Singular perturbation results for linear partial differential-algebraic equations of hyperbolic type
aa r X i v : . [ m a t h . A P ] F e b SINGULAR PERTURBATION RESULTS FOR LINEAR PARTIALDIFFERENTIAL-ALGEBRAIC EQUATIONS OF HYPERBOLIC TYPE ∗ R. ALTMANN † , C. ZIMMER † Abstract.
We consider constrained partial differential equations of hyperbolic typewith a small parameter ε >
0, which turn parabolic in the limit case, i. e., for ε = 0.The well-posedness of the resulting systems is discussed and the corresponding solutionsare compared in terms of the parameter ε . For the analysis, we consider the systemequations as partial differential-algebraic equation based on the variational formulationof the problem. Depending on the particular choice of the initial data, we reach first-and second-order estimates. Optimality of the lower-order estimates for general initialdata is shown numerically. Key words.
PDAEs, first-order hyperbolic systems, singular perturbation
AMS subject classifications. 35L50 , , Introduction
Singularly perturbed differential and evolution equations have been analyzed for manydecades already (see [KP03] for a review) and cover the entire spectrum of elliptic, para-bolic, as well as hyperbolic systems. Resulting estimates can be used, e. g., in the theoryof boundary layers or as a tool for the design of numerical algorithms [Esh87], which alsoserves as motivation for the present work.For the particular case of one-dimensional gas networks, i. e., a coupled system of hy-perbolic equations, perturbation results have been derived in [EK18b]. In general, suchnetwork structures can be modeled as constrained partial differential equations, where theconstraints are naturally given by the junctions within the network, reflecting fundamen-tal physical properties. Such an approach leads to partial differential-algebraic equations (PDAEs), cf. [EM13, LMT13, Alt15], which may be interpreted as differential-algebraicequations in Banach spaces. For the sake of completeness, we would like to mentionthat one may also consider the network as a domain on which the differential equationsare stated [Mug14]. In this approach, however, inhomogeneous boundary conditions stillaccount for constraints on the solution.In this paper, we focus on singularly perturbed (linear) PDAEs of first order (in time)that are hyperbolic, i. e., we consider hyperbolic partial differential equations includinga small parameter ε , which underlie an additional constraint. This includes the alreadymentioned propagation of pressure waves in a network of gas pipes [Osi87, BGH11, JT14,EKLS +
18] as well as electro-magnetic-energy propagation in power networks [MWTA00,GHS16]. The singular perturbation of the considered PDAEs is characterized througha small parameter ε . In particular, the system is of hyperbolic nature for ε > ε = 0. The first main result of this paper compares the twocorresponding solutions and shows that they only differ by a term of order ε , as long asthe initial data is chosen appropriately. At this point, we would like to emphasize that Date : February 8, 2021. ∗ Research funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –Project-ID 446856041. such a condition does not occur in the respective finite-dimensional setting, which wasanalyzed in [KKO99, Ch. 2.5]. The second main contribution considers a second-orderapproximation of the original solution. Again, sufficient conditions on the initial dataand regularity assumptions on the right-hand sides are discussed, which guarantee the fullapproximation order of two. To show that the presented estimates are indeed sharp, weexamine specific numerical examples.The paper is organized as follows. In Section 2 we introduce the functional ana-lytic setting for linear first-order PDAEs of hyperbolic type, including a small param-eter 0 < ε ≪
1. We discuss the existence of mild and classical solutions and presentstability estimates as well as a particular example, which fits into the presented frame-work. Section 3 then covers the limit case for ε = 0, which is of parabolic nature. Here,the existence of weak solutions under various regularity assumptions is studied. The mainresults of the paper are devoted to the comparison of the solutions of the original andthe limit equations, leading to first and second-order estimates in terms of the parame-ter ε . Finally, the theoretical approximation orders a numerically verified by a number ofexperiments in Section 4.Throughout this paper, we use for estimates the notion a . b for the existence of ageneric constant c > a ≤ cb .2. Hyperbolic PDAE Model
In this section, we introduce the system class of interest namely first-order hyperbolicsystems with a small parameter ε , which satisfy an additional constraint. In order to provethe existence of mild and classical solutions, we first discuss the solvability of a relatedstationary problem. Finally, an example from the field of gas dynamics on networks ispresented, which fits in the given framework.2.1. Function spaces and system equations.
