A generalized inf-sup stable variational formulation for the wave equation

In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of $H^1(Q$) with $Q$ being the space-time domain, the classical assumption is to consider the right-hand side $f$ in $L^2(Q)$. Here, we analyze a generalized setting of this variational formulation, which allows us to prove unique solvability also for $f$ being in the dual space of the test space, i.e., the solution operator is an isomorphism between the ansatz space and the dual of the test space. This new approach is based on a suitable extension of the ansatz space to include the information of the differential operator of the wave equation at the initial time $t=0$. These results are of utmost importance for the formulation and numerical analysis of unconditionally stable space-time finite element methods, and for the numerical analysis of boundary element methods to overcome the well-known norm gap in the analysis of boundary integral operators.

As for elliptic second-order partial differential equations, we consider the weak solution of the inhomogeneous wave equation in the energy space H 1 (Q) with respect to the spacetime domain Q := Ω × (0, T ). However, to ensure existence and uniqueness of a weak solution, we need to assume f ∈ L 2 (Q), see, e.g., [26,Theorem 5.1]. While this is a standard assumption to conclude sufficient regularity for the solution u, and therefore, to obtain linear convergence for piecewise linear finite element approximations u h , stability of common finite element discretizations require in most cases some CFL condition, which relates the spatial and temporal mesh sizes to each other, see, e.g., [25,26,29,31].
Although the variational formulation to find u in a suitable subspace of H 1 (Q) is welldefined for f being in the dual of the test space, this is not sufficient to establish unique solvability. This is mainly due to the missing information about the differential operator of the wave equation at t = 0 in the standard ansatz space. Hence, we are not able to define an isomorphism between the ansatz space and the dual of the test space. But such an isomorphism is an important ingredient in the analysis of equivalent boundary integral formulations for boundary value problems for the wave equation, and the numerical analysis of related boundary and finite element methods.
The proofs of the stability estimate (1.5) and Theorem 1.1 are based on an appropriate Fourier analysis, using the eigenfunctions of the spatial differential operator −∆ x , and the analysis of the related ordinary differential equation (1.7), which allows us also to present the essential ingredients for the new approach. So, for µ > 0, we consider the scalar ordinary differential equation The related variational formulation is to find u ∈ H 1 0, (0, T ) for a given right-hand side ,0 (0, T ) satisfies the terminal condition v(T ) = 0, and where the inner product ∂ t (·), ∂ t (·) L 2 (0,T ) makes both to Hilbert spaces. Note that the second initial condition ∂ t u(t) |t=0 = 0 enters the variational formulation (1.8) as natural condition. The dual space [H 1 ,0 (0, T )] ′ is characterized as completion of L 2 (0, T ) with respect to the Hilbertian norm where ·, · (0,T ) denotes the duality pairing as extension of the inner product in L 2 (0, T ), see, e.g., [27,Satz 17.3]. The continuity of the bilinear form (1.9) follows from we conclude unique solvability of (1.8) as well as the stability estimate   [18,28].
Since the variational formulation (1.8) is well-defined also for f ∈ [H 1 ,0 (0, T )] ′ , we are interested to establish, instead of (1.11), an inf-sup stability condition with a constant, which is independent of µ, and which later on can be generalized to the analysis of the variational problem (1.2).
The remainder of this paper is structured as follows: In Section 2, we present the main ideas in order to solve the ordinary differential equation (1.7). For this purpose, we introduce a suitable function space and prove a related inf-sup stability condition. Then, in Section 3, these results are generalized to analyze a variational formulation for the wave equation (1.1). The main result of this paper is given in Theorem 3.9, where we state bijectivity of the solution operator for the Dirichlet problem of the wave equation with zero boundary and initial conditions. Finally, in Section 4, we give some conclusions and comment on ongoing work.
In this work, C ∞ 0 (A) is the set of infinitely differentiable real-valued functions with compact support in any domain A ⊂ R d , d = 1, 2, 3, 4. The set C ∞ 0 (A) is endowed with the, usual for distributions, locally convex topology and is called the space of test functions on A. The set of (Schwartz) distributions [C ∞ 0 (A)] ′ is given by all linear and sequentially continuous functionals on C ∞ 0 (A). For given u ∈ L 2 (0, T ), we define the extension u ∈ L 2 (−T, T ) by The application of the differential operator µ to u is defined as distribution on (−T, T ), i.e., for all test functions ϕ ∈ C ∞ 0 (−T, T ), we define This motivates to consider the dual space [H 1 0 (−T, T )] ′ of H 1 0 (−T, T ), which is characterized as completion of L 2 (−T, T ) with respect to the Hilbertian norm where ·, · (−T,T ) denotes the duality pairing as extension of the inner product in L 2 (−T, T ), see, e.g., [27,Satz 17.3]. In other words, for [ , with this abstract inner product, [H 1 0 (−T, T )] ′ is a Hilbert space. Additionally, we define the subspace

