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Time-dependent Steklov--Poincar\'e operators and space-time Robin--Robin decomposition for the heat equation

Domain decomposition methods are a set of widely used tools for parallelization of partial differential equation solvers. Convergence is well studied for elliptic equations, but in the case of parabolic equations there are hardly any results for general Lipschitz domains in two or more dimensions. The aim of this work is therefore to construct a new framework for analyzing nonoverlapping domain decomposition methods for the heat equation in a space-time Lipschitz cylinder. The framework is based on a variational formulation, inspired by recent studies of space-time finite elements using Sobolev spaces with fractional time regularity. In this framework, the time-dependent Steklov--Poincar\'e operators are introduced and their essential properties are proven. We then derive the interface interpretations of the Dirichlet--Neumann, Neumann--Neumann and Robin--Robin methods and show that these methods are well defined. Finally, we prove convergence of the Robin--Robin method and introduce a modified method with stronger convergence properties.


Introduction
Domain decomposition methods enable the usage of parallel and distributed hardware and are commonly employed when approximating the solutions to elliptic equations. The basic idea is to first decompose the equation's spatial domain into subdomains. The numerical method then consists of iteratively solving the elliptic equation on each subdomain and thereafter communicating the results via the boundaries to the neighboring subdomains. An in-depth survey of the topic can be found in the monographs [25,29].
A recent development in the field is to apply this approach to parabolic equations. The decomposition into spatial subdomains is then replaced by the decomposition into space-time cylinders. In general, space-time decomposition schemes enable additional parallelization and less storage requirements when combined with a standard numerical method for parabolic problems. The methods have especially gained attention in the contexts of parallel time integrators; surveyed in [8], space-time finite elements; surveyed in [26], and parabolic problems with a spatial domain given by a union of domains with very different material properties [1,13].
There have been several studies concerning the convergence and other theoretical aspects of space-time decomposition schemes applied to parabolic equations on onedimensional or rectangular spatial domains; see [6,9,10,11,19,20]. However, there are hardly any convergence results for more general settings, e.g., with spatial Lipschitz domains in higher dimensions. The one exception is the convergence analysis considered in [12] for an optimized Schwarz waveform relaxation, where a methodspecific variational approach is proposed. The latter is similar to the analysis used by Lions for the Robin-Robin method applied to elliptic equations [23].
This lack of convergence results is in stark contrast to the elliptic setting, where the analysis of nonoverlapping domain decomposition schemes for Lipschitz domains in higher dimensions is standard. The basic elliptic framework is to reformulate the method as an iteration scheme posed on the intersections of the subdomains in terms of Steklov-Poincaré operators. This is the basis for the analysis of the Dirichlet-Neumann and Neumann-Neumann methods [25]. The Steklov-Poincaré framework even yields the convergence of the Robin-Robin method applied to nonlinear elliptic equations [4]. Note that the time-dependent Steklov-Poincaré operators were introduced without any analysis in [12].
Hence, the aim of this study is twofold. First, we strive to introduce a new variational framework for time-dependent Steklov-Poincaré operators and to derive the essential properties of these operators, e.g., coercivity and bijectivity. The latter property implies that the space-time generalizations of the Dirichlet-Neumann, Neumann-Neumann, and Robin-Robin methods are all well defined in the parabolic setting. Second, we will employ the new framework to give a first rigorous proof of the Robin-Robin method's convergence when applied to parabolic equations.
For sake of simplicity, we will restrict our attention to the heat equation with homogeneous Dirichlet conditions. However, our analysis does not depend on any symmetry of the underlying bilinear form, and a similar framework can be used for more general parabolic problems and boundary conditions. To limit the technicalities further, we consider the equation for all times t ∈ R, which allows a variational for- mulation with the same test and trial spaces. Our model problem is therefore the heat equation on the space-time cylinder Ω × R, i.e., where the spatial domain Ω ⊂ R d , d = 2, 3, is bounded with boundary ∂ Ω . Note that a homogeneous initial condition at time t 0 can be incorporated by prescribing a source term f that is zero for times t < t 0 . Next, we decompose the spatial domain Ω into nonoverlapping subdomains Ω i , i = 1, 2, with boundaries ∂ Ω i , and denote the interface separating the subdomains Ω i by Γ . That is, The space-time cylinder Ω × R is thereby decomposed into Ω i × R, i = 1, 2, as illustrated in Figure 1. The current setting is also valid for spatial subdomains Ω i given as unions of nonadjacent subdomains, i.e., With a fixed domain decomposition we can reformulate the heat equation as equations on Ω i × R, i = 1, 2, connected via transmission conditions on Γ × R. More precisely, we have the parabolic transmission problem where ν i denotes the unit outward normal vector of ∂ Ω i . Alternating between the decomposed space-time cylinders and the transmission conditions generates the spacetime generalizations of the Dirichlet-Neumann and Neumann-Neumann methods.
As ν 1 = −ν 2 on Γ , the transmission conditions are also equivalent to the Robin conditions ∇u 1 · ν i + su 1 = ∇u 2 · ν i + su 2 on Γ × R for i = 1, 2, where s is a non-zero parameter. The latter reformulation gives rise to the space-time Robin-Robin method, for which we will prove convergence. The method is given by finding (u n 1 , u n 2 ) for n = 1, 2, . . . such that where u 0 2 is a given initial guess, s > 0 is a fixed method parameter, and u n i approximates u| Ω i ×R . Note that the method per se is sequential, but the computation of each term u n i can be implemented in parallel if Ω i is a union of nonadjacent subdomains. The standard variational framework for parabolic problems, based on solutions in the space is unfortunately not well suited for our domain decompositions. The issue is that two functions u i , i = 1, 2, in the above solution space, which coincide on Γ × R in the sense of trace, can not be "pasted" together into a new function in H 1 R, H −1 (Ω ) ; compare with [3, Example 2.14]. Hence, the transmission problem (3) does not necessarily yield a solution to the heat equation (1) in this context. In order to remedy this, we consider a framework with solutions in which originates from [22] and resolves the above issue. This H 1/2 -setting also enables a trace theory valid for spatial Lipschitz domains [3] and the related bilinear form becomes coercive-equivalent [5]. This will be the starting point for our analysis. Note that the H 1/2 -setting has been used by [3] in the context of boundary element methods and was later employed for space-time finite elements [5,17,27]. The analysis is organized as follows: In Sections 2 and 3 we derive the properties of the required function spaces and operators. We introduce the variational formulations and prove equivalence between the heat equation and the transmission problem in Section 4. The time-dependent Steklov-Poincaré operators are analysed in Section 5 and three standard space-time domain decompositions are proven to be well defined. In Sections 6 and 7 we prove convergence of the space-time Robin-Robin method in L 2 R, H 1 (Ω 1 ) × L 2 R, H 1 (Ω 2 ) and the convergence of a modified version of the method in respectively.

