Optimal operator preconditioning for pseudodifferential boundary problems

Abstract

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain $\varOmega$, where $\varOmega$ is either in ${\mathbb {R}}^n$ or in a Riemannian manifold. For linear systems of equations arising from low-order Galerkin discretizations, we obtain condition numbers that are independent of the mesh size and of the choice of bases for test and trial functions. The basic ingredient is a classical formula by Boggio for the fractional Laplacian, which is extended analytically. In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach gives a unified, independent proof for a series of recent results by Hiptmair, Jerez-Hanckes, Nédélec and Urzúa-Torres. We also study the increasing relevance of the regularity assumptions on the mesh with the order of the operator. Numerical examples validate our theoretical findings and illustrate the performance of the proposed preconditioner on quasi-uniform, graded and adaptively generated meshes.

Appendices

A Proof of results for operator preconditioning on adaptive meshes

For the sake of presentation, we dedicate the next two subsections to briefly summarize some key concepts about adaptivity and the mesh conditions we need to fulfill for stability. Finally, we combine these preliminaries to state and and prove the new results on operator preconditioning in adaptively refined meshes.

1.1 A.1 Adaptivity preliminaries

We begin by reminding the reader of some of the concepts introduced in Sect. 5.2. Given an initial triangulation $\mathcal {T}_h^{(0)}$, the adaptive Algorithm A generates a sequence $\mathcal {T}_h^{(\ell )}$ of triangulations based on error indicators $\eta ^{(\ell )}(\tau ),\; \tau \in \mathcal {T}_h^{(\ell )}$, a refinement criterion and a refinement rule, by following the established sequence of steps:

$$\begin{aligned} \mathrm {SOLVE}\rightarrow \mathrm {ESTIMATE}\rightarrow \mathrm {MARK}\rightarrow \mathrm {REFINE}. \end{aligned}$$

There are different refinement rules that one can choose for the step REFINE. We now present some of the most common ones: red refinement, green refinement, and red-green refinement.

Definition 1

Let $\mathcal {T}_h^{(\ell )}$ be a triangulation. A triangle $\tau \in \mathcal {T }_h^{(\ell )}$ is red refined by connecting edge midpoints of $\tau$, thus splitting $\tau$ into 4 similar triangles.

Definition 2

Let $\mathcal {T}_h^{(\ell )}$ be a triangulation. A triangle $\tau \in \mathcal {T }_h^{(\ell )}$ is green refined by connecting an edge midpoint with the opposite vertex of $\tau$, thus splitting $\tau$ into 2 triangles.

Next, in order to define a red-green refinement, we introduce two related properties.

Definition 3

• (a) A triangulation $\mathcal {T}_h^{(\ell )}$ is called 1–irregular if the property

  $$\begin{aligned} \vert {\text {lev}}(\tau _k) - {\text {lev}}(\tau _m) \vert \le 1, \end{aligned}$$

  holds for any pair of triangles $\tau _k, \tau _m \in \mathcal {T}_h^{(\ell )}$ such that $\tau _k \cap \tau _m \ne \emptyset$.

  Here ${\text {lev}}(\tau _k)$ corresponds to the number of refinement steps required to generate $\tau _k$ from the initial triangulation $\mathcal {T}_h^{(0)}$.

• (b) The 2–neighbour rule: Red refine any triangle $\tau _k$ with 2 neighbours that have been red refined. Two triangles are neighbours, if they have a common edge.

Definition 4

A Red-green refinement for a triangulation $\mathcal {T}_h^{(\ell )}$ proceeds as follows:

• 1. Remove edges from any triangles that have been green refined.

• 2. All marked triangles are red refined.

• 3. Any triangles with 2 or more red refined neighbours are red refined, by 2–neighbour rule.

• 4. Any triangles that do not fulfil 1–irregularity rule are further refined.

• 5. Any triangles with hanging nodes generated during the refinement are green refined

See Figs. 6 and 7 for an illustration of the refinement rules. For a further description we refer to [8, 25].

