Quasi-Optimality Constants for Parabolic Galerkin Approximation in Space

Abstract

We consider Galerkin approximation in space of linear parabolic initial-boundary value problems where the elliptic operator is symmetric and thus induces an energy norm. For two related variational settings, we show that the quasi-optimality constant equals the stability constant of the L ²-projection with respect to that energy norm.

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1 Introduction

A Galerkin method S for a variational problem is quasi-optimal in a norm $\left \|\cdot \right \|$ if there exists a constant q such that

$$\displaystyle{ \left \|u - U_{S}\right \| \leq q\inf _{v}\left \|u - v\right \|, }$$

(1)

where u is any variational solution, U _S its associated Galerkin approximation and v varies in the discrete trial space. The quasi-optimality constant q _S is the best constant q in (1), and thus measures how well the Galerkin method S exploits the approximation potential offered by the discrete trial space. The determination or estimation of q _S is therefore the ideal first step in an a priori error analysis.

Here we are interested in Galerkin approximation in space for linear parabolic initial-boundary value problems like

$$\displaystyle{ \begin{array}{rl} \partial _{t}u -\varDelta u = f\text{ in }\varOmega \times (0,T), \\ u = 0\text{ on }\partial \varOmega \times (0,T),\quad u(\cdot,0) = w\text{ in }\varOmega.\end{array} }$$

(2)

Whereas for the stationary case, i.e. elliptic problems, quasi-optimality results like Céa’s lemma are very common, such results have been less explored for parabolic problems. A common assumption of such results is that the L ²-projection onto the underlying discrete space is H ¹-stable; see, e.g., [4, 5, 7], where the norm in (1) is either the one of H ¹(H ⁻¹) ∩ L ²(H ¹) or the one of L ²(H ¹). Recently, the authors [8] have clarified the role of this assumption by showing that it is also necessary. This follows by applying the inf-sup theory [2, 3] to two weak, essentially dual formulations: the standard weak formulation with trial space H ¹(H ⁻¹) ∩ L ²(H ¹) and the ultra-weak formulation with trial space L ²(H ¹).

This short note underlines the close relationship between parabolic quasi-optimality and the H ¹-stability of the L ²-projection. It improves the results of [8] in the special case of a time-independent symmetric elliptic operator. For the model problem (2) and both variational formulations, this improvement reads as follows: the quasi-optimality constant of a Galerkin approximation with values in a discrete subspace S of H ₀ ¹ is given by the operator norm in H ₀ ¹ of the L ²-projection onto S:

$$\displaystyle{ q_{\mathrm{std};S} = \left \|P_{S}\right \|_{\mathcal{L}(H_{0}^{1})} = q_{\mathrm{ult};S}. }$$

(3)

2 Petrov-Galerkin Framework and Quasi-Optimality

This section, which is taken from [8], provides the general framework for the derivation of our quasi-optimality results. Let $(H_{1},\left \|\cdot \right \|_{1})$ and $(H_{2},\left \|\cdot \right \|_{2})$ be two real Hilbert spaces. The dual space H ₂ ^∗ of H ₂ is equipped with the usual dual norm $\left \|\ell\right \|_{H_{2}^{{\ast}}} =\sup _{\left \|\varphi \right \|_{2}=1}\ell(\varphi )$ for ℓ ∈ H ₂ ^∗. Moreover, let b be a real-valued bounded bilinear form on H ₁ × H ₂ and set $C_{b}:=\sup _{\left \|v\right \|_{1}=\left \|\varphi \right \|_{2}=1}\left \vert b(v,\varphi )\right \vert$. We consider the problem

$$\displaystyle{ \text{given }\ell \in H_{2}^{{\ast}}\text{, find }u \in H_{ 1}\text{ such that }\forall \varphi \in H_{2}\quad b(u,\varphi ) =\ell (\varphi ) }$$

(4)

and say that it is well-posed if, for any ℓ ∈ H ₂ ^∗, there exists a unique solution that continuously depends on ℓ. This holds if and only if there hold the following two conditions involving the so-called inf-sup constant c _b , cf. [3]:

$$\displaystyle\begin{array}{rcl} & c_{b}:=\inf _{\left \|v\right \|_{ 1}=1}\;\sup _{\left \|\varphi \right \|_{2}=1}b(v,\varphi )> 0\ \ \ \ \ \ \ \ \text{(uniqueness)},&{}\end{array}$$