Preliminaries
The spatial domain Ω ⊂ R d and its decomposition (2) into Ω i , i = 1, 2, is assumed to satisfy the properties below.
For a description of Lipschitz domains, see [16,Chapter 6.2]. The assumptions are made in order ensure the existence of the spatial trace operator, as well as, to allow the usage of Poincaré's inequality. As a notational convention, operators only depending on space or time are "hatted" and their extensions to space-time are denoted without hats, e.g., Furthermore, c and C denote generic positive constants. Consider the spatial function spaces Here,T ∂ Ω i : H 1 (Ω i ) → H 1/2 (∂ Ω i ) denotes the spatial trace operator, see [16,Theorem 6.8.13]. The norm on V i and V 0 i is given by By Poincaré's inequality and Assumption 1 we have that v → ∇v L 2 (Ω i ) d is an equivalent norm on V i and V 0 i . The Hilbert spaces V , V i , and V 0 i are all separable. The space H 1/2 (∂ Ω i ) is defined as Denoting the extension by zero from Γ to ∂ Ω i byÊ i we define the Lions-Magenes space as Since H 1/2 (∂ Ω i ) is a separable Hilbert space, so is Λ . On V i the trace operator takes the formT (Γ ), L 2 (Γ )] 1/2 , which shows that Λ is independent of i = 1, 2. This reference is without proof, but a proof exists for the case of a smooth Ω in [21,Theorem 11.7]. The independence of Λ on i = 1, 2 is assumed to be true for the sake of presentation, but it is not a necessary assumption for our convergence analysis; see [4].
For the temporal function space H s (R), s ∈ [0, 1], we use the Fourier definition Here,F is the Fourier transform and L 2 C (R) is the complexification of the real Hilbert space L 2 (R). Note that this is equivalent to the Sobolev-Slobodetskii definition (5) for s = 1/2 and ∂ Ω i = R; see [28,Lemma 16.3]. We then introduce the temporal Hilbert transform byĤ From [15,Chapters 4,5] we have thatĤ : whereM sgn v(ξ ) = −i sgn(ξ )v(ξ ). The formula (7) combined with the definition (6) shows thatĤ : H s (R) → H s (R) is an isomorphism. We also introduce the temporal half-derivatives as∂ It is clear from the definition (6) that∂ 1/2 ± : H 1/2 (R) → L 2 (R) are bounded linear operators. The important relations between these operators are given in the lemma below.