Fig. 6

Example of red refined triangle (left) and green refined triangle (right) (color figure online)

Fig. 7

Example of a sequence of red-green refinement. Top element is marked by (o) and therefore is red refined in the first step. Bottom triangle then has a hanging node and is green refined in the consequent step (color figure online)

1.2 A.2 Mesh conditions

We recall that we aim to show (20), i.e.

$$\begin{aligned} \underset{\varphi _h\in \mathbb {W}_{h}}{\mathrm {sup}} \dfrac{\mathsf {d}(v_h, \varphi _h)}{\Vert \varphi _h\Vert _{H^{-s}(\varOmega )}} \ge {\beta _{\mathsf {d}}} \Vert v_h\Vert _{{\widetilde{H}}^s (\varOmega )}, \qquad \text {for all } v_h \in \widetilde{\mathbb {V}}_{h}, \end{aligned}$$

(see Sect. 5 for notation).

In the case of the discretizations based on dual meshes, this inf-sup stability is a consequence of three regularity conditions on the triangulation $\mathcal {T}_h$, see [48, Chapters 1–2]. We now proceed to introduce some notation to properly summarize this result.

Let $\mathcal {T}_h$ be a triangulation of $\varOmega \subset {\mathbb {R}}^n$. For each triangle $\tau _k \in \mathcal {T}_h$ we define its area $\varDelta _{ k} :=\int _{\tau _{k}} dx$; its local element size $h_{k} :=\varDelta _{k}^{ 1/ n}$; and its diameter $d_{k} :=\sup _{x, y \in \tau _{k}}|x-y|$.

Let $\varphi _j$ be a piecewise linear basis function in the span of $\widetilde{\mathbb {V}}_{h}$. We write $\omega _j := {\text {supp}} \, \varphi _j$ and define its associated local mesh size ${\hat{h}}_j$ as

$$\begin{aligned} {\hat{h}}_j&:= \frac{1}{\# I(j)} \sum _{m \in I(j)} h_{m}. \end{aligned}$$

Here, $I(j) :=\left\{ m \in \{1, \ldots , \#\mathcal {T}_h\} : \tau _{m} \cap \omega _{j} \ne \emptyset \right\} , \text { for } j=1, \ldots , N,$ is the index set of triangles $\tau _m\in \mathcal {T}_h$ where the basis function $\varphi _j$ is not identically zero.

Definition 5

For a triangulation $\mathcal {T}_h$, we define the following mesh conditions

(C1):

Shape regularity: there exists $c_R > 0$ such that for all $\tau _k \in \mathcal {T}_h$

$$\begin{aligned} 0<c_R<\frac{h_{k}}{d_{k}}<1. \end{aligned}$$

(C2):

Local quasi-uniformity: for all $\tau _k, \tau _m \in \mathcal { T}$ with $\tau _k \cap \tau _m \ne \emptyset$

$$\begin{aligned} \frac{h_k}{h_m} \le c_L, \end{aligned}$$

with $C_L$ a (uniform) positive constant.

(C3):

Local s-dependent condition: there exists $c_0>0$ such that for all $\tau \in \mathcal {T}_h$

$$\begin{aligned} \frac{51}{7}-\sqrt{\sum _{j \in J(m)} {\hat{h}}_{j}^{2 s} \sum _{j \in J(m)} \hat{h }_{j}^{-2 s}} \ge c_{0}>0, \end{aligned}$$

with $J(m) :=\left\{ i \in \{1, \ldots , N\} : \omega _{i}\cap \tau _{m} \ne \emptyset \right\}$ for $m=1,\ldots , \#\mathcal {T}_h$, the index set of basis functions $\varphi _i$ which are not identically zero on triangle $\tau _m$.