(5a)

$$\displaystyle\begin{array}{rcl} & \forall \varphi \in H_{2}\setminus \{0\}\ \exists v \in H_{1}\quad b(v,\varphi )> 0\ \ \ \ \ \ \ \text{(existence)}.&{}\end{array}$$

(5b)

If (5) is satisfied, we have the duality

$$\displaystyle{ \inf _{\left \|v\right \|_{ 1}=1}\sup _{\left \|\varphi \right \|_{2}=1}b(v,\varphi ) =\inf _{\left \|\varphi \right \|_{ 2}=1}\sup _{\left \|v\right \|_{1}=1}b(v,\varphi ). }$$

(6)

For notational simplicity, we take the viewpoint that a Petrov-Galerkin method for problem (4) is characterized by one pair of subspaces, instead of a family of pairs. Let M _i  ⊂ H _i , i = 1, 2, be nontrivial and proper subspaces. The Petrov-Galerkin method M = (M ₁, M ₂) for (4) reads

$$\displaystyle{ \text{given }\ell \in H_{2}^{{\ast}}\text{, find }U_{ M} \in M_{1}\text{ such that }\forall \varphi \in M_{2}\quad b(U_{M},\varphi ) =\ell (\varphi ). }$$

(7)

Problem (7) is well-posed if and only if there hold the semidiscrete counterparts of (5), involving the semidiscrete inf-sup constant c _M :

$$\displaystyle\begin{array}{rcl} & c_{M}:=\inf _{v\in M_{ 1}:\left \|v\right \|_{1}=1}\;\sup _{\varphi \in M_{2}:\left \|\varphi \right \|_{2}=1}b(v,\varphi )> 0,& {}\\ & \forall \varphi \in M_{2}\setminus \{0\}\ \exists v \in M_{1}\quad b(v,\varphi )> 0. & {}\\ \end{array}$$

A method M is quasi-optimal if there exists a constant q ≥ 1 such that, for any ℓ ∈ H ₂ ^∗, there holds

$$\displaystyle{ \left \|u - U_{M}\right \|_{1} \leq q\inf _{v\in M_{1}}\left \|u - v\right \|_{1}. }$$

(8)

The quasi-optimality constant q _M of the method M is the smallest constant verifying (8). The formula for q _M in [8, Theorem 2.1] or combining [2, 3] with [9] imply

$$\displaystyle{ \frac{c_{b}} {c_{M}} \leq q_{M} \leq \frac{C_{b}} {c_{M}}. }$$

(9)

3 Two Weak Formulations of Linear Parabolic Problems

In order to cast parabolic initial-boundary value problems in the form (4), we briefly recall two suitable weak formulations thereof.

Let V and W be two separable Hilbert spaces such that V ⊂ W ⊂ V ^∗ forms a Hilbert triplet. The scalar product in W as well as the duality pairing of V ^∗× V is denoted by $\left \langle \cdot,\cdot \right \rangle$. The norms are indicated by $\left \|\cdot \right \|_{V }$, $\left \|\cdot \right \|_{W}$, and $\left \|\cdot \right \|_{V ^{{\ast}}} =\sup _{\left \|v\right \|_{V }=1}\left \langle \cdot,v\right \rangle$.

Let $A \in \mathcal{L}(V,V ^{{\ast}})$ be a linear and continuous operator arising from a symmetric bilinear form a via $\left \langle Av,\varphi \right \rangle = a(v,\varphi )$. We assume that a is bounded and coercive, i.e.

$$\displaystyle{ \nu _{a}:=\inf _{\left \|v\right \|_{V }=1}\!a(v,v)> 0,\quad \ C_{a}:=\sup _{\left \|v\right \|_{V }=\left \|\varphi \right \|_{V }=1}\!\!\!\!a(v,\varphi ) <\infty. }$$

(10)

In view of (10) and the symmetry of a, the energy norm $\left \|\cdot \right \|_{a} = \left \langle A\cdot,\cdot \right \rangle ^{1/2}$ and the dual energy norm $\left \|\cdot \right \|_{a;{\ast}}:=\sup _{\left \|\varphi \right \|_{a}=1}\left \langle \cdot,\varphi \right \rangle$ are equivalent to $\left \|\cdot \right \|_{V }$ and $\left \|\cdot \right \|_{V ^{{\ast}}}$, respectively. Moreover, for every ℓ ∈ V ^∗ we have

$$\displaystyle{ \left \|\ell\right \|_{a;{\ast}} =\sup _{\left \|\varphi \right \|_{a}=1}\left \langle A\varphi,A^{-1}\ell\right \rangle = \left \|A^{-1}\ell\right \|_{ a} = \sqrt{\left \langle \ell, A^{-1}\ell\right \rangle }. }$$