Lemma 1
For v ∈ H 1/2 (R) one has the equalitieŝ ∂ 1/2 Moreover, for v ∈ H 1 (R) and w ∈ H 1/2 (R) one has the fractional integration by parts formula (∂ t v , w) L 2 (R) = (∂ 1/2 Proof Let v ∈ H 1/2 (R) and observe that The Fourier characterization of the operatorsĤ ,∂ 1/2 A similar argument, together with the fact that |F v| 2 is an even function, shows that for v ∈ H 1/2 (R). Finally, the fractional integration by parts formula follows by the The Fourier characterization of the operator∂ 1/2 is an equivalent norm on H 1/2 (R).

Tensor spaces
Inspired by the finite element analysis in [27], we identify our Bochner spaces in space-time as tensor spaces. A general introduction to tensor spaces can be found in [30,Chapter 3.4].
We denote the algebraic tensor product of two (real) separable Hilbert spaces X,Y by X ⊗Y . For elements of the form x ⊗ y the inner product is defined as and for arbitrary elements in X ⊗Y the definition is extended by linearity. The closure of X ⊗ Y with respect to this norm is denoted by X⊗Y . From these definitions it follows that X⊗Y = Y⊗ X.
If {x k } k≥0 and {y } ≥0 are orthonormal bases of X and Y , respectively, then {x k ⊗ y } k, ≥0 is an orthonormal basis of X⊗Y and every v ∈ X⊗Y can be represented as where {c k, } k, ≥0 are real coefficients. This follows from [30,Theorem 3.12].
We recall the following result on the extensions of operators to tensor spaces. The proof of Lemma 2 can be found in [2, Section 12.4.1].
Lemma 2 Let X k ,Y k , k = 1, 2, be separable Hilbert spaces, and A :