Theorem A1

([47, Theorems 2.1 and 2.2]) Let $\mathcal {T}_h$ be a triangulation of $\varOmega$ such that (C1), (C2) and (C3) are satisfied. Consider the primal-dual discretization $\widetilde{\mathbb {V}}_{h}= \mathbb {S}^1(\mathcal {T}_h)\cap {\widetilde{H}}^{s} (\varOmega )$ and $\mathbb {W}_{h}=\mathbb {S}^0(\check{\mathcal {T}}_h)$ for $0\le s\le 1$ (see Sect. 5.1).

Then, the discrete inf-sup condition (20) holds with a positive constant $\beta _{\mathsf {d}}$ independent of h.

1.3 A.3 Results on adaptively refined meshes

Now we turn our attention to study these conditions for a sequence of adaptive triangulations generated by Algorithm A. For this, we write the constants from conditions (C1), (C2) and (C3) associated to a triangulation $\mathcal {T}_h^{(\ell )}$ as $c_R^{(\ell )}, c_L^{(\ell )}$ and $c_0^{(\ell )}$, respectively.

The next Lemma is the complete version of Lemma 5 introduced in Sect. 5.2.

Lemma 6

Consider an initial triangulation $\mathcal {T}_h^{(0)}$ satisfying the mesh conditions from Definition 5, and such that its local quasi-uniformity constant $c_L^{(0)}$ verifies

$$\begin{aligned} c_L^{(0)} \le \frac{1}{2}\root 4 \vert s \vert \of {\frac{1129}{49}} \approx \frac{2.19^{1/\vert s \vert }}{2}. \end{aligned}$$

(27)

Let $\varXi := \lbrace \mathcal {T}_h^{(\ell )}\rbrace _{\ell \in {{\,\mathrm{\mathbb {N}}\,}}}$ be a family of meshes generated from $\mathcal {T}_h^{(0)}$ by the adaptive refinement described in Algorithm A, using red-green refinements. Then (C1), (C2) and (C3) hold for all $\mathcal {T}_h^{(\ell )} \in \varXi$ for some constants $c_R, c_L, c_0>0$, which are independent of $\ell \in {\mathbb {N}}$.

In particular, the inf-sup condition (20) holds for $\vert s \vert \le 1$, $\widetilde{\mathbb {V}}_{h}= \mathbb {S}^p(\mathcal {T}_h) \cap {\widetilde{H}}^{s}(\varOmega )$, $\mathbb {W}_{h}=\mathbb {S}^q(\mathcal {T}_h^{\prime })$, and for all $\mathcal {T}_h^{(\ell )}$ independent of $\ell$.

Proof

The proof proceeds by induction on $\ell$. By hypothesis, the initial triangulation $\mathcal {T}_h^{(0)}$ satisfies (C1) and (C2). Therefore, for the initial triangulation $\mathcal {T}_h^{(0)}$ we only need to check (C3).

For the sake of convenience, let us re-label the basis functions $j \in J(m)$ by $m_i$, with $i=1,\ldots , \#J(m)$. We note that $\max _{m} \#J(m)=3$ and that this is our worst case scenario. Therefore, it suffices to verify (C3) in this case:

$$\begin{aligned} \frac{51}{7}-\sqrt{\sum _{i = 1}^3 {\hat{h}}_{m_i}^{2 s} \sum _{i=1}^3 {\hat{h}}_{ m_i}^{-2 s}} \ge c_{0}>0. \end{aligned}$$

Without loss of generality, let ${\hat{h}}_{m_1} \ge {\hat{h}}_{m_2} \ge {\hat{h}}_{m_3}$. Then