(11)

Finally, given a final time T > 0 and a Hilbert space X, we set I: = (0, T) and denote with L ²(X):= L ²(I; X) the space of all Lebesgue-measurable and square-integrable functions of the form I → X. In addition, if Y is another Hilbert space, we set H ¹(X, Y ):={ v ∈ L ²(X)∣v′ ∈ L ²(Y )} and write H ¹(X) for H ¹(X, X).

3.1 Standard Weak Formulation

The standard weak formulation is very common, also for some nonlinear parabolic problems. In the above setting, it reads

$$\displaystyle{ \begin{array}{ccc} &\text{given }f \in L^{2}(V ^{{\ast}})\text{ and }w \in W,\text{ find }u \in H^{1}(V,V ^{{\ast}})\text{ such that }& \\ & u' + Au = f\text{ in }I,\quad u(0) = w &\end{array} }$$

(12)

and can be cast in the form (4) by choosing H ₁ = H ¹(V, V ^∗) and H ₂ = {φ = (φ ₀, φ ₁)∣φ ₀ ∈ W, φ ₁ ∈ L ²(V )} with norms

$$\displaystyle{ \left \|v\right \|_{1}^{2} = \left \|v(T)\right \|_{ W}^{2} +\!\int _{ I}\left \|v\right \|_{a}^{2} + \left \|v'\right \|_{ a;{\ast}}^{2},\quad \left \|\varphi \right \|_{ 2}^{2} = \left \|\varphi _{ 0}\right \|_{W}^{2} +\!\int _{ I}\left \|\varphi _{1}\right \|_{a}^{2}. }$$

(13)

Bilinear form and right-hand side are given, respectively, by

$$\displaystyle{ b(v,\varphi ) = b_{\mathrm{std}}(A;v,\varphi ):= \left \langle v(0),\varphi _{0}\right \rangle +\int _{I}\left \langle v',\varphi _{1}\right \rangle + \left \langle Av,\varphi _{1}\right \rangle }$$

(14)

and $\ell(\varphi ) = \left \langle w,\varphi _{0}\right \rangle +\int _{I}\left \langle \,f,\varphi _{1}\right \rangle$. We denote the constants of b _std by C _std etc.

The norm $\left \|\cdot \right \|_{1}$ in (13) slightly differs from the corresponding definition in [8] because it involves v(T) instead of v(0). This modification offers the following advantage, which was already observed in [1]: the norms in (13) mimic the energy norm for a linear elliptic problem in that the operator v ↦ b(v, ⋅ ) is an isometry. We provide a proof because its arguments will be used in what follows.

Proposition 1 (Isometry)

For every v ∈ H ₁ , we have $\left \|b(v,\cdot )\right \|_{H_{2}^{{\ast}}} = \left \|v\right \|_{1}.$

Proof

In view of $\int _{I}\left \langle v',v\right \rangle = \left \|v(T)\right \|_{W}^{2} -\left \|v(0)\right \|_{W}^{2}$, the symmetry of A and (11), we have the identity

$$\displaystyle{ \left \|v(0)\right \|_{W}^{2} +\int _{ I}\left \|A^{-1}v' + v\right \|_{ a}^{2} = \left \|v(T)\right \|_{ W}^{2} +\int _{ I}\left \|A^{-1}v'\right \|_{ a}^{2} + \left \|v\right \|_{ a}^{2} = \left \|v\right \|_{ 1}^{2} }$$