From Lemma 2 it follows that the parabolic trace operators
are bounded. Furthermore, we have the identity This is easily proven by validating the identity on the dense subset L 2 (R) ⊗V i and using the continuity of the operators. Note that the restrictions, as well as the extension by zero E i = I⊗Ê i , are all well defined operations due to Lemma 2.
For any separable Hilbert space X one has the identification The Sobolev-Bochner spaces below will make up the core of the analysis: Lemma 3 Suppose Assumption 1 holds. Then we have the identities Proof Throughout the proof let {x k } k≥0 be an orthonormal basis of L 2 (R). Let {y } ≥0 be an orthonormal basis of a separable Hilbert space Y given by the context, and the corresponding {z k } k≥0 are defined as in (8).
To prove the identity (9) recall that V 0 [16,Theorem 6.6.4]. This observation together with the continuity of T ∂ Ω i then yields that every v ∈ L 2 (R)⊗V 0 i satisfies It then follows by the orthonormality of {x k } k≥0 that Hence, z k ∈ V 0 i , k = 0, 1, . . . , and from (8) we can conclude that v ∈ L 2 (R)⊗V 0 i . The proof of (10) is similar after observing the definition To prove (11) first observe that E i can be interpreted as a map from L 2 (R)⊗Λ into L 2 (R)⊗ H 1/2 (∂ Ω i ), by Lemma 2. This immediately gives the inclusion from left to right. Conversely, assume that By (8), we have the representation Lemma 4 If Assumption 1 holds, then Z is dense in L 2 (Γ × R).
Proof The spaces H 1/4 (R) and Λ are dense in L 2 (R) and L 2 (Γ ), respectively; see [28,Lemma 15.10] and [4,Lemma 4.2]. Therefore, by [30,Theorem 3.12], the corresponding algebraic tensor space [30,Theorem 3.12], that our definition of T ∂ Ω i coincide with the definition given in [3,Lemma 2.4]. Hence, the restricted trace operator is well defined and bounded. If Lemma 6 Let X be a separable Hilbert space and s ∈ [0, 1]. The Hilbert transform Proof According to Lemma 2 the operatorĤ ⊗ I extends to a bounded linear op-

Lemma 7 If Assumption 1 holds then the restricted operators
Proof By different choices of s ∈ [0, 1] and separable Hilbert spaces X in Lemma 6 one obtains the isomorphisms Since the operators coincide on W i we have that H i : and by density the identity holds for v ∈ W i ⊂ L 2 (R)⊗V i .
Finally, let ϕ ∈ [0, π/2] be a parameter to be chosen later and define

Variational space-time formulations
By Lemma 2 the extensions of the spatial gradient are all bounded linear operators. Note that we leave out the dependence on i = 1, 2 on the above operators for the sake of readability. The bilinear forms a : W ×W → R and a i : respectively. For f ∈ W * the weak, or variational, formulation of the heat equation (1) is to find u ∈ W such that Similarly Here, X * denotes the dual of the Hilbert space X and ·, · denotes the dual pairing in X * × X. The weak problem (14) can be derived by multiplying (1) by v ∈ W , integrating and using the fractional integration by parts formula from Lemma 1 extended to the tensor setting. The weak formulations (15) follow in the same way. A bilinear form a : and coercive-equivalent in X with the isomorphism A : The following result follows the argumentation in [17, Section 2.8], which is a summary of the coercive-equivalency idea from [5].
Lemma 8 Let Assumption 1 be valid. The bilinear forms a i are then bounded in W i , coercive in L 2 (R)⊗ H 1 (Ω i ), and coercive-equivalent in W i with the isomorphisms H ϕ i for a sufficiently small ϕ > 0. An analogous result holds for the bilinear form a. In particular there exists unique solutions to (14) and (15), respectively.
Proof We will only consider the case with the bilinear form a i , as the same proof holds for a. Here, the boundedness of a i on W i follows directly by the Cauchy-Schwarz inequality together with the boundedness of the operators ∇ and ∂ 1/2 ± . To prove the coercivity properties first observe that the identities in Lemma 1 can trivially be validated on H 1/2 (R) ⊗ L 2 (Ω i ) and extended to H 1/2 (R)⊗ L 2 (Ω i ) by density and the boundedness of the related operators. That is, Second, as Assumption 1 is valid, a similar density argument together with the observations regarding the equivalent norms in Section 2 yields the "extended" Poincaré's inequality and the fact that v → ∂ 1/2 is an equivalent norm on W i . Third, by Lemma 6, ∇H i is a bounded operator from With these observations we have the bound for every v ∈ W i . Thus, by choosing a sufficiently small ϕ > 0 we have the sought after coercive-equivalency and choosing ϕ = 0 yields the coercivity bound The derived properties of a i yield the existence and uniqueness of a solution for (15), which follows by [17,Corollary 2.2].
The next lemma is required in order to consider non-homogeneous boundary values on the interface.