$$\begin{aligned} \begin{aligned} \sum _{i = 1}^3 {\hat{h}}_{m_i}^{2 s} \sum _{i=1}^3 {\hat{h}}_{m_i}^{-2 s}&= \textstyle {3+ \left( \frac{{\hat{h}}_{m_1}}{{\hat{h}}_{m_2}} \right) ^{2|s|} + \left( \frac{{\hat{h}}_{m_2}}{{\hat{h}}_{m_3}} \right) ^{2|s|} + \left( \frac{{\hat{h}}_{ m_3}}{{\hat{h}}_{m_1}} \right) ^{2|s|} }\\&\qquad +\textstyle { \left( \frac{\hat{h }_{m_1}}{{\hat{h}}_{m_3}} \right) ^{2|s|} + \left( \frac{{\hat{h}}_{m_2}}{{\hat{h}}_{m_1}} \right) ^{2|s|} + \left( \frac{{\hat{h}}_{m_3}}{{\hat{h}}_{m_2}} \right) ^{2|s|} }\\&\le 3+\textstyle {2\left( \left( \frac{{\hat{h}}_{m_1}}{{\hat{h}}_{m_3}} \right) ^{ 2|s|} + \left( \frac{{\hat{h}}_{m_2}}{{\hat{h}}_{m_2}} \right) ^{2|s|} + \left( \frac{ {\hat{h}}_{m_3}}{{\hat{h}}_{m_1}} \right) ^{2|s|} \right) \le 7+2\left( \frac{\hat{ h}_{m_1}}{{\hat{h}}_{m_3}} \right) ^{2|s|},}\\ \end{aligned} \end{aligned}$$

where we use the rearrangement inequality. We conclude that (C3) is satisfied for $\mathcal {T}_h^{(0)}$ provided that

$$\begin{aligned} \textstyle {\left( \frac{{\hat{h}}_{m_1}}{{\hat{h}}_{m_3}} \right) ^{2|s|} < \frac{1129}{49}.} \end{aligned}$$

(28)

A simple calculation using the mesh conditions yields $\frac{{\hat{h}}_{m_1}}{ {\hat{h}}_{m_3}} \le (c_L^{(0)})^2$, so that (28) holds and (C3) is satisfied for $\mathcal {T}_h^{(0)}$.

For the inductive step, assume that conditions (C1)–(C3) are satisfied on an adaptively refined triangulation $\mathcal {T}_h^{(\ell )}$ using red-green refinements subject to 1–irregularity and 2–neighbour rules. In order to generate a new triangulation $\mathcal {T}_h^{(\ell +1)}$, the appropriate triangles are marked.

We note that red-refinement does not change the shape regularity constant, but green refinement worsens the shape regularity constant by at most a factor of $\frac{1}{\sqrt{2}}$. However, due to the removal of green edges, the constant does not degenerate as $\ell \rightarrow \infty$. Thus condition (C1) is satisfied with $c_R^{(\ell +1)} \ge \frac{1}{\sqrt{2}} c_R^{(0)}$ for $\mathcal {T}_h^{(\ell +1)}$.

Condition (C2) remains satisfied due to the 1–irregularity condition in the refinement procedure. This restriction guarantees that $\frac{h_i}{h_j} \le c_L^{(\ell +1)} \le 2 c_L^{(0)}$.

As for the initial triangulation $\mathcal {T}_h^{(0)}$, we know that condition (C3) is satisfied for $\mathcal {T}_h^{(\ell +1)}$ when (28) holds. Due to the 1–irregularity condition, we have that $\frac{{\hat{h}}_{m_1}}{\hat{ h}_{m_3}} \le (2 c_L^{(0)})^2$, so the estimate (28) is satisfied provided $c_L^{(0)} < \frac{1}{2}{\left( \frac{1129}{49} \right) }^{{1}/{4 \vert s \vert }}$.

We conclude that (C1), (C2), (C3) are satisfied for $\lbrace \mathcal {T}_h^{ (\ell )} \rbrace _{\ell =0}^{\infty }$ independently of $\ell$. $\square$

Remark 9

• a) We note that the estimates in Lemma 6 are not sharp. Still, the local quasi-uniformity assumption on the initial triangulation $\mathcal {T }^{(0)}$ becomes more restrictive as $\vert s \vert$ increases. Thus, the initial mesh needs to be of increasingly higher regularity for higher values of |s|.