(15)

for every v ∈ H ₁. On the one hand, this gives, for every v ∈ H ₁, φ ∈ H ₂,

$$\displaystyle{ \begin{array}{rl} b(v,\varphi )& = \left \langle v(0),\varphi _{0}\right \rangle +\int _{I}\left \langle v',\varphi _{1}\right \rangle + \left \langle Av,\varphi _{1}\right \rangle \\ & \leq \left (\left \|v(0)\right \|_{W}^{2} +\int _{I}\left \|A^{-1}v' + v\right \|_{a}^{2}\right )^{1/2}\left \|\varphi \right \|_{2} = \left \|v\right \|_{1}\left \|\varphi \right \|_{2},\end{array} }$$

which implies $\left \|b(v,\cdot )\right \|_{H_{2}^{{\ast}}} \leq \left \|v\right \|_{1}$. On the other hand, choosing

$$\displaystyle{ \varphi _{0} = v(0),\qquad \varphi _{1} = v' + A^{-1}v }$$

(16)

and using again (15), we get $\left \|\varphi \right \|_{2} = \left \|v\right \|_{1}$ and

$$\displaystyle{ \begin{array}{rl} b(v,\varphi ) = \left \|v(0)\right \|_{W}^{2} +\int _{I}\left \langle v',v\right \rangle + \left \langle v',A^{-1}v\right \rangle + \left \langle Av,v\right \rangle = \left \|v\right \|_{1}^{2}. \end{array} }$$

Hence, $\left \|b(v,\cdot )\right \|_{H_{2}^{{\ast}}} \geq \left \|v\right \|_{1}$. □ 

Corollary 2 (Standard bilinear form)

The bilinear form b in (14) is continuous and satisfies the inf-sup condition with $C_{\mathrm{std}} = c_{\mathrm{std}} = 1$.

Proof

The equalities follow readily from Proposition 1. The proof of the non-degeneracy condition (5b) can be found in [8, Prop. 3.1]. □ 

3.2 Ultra-Weak Formulation

Discontinuous Galerkin methods, applications in optimization and stochastic PDEs motivate to consider solution notions with less regularity in time. In order to obtain such a solution notion for (12), one may multiply the differential equation with a test function

$$\displaystyle{ \varphi \in H_{T}^{1}(V,V ^{{\ast}}):=\{\varphi \in L^{2}(I;V )\mid \varphi ' \in L^{2}(I,V ^{{\ast}}),\varphi (T) = 0\}, }$$

integrate in time and by parts. This results in the ultra-weak formulation, which can be cast in the form (4) by choosing H ₁ = L ²(V ), and H ₂ = H _T ¹(V, V ^∗), with norms

$$\displaystyle{ \left \|v\right \|_{1}^{2} =\int _{ I}\left \|v\right \|_{V }^{2},\qquad \left \|\varphi \right \|_{ 2}^{2} =\int _{ I}\left \|\varphi \right \|_{a}^{2} + \left \|\varphi '\right \|_{ a;{\ast}}^{2}. }$$

Here, bilinear form and right-hand side are given, respectively, by

$$\displaystyle{ b(v,\varphi ) = b_{\mathrm{ult}}(A;v,\varphi ):=\int _{I} -\left \langle \varphi ',v\right \rangle + \left \langle Av,\varphi \right \rangle }$$

(17)

and $\ell(\varphi ) = \left \langle w,\varphi (0)\right \rangle +\int _{I}\left \langle \,f,\varphi \right \rangle + \left \langle \varphi ',f_{1}\right \rangle$, with f ∈ L ²(V ^∗), f ₁ ∈ L ²(V ) and w ∈ W. We denote the constants of b _ult by C _ult etc. Every solution of the standard weak formulation is one of the ultra-weak formulation.

Corollary 3 (Ultra-weak bilinear form)

The bilinear form b in (17) is continuous and satisfies the inf-sup condition with $C_{\mathrm{ult}} = c_{\mathrm{ult}} = 1$.

Proof

We exploit the duality with the standard weak formulation. Setting ι v(t):= v(T − t), t ∈ I = (0, T) and using the symmetry of A, we have

$$\displaystyle{ \forall v_{1} \in L^{2}(V ),v_{ 2} \in H_{T}^{1}(V,V ^{{\ast}})\quad b_{\mathrm{ ult}}(A;v_{1},v_{2}) = b_{\mathrm{std}}(A;\iota v_{2},\iota v_{1}); }$$

(18)

see [8, Lemma 4.1]. Since Proposition 1 holds also with H ₀ ¹(V, V ^∗):={ v ∈ H ¹(V, V ^∗)∣v(0) = 0} in place of H ¹(V, V ^∗), we thus deduce C _ult = C _std = 1 and c _ult = c _std = 1 with the help of (6). □ 

4 Galerkin Approximation in Space and Quasi-Optimality Constants

We review Galerkin approximation in space for the standard and the ultra-weak formulation and then derive identities for the corresponding quasi-optimality constants.