Lemma 9
Suppose that Assumption 1 holds. For any η ∈ Z there exists u i ∈ W i such that T i u i = η and It follows from (10) that u i ∈ W i . The bound u i W i ≤ C η Z follows by [3, p. 515] together with the definitions of the related norms.
According to Lemma 9 there exists a bounded linear operator F i : with left inverse T i , i.e., T i F i η = η for η ∈ Z. From Lemma 8 there also exists a bounded linear operator G i : These two operators are the main ingredients for stating the transmission problem and then reformulating the problem on the space-time interface Γ × R. Before moving on to the transmission problem it is necessary to prove that we can "paste" together functions in our H 1/2 -framework. This is the purpose of the following lemma.

Lemma 10 Suppose that Assumption 1 holds. Let
In order to prove the L 2 (R)⊗V -regularity of v, let {x k } k≥0 be an orthonormal basis of L 2 (R) and {(y i ) } ≥0 be orthonormal bases of V i , i = 1, 2. The corresponding elements {(z i ) k } k≥0 ⊂ V i are defined as in (8). This yields the representation That is,T 1 (z 1 ) k =T 2 (z 2 ) k , for all k = 0, 1, . . ., and from [4, Lemma 4.6] it follows that is a Cauchy sequence in L 2 (R)⊗V . Hence, v is also an element in L 2 (R)⊗V .
The H 1/2 (R)⊗ L 2 (Ω )-regularity of v follows in a similar fashion, by expanding the v i elements in terms of an orthonormal basis {x k } k≥0 of H 1/2 (R) and an orthonormal basis {(y i ) k } k≥0 of L 2 (Ω i ) together with the observation that z k 2 L 2 (Ω ) = (z 1 ) k 2 L 2 (Ω 1 ) + (z 2 ) k 2 L 2 (Ω 2 ) . In conclusion, v ∈ W . Conversely, let v ∈ W . From (10) we get that v i = v| Ω i ×R is an element in W i and, by Lemma 5, T i v i ∈ Z. Let {x k } k≥0 be an orthonormal basis of L 2 (R) and {y } ≥0 an orthonormal basis of V . The related elements {z k } k≥0 ⊂ V are given by (8) and we obtain [4,Lemma 4.5] it follows thatT 1 (z 1 ) k =T 2 (z 2 ) k for every k = 0, 1, . . . Therefore and the sought after equality is obtained.
The existence and uniqueness theory holds for any f ∈ W * , but for the convergence analysis we need a stricter requirement. For simplicity we assume the following:

Assumption 2
The source term f is an element in L 2 (Ω × R).
With Assumption 2 we can define f i = f | Ω i ×R , which we will assume to hold for the remainder of the analysis. The weak transmission problem is then to find (u 1 , u 2 ) ∈ W 1 ×W 2 such that Lemma 11 Suppose that Assumptions 1 and 2 hold. Then the weak heat equation is equivalent to the weak transmission problem in the following way: If u solves (14) then (u 1 , u 2 ) = ( u| Ω 1 ×R , u| Ω 2 ×R ) solves (18). Conversely, if (u 1 , u 2 ) solves (18) then u = {u i on Ω i × R , i = 1, 2} solves (14). In particular, there exists a unique solution to (18).
The weak form of the space-time Robin-Robin method is the following: Let u 0 2 ∈ W 2 be an initial guess and s > 0 be the fixed method parameter. For each n = 1, 2, . . . find functions (u n 1 , u n 2 ) ∈ W 1 ×W 2 such that The weak forms of the Dirichlet-Neumann and Neumann-Neumann methods can be given in a similar form, but are not stated here since they are the same as their elliptical counterparts; see [25,Chapter 1.3].