• b) Let $\varGamma {\subset } {\mathbb {R}}^n$ be a polyhedral domain which satisfies an interior cone condition. Then the assumptions in Lemma 6 can be satisfied for a sufficiently fine $\mathcal {T}_h^{(0)}$.

Remark 10

Similar results can be shown for alternative refinement strategies, such as the newest vertex bisection [19, Section 2.2]. See [54, Chapter 4] for details.

B Proof of Proposition 2

The idea for the proof is like in [12] where the case $\mathbb {W}_{h}= \widetilde{\mathbb {V}}_{h}$ is shown. Here we generalize the proof to different discrete test and trial space. For the sake of brevity we will discuss the case when $s\in (1/2,1]$ and remark that the proof for $s\in [-1,-1/2)$ follows analogously. We remind the reader that in this setting ${\widetilde{H}}^s(\varOmega ) \equiv H^s_0(\varOmega )\ne H^s(\varOmega )$, but that $\Vert u \Vert _{\widetilde{H }^s(\varOmega )} \equiv \Vert u \Vert _{{H}^s(\varOmega )}, \, \forall u \in \widetilde{ H}^s(\varOmega )$.

Let $\mathcal {T}_h$, $\mathbb {S}^p(\mathcal {T}_h), p\in {{\,\mathrm{\mathbb {N}}\,}}$ be as in Sect. 5. Moreover, we recall that for this setting we consider the finite element spaces $\widetilde{\mathbb {V}}_{h}=\mathbb {S}^1(\mathcal {T}_h )\cap {\widetilde{H}}^{s}(\varOmega )$ and $\mathbb {W}_{h}\subset {H}^{-s}(\varOmega )$. Additionally, we denote $\mathbb {V}_{h}=\mathbb {S}^1(\mathcal {T}_h)\subset {H}^{s}(\varOmega )$ and note that $\widetilde{\mathbb {V}}_{h}\subset \mathbb {V}_{h }$. Indeed, $\widetilde{\mathbb {V}}_{h}$ is the space of affine continuous functions that are zero on the boundary, while $\mathbb {V}_{h}$ is analogous to $\widetilde{\mathbb {V}}_{h}$, but admits non-zero values on $\partial \varOmega$.

Let us introduce the generalized $L^2$-projection ${\widetilde{Q}}_h:L^2(\varOmega ) \rightarrow \widetilde{\mathbb {V}}_{h}$ for a given $u\in L^2(\varOmega )$, as the solution of the variational problem

$$\begin{aligned} \left\langle {\widetilde{Q}}_h u\,,\, \psi _h\right\rangle _{\varOmega } = \left\langle u\,,\,\psi _h\right\rangle _{\varOmega }, \quad \forall \psi _h \in \mathbb {W}_{h}. \end{aligned}$$

(29)

From [48, Chapter 2], [36], we know that it satisfies

$$\begin{aligned} \Vert {\widetilde{Q}}_h u \Vert _{{\widetilde{H}}^s(\varOmega )} \le {\beta _{\mathsf {d}}^{-1}} \Vert u \Vert _{{\widetilde{H}}^s(\varOmega )}, \qquad \forall u\in {\widetilde{H}}^s(\varOmega ). \end{aligned}$$

(30)

where ${\beta _{\mathsf {d}}}$ is the inf-sup constant from (20).

Given that we are interested in the case where we have a space mismatch, i.e. when $u \in H^s(\varOmega )$ but $u \notin {\widetilde{H}}^s(\varOmega )$, we additionally prove the following:

Lemma 7

The projection ${\widetilde{Q}}_h$ satisfies

$$\begin{aligned} \Vert {\widetilde{Q}}_h u_h \Vert _{H^s(\varOmega )} \le ( 1 + {\frac{c_2}{s-{1/2} }}\,h^{1/2-s} )\Vert u_h \Vert _{H^s(\varOmega )}, \qquad \forall u_h\in \mathbb {V }_{h}, \end{aligned}$$

(31)

with $c_2>0$ and independent of h.