For a general formulation of constrainedhyperbolic systems of first order, we introduce the three Hilbert spaces P , M , and Λ . Thecorresponding solution components will be denoted by p , m , and λ , respectively. In general,one may think of p modeling a potential whereas m is a flow variable . The variable λ servesas Lagrange multiplier for the incorporation of the given constraint. We assume that P forms a Gelfand triple with pivot space H , i. e., P ֒ → H ∼ = H ∗ ֒ → P ∗ where all embeddingsare dense. On the other hand, M is assumed to be identifiable with its own dual space,i. e., M ∼ = M ∗ . For the applications in mind, M equals an L -space, see Section 2.4.As we consider time-dependent problems, appropriate solution spaces are given bySobolev-Bochner spaces; see [Rou05, Ch. 7] for an introduction. Denoting the space of qua-dratic Bochner integrable functions with values taken in a Banach space X by L (0 , T ; X ),we use the notion H m (0 , T ; X ), m ∈ N , for functions with higher regularity in time. More-over, we define for two Sobolev spaces X ֒ → X the space W (0 , T ; X , X ) := (cid:8) v ∈ L (0 , T ; X ) | ˙ v exists in L (0 , T ; X ) (cid:9) . Within this paper, we consider PDAEs of the form˙ p + A p − K ∗ m + B ∗ λ = g in P ∗ , (2.1a) ε ˙ m + K p + D m = f in M ∗ , (2.1b) B p = h in Λ ∗ (2.1c)with initial conditions for p (0) and m (0). Note that (2.1a) and (2.1b) are differentialequations, whereas (2.1c) reflects a constraint on p , which defines the PDAE structure.Further note that all three equations are formulated in the respective dual spaces, which INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 3 display the corresponding space of test functions. The precise assumptions are summarizedin the following.
Assumption . All operators A : H → H ∗ , K : P → M ∗ , D : M → M ∗ , B : P → Λ ∗ . are linear and bounded. The operator norm of A is denoted by C A ≥ K , D , and B . Further, we assume that the operator D is elliptic with c D k m k M ≤ hD m, m i and that B is inf-sup stable, i. e., there exists a positive constant β withinf µ ∈ Λ \{ } sup q ∈P\{ } hB q, µ ik q k P k µ k Λ = β > . Finally, we introduce the constant c K >
0, which satisfies c K k q ker k P ≤ kK q ker k M ∗ for all q ker ∈ P ker := ker B ⊆ P .As usual for constrained systems, the inf-sup stability of the constraint operator B isa crucial property for the well-posedness of the PDAE (2.1). This is due to the saddlepoint structure of the system equations. Further note that B is automatically inf-supstable if it is surjective and Λ ∗ is a finite-dimensional space; see [AZ18b]. Contrariwise,the inf-sup condition implies surjectivity of the operator B as well as injectivity of itsdual B ∗ . The classical result presented in [Bra07, Lem. III.4.2] implies the existence of aright inverse B − : Λ ∗ → P and the estimates kB q k Λ ∗ ≥ β k q k P and kB − h k P ≤ β − k h k Λ ∗ for all q ∈ P ⊥ ker and h ∈ Λ ∗ .The right-hand sides in (2.1) are of the form f : [0 , T ] → M ∗ , g : [0 , T ] → P ∗ , h : [0 , T ] → Λ ∗ . Finally, the parameter ε > ε ˙ m in (2.1b) takesthe role of a singular perturbation. The limit case for ε = 0 will be subject of Section 3.1.An initial value of p is called consistent if the difference p (0) − B − h (0) is an element of theclosure of P ker in H , which we denote by H ker in the sequel. For p (0) ∈ P , the consistencyconditions turns into B p (0) = h (0).2.2. An auxiliary problem.
It turns out that the following auxiliary problem is helpfulfor the upcoming analysis,( A + C A id) p − K ∗ m + B ∗ λ = g in P ∗ , (2.2a) K p + D m = 0 in M ∗ , (2.2b) B p = 0 in Λ ∗ . (2.2c)Note that the system does not include time derivatives of the variables but that the right-hand side may still be time-dependent. To show the existence of a unique solution ( p, m, λ )we first define an elliptic operator L . Lemma 2.2.
Given Assumption 2.1, the operator L := K ∗ D − K : P → P ∗ is linear,continuous, and non-negative. Furthermore, its restriction to P ker is elliptic, i. e., thereexists a constant c L > such that for all q ker ∈ P ker we have that hL q ker , q ker i ≥ c L k q ker k P . SINGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE
Proof.
By the ellipticity of D , the operator L is well-defined and its linearity and continuityare obvious. For the non-negativity, we apply the ellipticity of D , leading to hL q, q i P ∗ , P = hD − K q, K q i M , M ∗ = hD − K q, DD − K q i M , M ∗ ≥ c D kD − K q k M ≥ q ∈ P . In addition, for q ker ∈ P ker we can apply the properties of K . This yields hL q ker , q ker i P ∗ , P ≥ c D kD − K q ker k M ≥ c D C D kDD − K q ker k M ∗ ≥ c D c K C D k q ker k P , which shows the claimed ellipticity. (cid:3) Based on the newly introduced operator from the previous lemma, we now prove theexistence of a solution to (2.2).