Time-dependent Steklov-Poincaré operators
The goal is now to reformulate the transmission problem and the space-time domain decomposition methods to problems and methods given on the interface Γ × R. In order to do so, we first introduce the time-dependent Steklov-Poincaré operators S i : Z → Z * and the functionals χ i ∈ Z * by respectively. We also write S = S 1 + S 2 and χ = χ 1 + χ 2 . We can then introduce the weak Steklov-Poincaré equation Sη = χ in Z * or, equivalently, Remark 3 The Steklov-Poincaré operators do not depend on the choice of extension F i . For an arbitrary extension R i : Z → W i such that T i R i µ = µ we have, by (9), that F i µ − R i µ ∈ W 0 i . Combining this with (16) implies that The Steklov-Poincaré operators have the corresponding properties as the bilinear forms a i in Lemma 8.

Lemma 12 Suppose that Assumptions 1 and 2 hold. Then the time-dependent Steklov-Poincaré operator S i is bounded and satisfies the coercivity bound
Moreover, S i is coercive-equivalent in Z with the isomorphism H ϕ Γ for ϕ > 0 small enough. That is, S i η, H ϕ Γ η ≥ c η 2 Z for all η ∈ Z. Analogous results hold for S.
Proof Throughout the proof, let η, µ ∈ Z be arbitrary elements. The boundedness of S i is proved by Lemmas 8 and 9, since We also obtain the coercivity bound using Lemma 8. From (13) we have (9). This together with (16) implies that Then Lemmas 5 and 8 give that The bounds for S follow by summing the bounds for S 1 and S 2 .
From Lemma 4 we have the Gelfand triple We introduce the Riesz isomorphism J : L 2 (Γ × R) → L 2 (Γ × R) * and note that the Gelfand triple implies the identity We trivially also have the bounds and Jη, η ≥ η 2 Theorem 1 Suppose that Assumptions 1 and 2 hold and let s ≥ 0. Then the operators sJ + S i , sJ + S : Z → Z * are isomorphisms. In particular, there exists a unique solution to the weak Steklov-Poincaré equation (20).
Proof For simplicity we assume that s = 0. Then S i is bijective if and only if for any z ∈ Z * there exists a unique solution η ∈ Z to S i η, µ = z, µ for all µ ∈ Z.
Since H ϕ Γ : Z → Z is an isomorphism this is equivalent to Note that µ → z, H ϕ Γ µ is also an element in Z * . Moreover, by Lemma 12 the bilinear form (η, µ) → S i η, H ϕ Γ µ is bounded and coercive in Z. Therefore, by the Lax-Milgram lemma [18, Chapter 6, Theorem 6] there exists a unique solution η to (25). It follows from Lemma 12 that S i and S −1 i are bounded and therefore S i is an isomorphism. Since we have (23) and (24) the proof can be generalized to any s ≥ 0. The statement for S is proven similarly to S i . Lemma 13 Suppose that Assumptions 1 and 2 hold. The weak transmission problem and the weak Steklov-Poincaré equation are equivalent in the following way: If (u 1 , u 2 ) solves (18) then η = T i u i solves (20). Conversely, if η solves (20) then The proof of Lemma 13 is immediate after writing out the definitions of the Steklov-Poincaré operators S i , i = 1, 2.
We will now introduce the iterations on the interface Γ × R and show equivalence to the corresponding domain decomposition methods. The first one is the weak Peaceman-Rachford iteration, which is given by finding (η n 1 , η n 2 ) ∈ Z × Z, for n = 1, 2, . . . , such that Here, η 0 2 ∈ Z is an initial guess. Lemma 14 Suppose that Assumptions 1 and 2 hold. Then the weak forms of the Robin-Robin method and the Peaceman-Rachford iteration are equivalent in the following way: If (u n 1 , u n 2 ) n≥1 solves (19) then (η n 1 , η n 2 ) n≥1 , defined as η n i = T i u n i , solves (26) with η 0 2 = T 2 u 0 2 . Conversely, if (η n 1 , η n 2 ) n≥1 solves (26) then (u n 1 , u n 2 ) n≥1 , defined as u n i = F i η n i + G i f i , solves (19) with u 0 2 = F 2 η 0 2 + G 2 f 2 .
Proof Let (u n 1 , u n 2 ) n≥1 ⊂ W 1 ×W 2 be a weak Robin-Robin approximation solving (19) and define η n i = T i u n i ∈ Z. This together with the first and third equation in (19) yields the identification u n i = F i η n i + G i f i . Entering these observations into the second and fourth equation in (19) gives the equalities for all µ ∈ Z. By the definitions of S i and χ i , the equalities (27) are equivalent to (26), i.e., (η n 1 , η n 2 ) n≥1 solves (26) with η 0 2 = T 2 u 0 2 . Conversely, let (η n 1 , η n 2 ) n≥1 ⊂ Z × Z be a weak Peaceman-Rachford approximation solving (26) and define u n i = F i η n i + G i f i . This definition ensures that the first and third equation in (19) are fulfilled, as well as T i u n i = η n i . As already observed, one Peaceman-Rachford iterate (26) is equivalent to (27). Inserting the definition of u n i into (27) then yields that the second and fourth equation in (19), i.e., (u n 1 , u n 2 ) n≥1 solves (19) Corollary 1 Suppose that Assumptions 1 and 2 hold. Then the weak forms of the Robin-Robin method and the Peaceman-Rachford iteration are well defined, that is, (19) and (26) have unique solutions at each iteration.
Proof It follows from Theorem 1 that (26) has a unique solution at each iteration and, by Lemma 14, so does (19).
The theory above applies to the weak forms of the Dirichlet-Neumann and Neumann-Neumann methods as well. To this end, we introduce two preconditioned fixed point iterations on the interface Γ × R. The first one is given by finding η n ∈ Z, for n = 1, 2, . . . , such that Here, η 0 is an initial guess and s 0 ∈ (0, 1) is a method parameter. The second one is given by finding (λ n 1 , λ n 2 , η n ) ∈ Z × Z × Z, for n = 1, 2, . . . , such that Again, η 0 is an initial guess and s 1 , s 2 > 0 are method parameters.
Lemma 15 Suppose that Assumptions 1 and 2 hold. The weak Dirichlet-Neumann method is equivalent to the interface iteration (28) and the weak Neumann-Neumann method is equivalent to the interface iteration (29).
Here, the definition of the methods being equivalent and the proofs thereof follow in the same fashion as for Lemma 14. By Theorem 1 and Lemma 15, we now trivially have the following existence and uniqueness results.
Corollary 2 Suppose that Assumptions 1 and 2 hold. The weak forms of the Dirichlet-Neumann and Neumann-Neumann methods, and their corresponding interface iterations (28) and (29), respectively, are all well defined.
The standard convergence theories of the Dirichlet-Neumann and Neumann-Neumann methods [25,Chapter 4] do not apply for the heat equation since the time-dependent Steklov-Poincaré operators are not symmetric. In the following sections we instead prove convergence results for Robin-Robin-type methods.