Proof

Set $u^0_h \in \widetilde{\mathbb {V}}_{h}$ to be the function defined by

$$\begin{aligned} u^0_h := {\left\{ \begin{array}{ll} u_h, \quad \text { in all interior nodes},\\ 0, \qquad \text {on }\partial \varOmega . \end{array}\right. } \end{aligned}$$

(32)

Then, by definition

$$\begin{aligned} \Vert u_h - {\widetilde{Q}}_h u_h \Vert _{L^2(\varOmega )}&=\Vert u_h - u_h^0 \Vert _{ L^2(\varOmega )}\le h^{1/2}\Vert u_h \Vert _{L^2(\partial \varOmega )}, \end{aligned}$$

where the last inequality holds by basic computations (c.f. [12, Equation 1.3.27]).

From the trace theorem, we have that $\Vert u_h \Vert _{L^2(\partial \varOmega )} \le \dfrac{c_{tt}}{s-1/2} \Vert u_h \Vert _{H^s(\varOmega )}$, with $c_{tt}>0$ independent of h.

Therefore, combining all the above, we obtain

$$\begin{aligned} \Vert {\widetilde{Q}}_h u_h \Vert _{H^s(\varOmega )}&\le \Vert u_h \Vert _{H^s(\varOmega )} + \Vert {\widetilde{Q}}_h u_h - u_h \Vert _{H^s(\varOmega )}\\&\le \Vert u_h \Vert _{H^s(\varOmega )} + {c_1} h^{-s}\Vert {\widetilde{Q}}_h u_h - u_h \Vert _{L^2(\varOmega )}\\&\le \left( 1 + \frac{{c_1 c_{tt}}}{s-1/2}h^{1 /2-s} \right) \Vert u_h \Vert _{H^s(\varOmega )}. \end{aligned}$$

$\square$

Now, let us also introduce the finite element space $\widetilde{\mathbb {W}}_{h} \subset {\widetilde{H}}^{-s}(\varOmega )$. We consider the generalized $L^2$-projection ${\widetilde{P}}_h:L^2(\varOmega )\rightarrow \widetilde{\mathbb {W}}_{h}$ for a given $\varphi \in L^2(\varOmega )$, as the solution of the variational problem

$$\begin{aligned} \left\langle {\widetilde{P}}_h \varphi \,,\, v_h\right\rangle _{\varOmega } = \left\langle \varphi \,,\,v_h\right\rangle _{\varOmega }, \quad \forall v_h \in \mathbb {V}_{h}. \end{aligned}$$

(33)

Then, in analogy with Lemma 7, we have that

Lemma 8

The projection ${\widetilde{P}}_h$ satisfies

$$\begin{aligned} \Vert {\widetilde{P}}_h \varPhi _h \Vert _{H^{-s}(\varOmega )} \le c_3( 1 + {\frac{c_2}{s-{1/2} }}\,h^{1/2 -s} )\Vert \varPhi _h \Vert _{H^{-s}(\varOmega )}, \qquad \forall \varPhi _h\in \mathbb {W}_{h}, \end{aligned}$$

(34)

with $c_2,c_3>0$ and independent of h.

Proof

Let us use the norms’ properties and write

$$\begin{aligned} \Vert {\widetilde{P}}_h \varPhi _h \Vert _{H^{-s}(\varOmega )} \le \Vert {\widetilde{P}}_h \varPhi _h \Vert _{{\widetilde{H}}^{-s}(\varOmega )} = \underset{0\ne u \in H^{s}(\varOmega )}{\mathrm {sup}} \dfrac{\left\langle {\widetilde{P}}_h \varPhi _h\,,\,u\right\rangle _{\varOmega }}{\Vert u \Vert _{H^{s} (\varOmega )}}. \end{aligned}$$