Let S be a finite-dimensional, nontrivial, and proper subspace of V. Observe that S is also a subspace of W and, with the identification S ^∗ = S, also of V ^∗. As a subspace of V ^∗, we can equip S = S ^∗ with

$$\displaystyle{ \left \|\ell\right \|_{a;{\ast}} =\sup _{\varphi \in V } \frac{\left \langle \ell,\varphi \right \rangle } {\left \|\varphi \right \|_{a}}\quad \text{as well as}\quad \left \|\ell\right \|_{a;S^{{\ast}}}:=\sup _{\varphi \in S} \frac{\left \langle \ell,\varphi \right \rangle } {\left \|\varphi \right \|_{a}}. }$$

The following relationship, which can be found, e.g., in [8, Proposition 2.5], will be crucial:

$$\displaystyle{ \sup _{\ell\in S} \frac{\left \|\ell\right \|_{a;{\ast}}} {\left \|\ell\right \|_{a;S^{{\ast}}}} = \left \|P_{S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})}:=\sup _{\left \|w\right \|_{a}=1}\left \|P_{S}w\right \|_{a}, }$$

(19)

where P _S is the W-orthogonal projection onto S satisfying $\left \langle P_{S}w,\varphi \right \rangle = \left \langle w,\varphi \right \rangle$ for all φ ∈ S and every w ∈ W.

4.1 Standard Weak Formulation

We first consider the standard weak formulation and define the spaces H ₁, H ₂, their norms and the bilinear form b as in Sect. 3.1. The Galerkin approximation with values in S is characterized by (7) with M = (M ₁, M ₂) where

$$\displaystyle{ M_{1} = H^{1}(S) \subset H_{ 1},\qquad M_{2} = S \times L^{2}(S) \subset H_{ 2}. }$$

(20)

In order to determine the associated inf-sup constant c _std; S in (5a), we first derive a discrete counterpart of Proposition 1. To this end, we define on M ₁ the following S-dependent variant of $\left \|\cdot \right \|_{1}$:

$$\displaystyle{ \left \|v\right \|_{1;S}^{2}:= \left \|v(T)\right \|_{ W}^{2} +\int _{ I}\left \|v\right \|_{a}^{2} + \left \|v'\right \|_{ a;S^{{\ast}}}^{2}, }$$

where we replaced the dual norm $\left \|\cdot \right \|_{a;{\ast}}$ of the time derivative with the discrete dual norm $\left \|\cdot \right \|_{a;S^{{\ast}}}$. This gives rise to

$$\displaystyle{ \tilde{c}_{\mathrm{std};S}:=\inf _{v\in M_{ 1}}\;\sup _{\varphi \in M_{2}} \frac{b(v,\varphi )} {\left \|v\right \|_{1;S}\left \|\varphi \right \|_{2}},\quad \tilde{C}_{\mathrm{std};S}:=\sup _{v\in M_{1}}\;\sup _{\varphi \in M_{2}} \frac{b(v,\varphi )} {\left \|v\right \|_{1;S}\left \|\varphi \right \|_{2}} }$$

and

$$\displaystyle{ \inf _{v\in M_{1}} \frac{\left \|v\right \|_{1;S}} {\left \|v\right \|_{1}} \,\tilde{c}_{\mathrm{std};S} \leq c_{\mathrm{std};S} \leq \inf _{v\in M_{1}} \frac{\left \|v\right \|_{1;S}} {\left \|v\right \|_{1}} \,\tilde{C}_{\mathrm{std};S}. }$$

(21)

Proposition 4 (Discrete isometry)

For every v ∈ M ₁ , we have

$$\displaystyle{ \left \|b(v,\cdot )\right \|_{M_{2}^{{\ast}}}:=\sup _{\varphi \in M_{2}} \frac{b(v,\varphi )} {\left \|\varphi \right \|_{2}} = \left \|v\right \|_{1;S}. }$$

Proof

In order to proceed as in the proof of Proposition 1, we introduce the discrete counterpart of A, namely the operator A _S : S → S ^∗ given by $\left \langle A_{S}v,\varphi \right \rangle = a(v,\varphi )$, for every v, φ ∈ S. In analogy to (11), we have $\left \langle \ell,A_{S}^{-1}\ell\right \rangle =\| A_{S}^{-1}\ell\|_{a}^{2} = \left \|\ell\right \|_{a;S^{{\ast}}}^{2}$. We thus conclude as in the proof of Proposition 1, upon replacing φ = (φ ₀, φ ₁) in (16) with φ ₀ = v(0) ∈ S, φ ₁ = v + A _S ⁻¹ v′ ∈ L ²(S). □ 