Convergence of the Robin-Robin method
The variational framework appears to be too general for analyzing the convergence of the Peaceman-Rachford iteration. Therefore, we introduce the Steklov-Poincaré operators as affine unbounded operators on L 2 (Γ × R) before we prove that the iteration converges. To this end, let and define the unbounded affine operators In L 2 (Γ × R) the Steklov-Poincaré equation is to find η ∈ D(S ) such that and the Peaceman-Rachford iteration takes the following form: For each n = 1, 2, . . .
Here, η 0 2 ∈ D(S 2 ) is a given initial guess. We now verify that (30) is indeed a restriction of the weak Steklov-Poincaré equation (20).
Proof From (30) we have Therefore, we get by (22) that A similar result holds for the Peaceman-Rachford iteration. The proof is left out since it is similar to the proof of Lemma 16.
Lemma 18 Suppose that Assumptions 1 and 2 hold. Then S i , i = 1, 2, satisfy the monotonicity property for all η, µ ∈ D(S i ).
(32) Moreover, for any s ≥ 0 the operators sI + S i : D(S i ) → L 2 (Γ × R), i = 1, 2, are bijective. Similar results hold for S . In particular, there exists a unique solution to (30) and the iteration (31) is well defined.
Proof The monotonicity follows from (21) and (22), since for all η, µ ∈ D(S i ). Let µ ∈ L 2 (Γ × R) be arbitrary. Then χ i + Jµ ∈ Z * and, by Theorem 1, there exists a unique η ∈ Z such that (sJ Thus, we have shown that (sI + S i ) is bijective. The proof for S is similar and is therefore left out.
In order to proceed with the convergence analysis, we require the following regularity of the solution to the weak heat equation (14).
are elements in L 2 (Γ × R) * , where u ∈ W is the solution to (14).