Then, using the definition of ${\widetilde{Q}}_h$ and the estimates above, we get

$$\begin{aligned} \Vert {\widetilde{P}}_h \varPhi _h \Vert _{H^{-s}(\varOmega )}&\le (1 +{\frac{c_2}{s-{1/2}}}\,h^{1/2-s}) \underset{0\ne u \in H^{s}(\varOmega )}{\mathrm {sup}} \dfrac{ \left\langle {\widetilde{P}}_h \varPhi _h\,,\,{\widetilde{Q}}_h u\right\rangle _{\varOmega }}{\Vert {\widetilde{Q}}_h u \Vert _{H^{s}(\varOmega )}}\\&\le (1 + {\frac{c_2}{s-{1/2}}}\,h^{1/2-s}) \underset{0\ne u_h \in \widetilde{\mathbb {V}}_h}{\mathrm {sup}} \dfrac{\left\langle {\widetilde{P}}_h \varPhi _h\,,\,u_h \right\rangle _{\varOmega }}{\Vert u_h \Vert _{H^{s}(\varOmega )}}. \end{aligned}$$

Now, by definition of ${\widetilde{P}}_h$, and since $\widetilde{\mathbb {V}}_h \subset \mathbb {V}_h$, we have

$$\begin{aligned} \Vert {\widetilde{P}}_h \varPhi _h \Vert _{H^{-s}(\varOmega )}&\le (1 +{\frac{c_2}{s -{1/2}}}\,h^{1/2-s}\,) \underset{0\ne u_h \in \widetilde{\mathbb {V}}_h}{ \mathrm {sup}} \dfrac{\left\langle \varPhi _h\,,\,u_h\right\rangle _{\varOmega }}{\Vert u_h \Vert _{H^{s}(\varOmega ) }}\\&\le {c_3}(1 + {\frac{c_2}{s-{1/2}}}\,h^{1/2-s}\,) \Vert \varPhi _h \Vert _{H^{-s}(\varOmega )}. \end{aligned}$$

$\square$

Lemma 9

Let $s\in (1/2,1)$. Then, the following inf-sup condition holds

$$\begin{aligned} \underset{\phi _h \in \widetilde{\mathbb {W}}_h}{\mathrm {sup}}\dfrac{\left\langle v_h\,,\, \phi _h\right\rangle _{\varOmega }}{\Vert \phi _h \Vert _{H^{-s}(\varOmega )}}\ge {\frac{\beta _{\mathsf {d}} }{c_3}}\left( 1 + {\frac{c_2}{s-{1/2}}} \,h^{1/2-s} \right) ^{-1} \Vert v_h \Vert _{{\widetilde{H}}^{s}(\varOmega )}, \quad \forall v_h \in \widetilde{\mathbb {V} }_{h}, \end{aligned}$$

(35)

with ${c_3,c_2}>0$ and independent of h.

Proof

Let us introduce the operator $\Pi _h^s: {\widetilde{H}}^s(\varOmega ) \rightarrow \mathbb {W}_h\subset H^{-s}(\varOmega )$ for $s\in (0,1]$, defined by the variational formulation

$$\begin{aligned} \left\langle \Pi _h^s u\,,\, v_h\right\rangle _{\varOmega } = (u, v_h)_{{\widetilde{H}}^s(\varOmega )}, \quad \forall v_h \in \widetilde{\mathbb {V}}_{h}, \end{aligned}$$

(36)

where $(\cdot ,\cdot )_{{\widetilde{H}}^s(\varOmega )}$ denotes the ${\widetilde{H}}^s( \varOmega )$-inner product. This operator is analogous to [48, Equation 1.75] [34, Equation 4.22], and thus it verifies

$$\begin{aligned} \Vert \Pi _h^s u \Vert _{{H}^{-s}(\varOmega )} \le {\beta _{\mathsf {d}}^{-1}} \Vert u \Vert _{{\widetilde{H}}^s(\varOmega )}, \qquad \forall u\in {\widetilde{H}}^s(\varOmega ). \end{aligned}$$

(37)