be any continuous and injective extension operator with its adjoint operator
An example for such an extension operator is given by reflection in t = 0, i.e., consider the function v, defined for which leads to the constant c E = 2 in this particular case. With this, we prove the following lemma.

Proof.
First, we prove that . It holds true that z g|(−T,0) = 0, since we have i.e., is surjective. Third, the equality (2.4) follows from (2.5) and (2.6) for v = Rz for any z ∈ H 1 0 (−T, T ). The last assertion of the lemma is straightforward.
The last lemma gives immediately the following corollary.
i.e., the assertion is proven.
Next, we introduce ′ is unique and therefore, in the following, we identify the distribution µ u : Next, we state properties of the space H(0, T ). Clearly, (H(0, T ), · H(0,T ) ) is a normed vector space, and it is even a Banach space.
, induces the norm · H(0,T ) . Hence, the space (H(0, T ), ·, · H(0,T ) ) is even a Hilbert space, but this abstract inner product is not used explicitly in the remainder of this work.
Proof. First, we prove that for all z ∈ H 1 0 (−T, T ), where a µ (·, ·) is the bilinear form (1.9). The continuity of f u follows from for all z ∈ H 1 0 (−T, T ), where the estimate (1.10) is used. Using the definition (2.3), and integration by parts, this gives The equality (2.9) follows from the density of C ∞ 0 (−T, T ) in H 1 0 (−T, T ). The estimate (2.8) is proven by when using the norm representation (2.7), the equality (2.9), and the bound (1.10).
Next, by completion, we define the Hilbert space endowed with the Hilbertian norm · H(0,T ) , i.e., For each u n ∈ H 1 0, (0, T ), we define w n ∈ H 1 ,0 (0, T ) as unique solution of the backward problem i.e., w n ∈ H 1 ,0 (0, T ) solves the variational problem with the bilinear form (1.9). In particular for v = u n , this gives a µ (u n , w n ) = u n 2 L 2 (0,T ) .
Analogously to the estimate (1.13) for the solution of (1.7), we conclude for the solution w n of (2.10), i.e., For the zero extension u n ∈ L 2 (−T, T ) of u n ∈ H 1 0, (0, T ), we obtain, when using the norm representation (2.7), and (2.9), that and the assertion follows by completion for n → ∞.
is complete, i.e., a Hilbert space.
Next, we state the new variational setting for the scalar ordinary differential equation (1.7). For given f ∈ [H 1 ,0 (0, T )] ′ , we consider the variational formulation to find u ∈ H 0, (0, T ) such that i.e., the operator equation With the properties of the bilinear form a µ (·, ·), the unique solvability of the variational formulation (2.14), i.e., the main theorem of this section, is proven.
Proof. The assertion follows immediately from Lemma 2.10 and (2.13). For the first-order distributional derivative is identified with the function i.e., it is a regular distribution. To compute the second-order distributional derivative of u, we consider for all ϕ ∈ C ∞ 0 (−T, T ). Hence, solves the variational formulations (1.8) and (2.14) for the right-hand side which realizes the initial condition ∂ t u(t) |t=0 = v 0 := √ µ.
Remark 2.14 The variational formulation (2.14) is the weak formulation of the differential equation which can be written as coupled system, using u(t) = u(t) for t ∈ (0, T ), and u − (t) = u(t) for t ∈ (−T, 0), together with the transmission interface conditions

