8

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Keywords

  • Helmholtz equation at high wavenumber
  • Stability
  • Convergence
  • Discontinuous Galerkin methods
  • Ultra-weak variational formulation
  • Polynomial hp-finite elements

Mathematics Subject Classification (2000)

  • 35J05
  • 65N12
  • 65N30