Next, we have that for any $v_h \in \widetilde{\mathbb {V}}_h$

$$\begin{aligned} \Vert v_h \Vert _{{\widetilde{H}}^{s}(\varOmega )}&= \dfrac{(v_h, v_h)_{\widetilde{H }^{s}(\varOmega )}}{\Vert v_h \Vert _{{\widetilde{H}}^{s}(\varOmega )}} = \dfrac{\left\langle v_h\,,\,\Pi _hv_h\right\rangle _{\varOmega }}{\Vert v_h \Vert _{{\widetilde{H}}^{s}( \varOmega )}} \\&\le \beta _{\mathsf {d}}^{-1}\dfrac{\left\langle v_h\,,\,\Pi _hv_h\right\rangle _{\varOmega }}{\Vert \Pi _h v_h\Vert _{H^{-s}(\varOmega )}} = \beta _{\mathsf {d}}^{-1} \dfrac{\left\langle v_h\,,\,\widetilde{ P}_h\Pi _hv_h\right\rangle _{\varOmega }}{\Vert \Pi _hv_h\Vert _{{H}^{-s}(\varOmega )}}, \end{aligned}$$

where in the last step we used that $\Pi _hv_h \in \mathbb {W}_h$ and the definition of ${\widetilde{P}}_h$.

Now, let us use our previous estimates to derive

$$\begin{aligned} \Vert v_h \Vert _{{\widetilde{H}}^{s}(\varOmega )} \le \frac{c_3}{\beta _{\mathsf {d}}}(1 + \frac{c_2}{s-1/2}\, h^{1/2-s})\dfrac{\left\langle v_h\,,\,{\widetilde{P}}_h\Pi _hv_h\right\rangle _{\varOmega }}{\Vert {\widetilde{P}}_h\Pi _hv_h\Vert _{{H}^{-s}(\varOmega )}}. \end{aligned}$$

Set $\varphi _h := {\widetilde{P}}_h\Pi _hv_h$ and note that $\varphi _h\in \widetilde{ \mathbb {W}}_h$. Therefore, this gives

$$\begin{aligned} \Vert v_h \Vert _{{\widetilde{H}}^{s}(\varOmega )}&\le \frac{c_3}{\beta _{\mathsf {d}}}\left( 1 + \frac{c_2}{s-1/2}\, h^{1/2-s}\,\right) \dfrac{\left\langle v_h\,,\,\varphi _h\right\rangle _{\varOmega }}{\Vert \varphi _h\Vert _{{H}^{-s}(\varOmega )}} \\&\le \frac{c_3}{\beta _{\mathsf {d}}}\left( 1 + \frac{c_2}{s-1/2}\, h^{1/2-s}\,\right) \underset{\phi _h \in \widetilde{\mathbb {W}}_h}{\mathrm {sup}}\dfrac{\left\langle v_h\,,\,\phi _h\right\rangle _{\varOmega }}{\Vert \phi _h\Vert _{{H}^{-s}(\varOmega )}}. \end{aligned}$$

Finally, move the factors to the other side and one gets the desired result. $\square$

Proof of Proposition 2

First notice that in this context the inf-sup constant of $\mathsf {d}$ is

$$\begin{aligned} {\tilde{\beta }_\mathsf {d}:= \frac{\beta _{\mathsf {d}}}{c_3}\left( 1 + \frac{c_2}{s-1/2}\, h^{1/2-s} \right) ^{-1}}. \end{aligned}$$

Then, we plug this in (22) and get

$$\begin{aligned} \kappa \left( {\mathbf {D}}^{-1} \tilde{{\mathbf {C}}}_s {\mathbf {D}}^{-T} {\mathbf {A}} \right) \le \dfrac{C_{\gamma } C_A \Vert \mathsf {d}\Vert ^2 c_3^2\left( 1 + \dfrac{c_2 }{s-1/2}\, h^{1/2-s} \right) ^2}{\beta _A \beta _{\gamma } \beta _{\mathsf {d}}^2} \sim \mathcal {O}(h^{1-2s}). \end{aligned}$$

(38)

$\square$