A generalized variational formulation for the wave equation
In this section, we generalize the approach, as introduced for the solution of the ordinary differential equation ( where ·, · Q denotes the duality pairing as extension of the inner product in L 2 (Q). Note that [H 1,1 0; ,0 (Q)] ′ is a Hilbert space, see Section 2. For given u ∈ L 2 (Q), we define the extension u ∈ L 2 (Q − ) by The application of the wave operator := ∂ tt − ∆ x to u is defined as a distribution on Q − , i.e., for all test functions ϕ ∈ C ∞ 0 (Q − ), we define This motivates to consider the dual space [H 1 0 (Q − )] ′ of H 1 0 (Q − ), which is characterized as completion of L 2 (Q − ) with respect to the Hilbertian norm where the inner product , and ·, · Q − denotes the duality pairing as extension of the inner product in L 2 (Q − ), see [27,Satz 17.3]. Note that [H 1 0 (Q − )] ′ is a Hilbert space, see Section 2. In addition we define the subspace for (x, t) ∈ Q − , and a given function v ∈ H 1,1 0;,0 (Q), which leads to a constant c E = 2 in this particular case. With this, we prove the following lemma as the counter part of Lemma 2.1.
The last lemma gives immediately the following corollary. Proof. Let g ∈ H −1 |Q (Q − ) be arbitrary but fixed. With Lemma 3.1, we have i.e., the assertion is proven.
Next, we introduce ′ is unique and therefore, in the following, we identify the distribution u : Next, we state properties of the space H(Q). Clearly, (H(Q), · H(Q) ) is a normed vector space and it is even a Banach space.   (3.7) is proven by Proof. For 0 = u ∈ H 0, (Q), there exists a non-trivial sequence (u n ) n∈N ⊂ H 1,1 0;0, (Q), u n ≡ 0, with lim n→∞ u − u n H(Q) = 0.
Analogously to the estimate (1.5) for the solution of (1.1), we conclude For the zero extension u n ∈ L 2 (Q − ) of u n ∈ H 1,1 0;0, (Q), we obtain, when using the norm representation (3.6) and (3.8), that and the assertion follows by completion for n → ∞.
While for the ordinary differential equation ( 2). Next, the following functions are given to get a first impression of the solution space H 0, (Q).

Conclusions and outlook
In this paper, we presented a new approach to set up a bijection for the solution of the wave equation, when the right-hand side is considered in the dual space of the test space of the variational formulation. For this, we had to enlarge the ansatz space to prove a related inf-sup stability condition. Based on these results, we aim to derive a space-time finite element method for the numerical solution of the wave equation, and of related problems, which is unconditionally stable, and which also allows for an adaptive resolution of the solution simultaneously in space and time, and for an efficient solution, which is also parallel in time. First numerical results are very promising, see [15], and the related numerical analysis is ongoing work, and will be published elsewhere.
The presented results on the existence and uniqueness of solutions for the wave equation, in particular the bijectivity results for the solution operator in related function spaces, are of utmost importance for the analysis of related boundary integral equations for the approximate solution of the wave equation by boundary element methods. Using the appropriate Dirichlet and Neumann trace operators, we are able to analyze the mapping properties of related boundary integral operators [23], i.e., boundedness and coercivity, to close the existing gap in using different norms, see, e.g., [22]. Note that this norm gap also results in error estimates, which are not optimal, see also [24] for first numerical results.
We end this paper with an outlook for possible extensions of the approach in Section 3. Since the constructions of the spaces H(Q), H 0, (Q) and the proofs in this section mainly rely on the treatment of the second-order temporal differential operator ∂ tt + µ with a parameter µ, a generalization of the results of this section to differential operators ∂ tt + A x , acting on vector fields or scalar fields is possible, where the second-order spatial differential operator A x has to fulfill certain properties, e.g., boundedness and ellipticity. A more detailed discussion is left for future work.