Consequently, the counterparts of the identities in Corollary 2 are

$$\displaystyle{ \tilde{c}_{\mathrm{std};S} =\tilde{ C}_{\mathrm{std};S} = 1, }$$

(22)

which imply a symmetric error estimate for $\left \|\cdot \right \|_{1;S}$, similar to the one in [6]. For $\left \|\cdot \right \|_{1}$ instead, we have:

Theorem 5 (Quasi-optimality in $H^{1}(V,\left \|\cdot \right \|_{a};V ^{{\ast}},\left \|\cdot \right \|_{a;{\ast}})$)

The quasi-optimality constant of the Galerkin method (20) is given in terms of the W-projection onto S by

$$\displaystyle{ q_{\mathrm{std};S} = \left \|P_{S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})}. }$$

Proof

Identity (19) entails that the ratio of the two norms in the trial space is

$$\displaystyle{ \sup _{v\in M_{1}} \frac{\left \|v\right \|_{1}} {\left \|v\right \|_{1;S}} = \left \|P_{S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})}, }$$

(23)

see [8, Proposition 2.5 and (3.14)]. We thus deduce

$$\displaystyle{ q_{\mathrm{std};S} = c_{\mathrm{std};S}^{-1} =\sup _{ v\in M_{1}} \frac{\left \|v\right \|_{1}} {\left \|v\right \|_{1;S}} = \left \|P_{S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})}. }$$

(24)

by using Corollary 2 in (9) and (22) in (21). □ 

Remark 6 (Non-symmetric case)

If a is not symmetric, Theorem 5 can be generalized to

$$\displaystyle{ \kappa _{a}^{-1}\left \|P_{ S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})} \leq q_{\mathrm{std};S} \leq \kappa _{a}\left \|P_{S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})}, }$$

where $\left \|\cdot \right \|_{a}$ is given by the symmetric part of a and κ _a depends on C _a and ν _a , with κ _a  = 1 whenever a is symmetric. To this end, the bilinear form is split into its symmetric and skew-symmetric part, where the latter part is treated as a perturbation. An alternative and more general approach is offered by [8]. That analysis appears to be simpler but we only have $\kappa _{a} = \sqrt{2}$ if a is symmetric and one adopts the above energy-norm setting.

4.2 Ultra-Weak Formulation

We turn to Galerkin approximation based upon the ultra-weak formulation. Let the spaces H ₁, H ₂, their norms and the bilinear form b be given as in Sect. 3.2. The corresponding Galerkin approximation with values in S is characterized by (7) with M = (M ₁, M ₂) where

$$\displaystyle{ M_{1} = L^{2}(S) \subset H_{ 1},\quad M_{2} = H_{T}^{1}(S):= H^{1}(S) \cap H_{ T}^{1}(V,V ^{{\ast}}) \subset H_{ 2}. }$$

(25)

Also, the Galerkin approximation of the ultra-weak formulation generalizes the Galerkin approximation of the standard weak formulation. Moreover:

Theorem 7 (Quasi-optimality in $L^{2}(V,\left \|\cdot \right \|_{a})$)

The quasi-optimality constant of the ultra-weak Galerkin method (25) is determined in terms of the W-projection onto S by

$$\displaystyle{ q_{\mathrm{ult};S} = \left \|P_{S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})}. }$$

Proof

We exploit again duality. To this end, notice first that Proposition 4 and (23) hold also if H ¹(S) is replaced by H ₀ ¹(S):={ v ∈ H ¹(S)∣v(0) = 0}. Hence, the discrete inf-sup constant does not change under this replacement and (18) yields c _ult; S  = c _std; S . We thus obtain

$$\displaystyle{ q_{\mathrm{ult};S} = c_{\mathrm{ult};S}^{-1} = c_{\mathrm{ std};S}^{-1} = \left \|P_{ S}\right \|_{\mathcal{L}(V,\left \|\cdot \right \|_{a})} }$$

by using Corollary 3 in (9) and (24). □ 

Theorems 5 and 7 with W = L ²(Ω), V = H ₀ ¹(Ω) and A = −Δ yield (3).