Remark 4
The assumption is somewhat implicit, but can be interpreted as the solution having a normal derivative ∇u · ν i on the space-time interface belonging to L 2 (Γ × R). This holds, for instance, if the solution satisfies the additional regularity To see this, first observe that a Lipschitz manifold ∂ Ω i has a normal vector ν i in L ∞ (∂ Ω i ) d . The additional regularity of u then yields that each term ∂ j u (ν i ) j of the normal derivative becomes an element in L 2 (Γ × R).
Proof Let η ∈ D(S ) be the solution to (30). By Lemmas 11 and 13, we have the identification u| Ω i ×R = F i η + G i f i , where u ∈ W is the solution to the weak heat equation (14). Assumption 3 then yields that is an element in L 2 (Γ × R) * , i.e., η ∈ D(S i ) for i = 1, 2.
Lemma 20 is a consequence of the abstract result [24,Proposition 1]. The latter requires the monotonicity (32) and the fact that the solution to (30) satisfies η ∈ D(S 1 ) ∩ D(S 2 ), which follows from Lemma 19. A simpler proof of Lemma 20 can be found in [4,Lemma 8.8].
We are now in a position to prove that the Robin-Robin method converges.
Proof From (32) and (33) we obtain as n tends to infinity. By Lemmas 11, 13 and 14, one has the identities This together with the last bound in the limit above yields that as n tends to infinity.
Remark 5 Note that the Robin-Robin method is unlikely to have a geometric convergence rate in the present continuous framework. This is indicated already in the elliptic case, by observing that the spatially discrete method has a convergence rate of the form 1 − O( √ h); see [7]. That is, a rate deteriorating to one as the spatial discretization parameter h tends to zero.

Modified Robin-Robin method
One can achieve a slightly stronger convergence result if one allows the evaluation of the (non-local) Hilbert transform in the method. Instead of the standard Robin-Robin method, we therefore also consider the method given by finding, for each n = 1, 2, . . . , functions (u n 1 , u n 2 ) ∈ W 1 ×W 2 such that Here, u 0 2 ∈ W 2 is an initial guess and s, ϕ > 0 are suitably chosen method parameters. We refer to (34) as the weak formulation of the modified Robin-Robin method. Moreover, we consider the weak modified Peaceman-Rachford iteration, as finding functions (η n 1 , η n 2 ) ∈ Z × Z, for each n = 1, 2, . . . , such that Here η 0 2 ∈ D(S ϕ 2 ) is an initial guess. We have an analogous result to Lemma 17 so that (36)   Theorem 3 Suppose that Assumptions 1 to 3 hold. Let u be the solution to the weak heat equation (14) and η be the solution to the L 2 -Steklov-Poincaré equation (30). The iterates (η n 1 , η n 2 ) n≥1 of the L 2 -modified Peaceman-Rachford iteration (36) converges to η, i.e., η n 1 − η Z + η n 2 − η Z → 0, as n tends to infinity. Moreover, the weak modified Robin-Robin approximation (u n 1 , u n 2 ) n≥1 converges to (u 1 , u 2 ) = ( u| Ω 1 ×R , u| Ω 2 ×R ), i.e., u n 1 − u 1 W 1 + u n 2 − u 2 W 2 → 0, as n tends to infinity.