Lemma 2.3 (Existence result for the auxiliary problem) . Given Assumption 2.1 and aright-hand side g ∈ H m (0 , T ; P ∗ ) for some m ∈ N , system (2.2) has a unique solution (cid:0) p, m, λ (cid:1) ∈ H m (0 , T ; P ker ) × H m (0 , T ; M ) × H m (0 , T ; Λ ) , which depends continuously on the right-hand side g .Proof. We show that for g ∈ P ∗ (independent of time) system (2.2) has a unique solution (cid:0) p, m, λ (cid:1) ∈ P ker × M × Λ . The result for a time-dependent right-hand side g ∈ H m (0 , T ; P ∗ ) then follows immediatelyby considering system (2.2) pointwise in time. The resulting solution is H m -regular in time,since all involved operators are time-independent.Now consider g ∈ P ∗ . Since the operator D is invertible, we can insert equation (2.2b)into (2.2a), which results in the system( L + A + C A id) p + B ∗ λ = g in P ∗ , B p = 0 in Λ ∗ . By standard arguments [BF91, Ch. II.1.1] this system has a unique solution p ∈ P ker , λ ∈ Λ , which is bounded in terms of g . The existence of m and the stability bound k p k P + k m k M + k λ k Λ . k g k P ∗ then follow by m = −D − K p . (cid:3) As another preparation for the existence results in the upcoming subsection, we considerthe following lemma.
Lemma 2.4.
Consider Assumption 2.1 and define the (unbounded) operator (2.3) A γ := " − γ A K ∗ −K −D /γ : D ( A γ ) ⊆ ( H ker × M ) → H ker × M for an arbitrary positive parameter γ > . Then, A γ generates a C -semigroup with thedomain D ( A γ ) = P ker × (cid:8) m ∈ M | ∃ h ∈ H ker : ( h , q ker ) H = hK ∗ m, q ker i for all q ker ∈ P ker (cid:9) . Proof.
Without loss of generality, we assume that A is non-negative. Otherwise, we con-sider A γ − γC A id H ker ×M and use [Paz83, p. 12]. This then updates the operator D in the(2 , A γ to D + γ C A id M , which is still elliptic on M for every γ > A q ker is an element of H ∗ ⊆ H ∗ ker ∼ = H ker for every q ker ∈ P ker , theoperator A γ is bounded on its domain D ( A γ ). We show that A γ is a densely defined,closed, and dissipative operator with a dissipative adjoint A ∗ γ . The statement then followsby [Paz83, Ch. 1.4, Cor. 4.4]. By Lemma 2.3 the operator A γ is linear, bounded, mapsfrom P ker × M to P ∗ ker × M ∗ , and has a bounded inverse. In particular, it holds that INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 5 A − γ ( g, f ) ∈ D ( A γ ) for all ( g, f ) ∈ H ker × M ֒ → P ∗ ker × M ∗ . This proves the closeness by asimple calculation. Furthermore, the operator A γ is dissipative by Definition 4.1 in [Paz83,Ch. 1.4]. To see this, consider a given ( p, m ) ∈ D ( A γ ) ֒ → P ker × M and choose the sameelement as test function under the embedding P ker × M ֒ → H ker × M . The adjoint A ∗ γ isdissipative as well, since the adjoint operators A ∗ : P → P ∗ and D ∗ : M → M ∗ have thesame properties as A and D , respectively.It remains to show that A γ is densely defined. Since D ( A γ ) is independent of γ , wemay fix γ = 1 for the remainder of the proof. Let ( h ker , m ) ∈ H ker × M be arbitrary. Bythe embeddings given by the Gelfand triple P ker , H ker , P ∗ ker , there exist for every δ > p ker ∈ P ker and g ′ ∈ H ker with k p ker − h k H < δ and k g ′ − ( A p ker − K ∗ m ) k P ∗ ker < δ .Let ( p ′ ker , m ′ ) ∈ P ker × M be the unique solution of A p ′ ker − K ∗ m ′ = g ′ in P ∗ ker , K p ′ ker + D m ′ = K p ker + D m in M ∗ . By construction, we then have ( p ′ ker , m ′ ) ∈ D ( A ) = D ( A γ ). We finally choose ( p ′ ker , m ′ )as approximation of ( h, m ) and conclude with the boundedness of − A − that k h − p ′ ker k H + k m − m ′ k M . k h − p ker k H + k p ker − p ′ ker k P + k m − m ′ k M . k h − p ker k H + k g ′ − ( A p ker − K ∗ m ) k P ∗ ker < δ. (cid:3) With the previous two lemmata, we are now in the position to discuss the uniquesolvability of the PDAE (2.1).2.3.
Existence of solutions.
In this subsection, we first discuss the existence of mildsolutions and turn to classical solutions afterwards. We emphasize that the property of ε being small is not needed for the here presented existence results. Proposition 2.5 (Existence of a mild solution) . Consider Assumption 2.1 and right-hand sides f ∈ L (0 , T ; M ∗ ) , g = g + g with g ∈ H (0 , T ; P ∗ ) , g ∈ L (0 , T ; H ∗ ) , and h ∈ H (0 , T ; Λ ∗ ) . Further assume initial data p (0) ∈ H with p (0) − B − h (0) ∈ H ker and m (0) ∈ M . In this case, there exists a unique mild solution ( p, m, λ ) of (2.1) with p ∈ C ([0 , T ] , H ) ∩ H (0 , T ; P ∗ ker ) and m ∈ C ([0 , T ] , M ) . Moreover, the Lagrange multiplier λ exists in a distributional sense with a regular primitivein the space C ([0 , T ] , Λ ) and it satisfies that ˙ p + B ∗ λ ∈ L (0 , T ; P ∗ ) . Proof.
Let ( p, m, λ ) ∈ H (0 , T ; P ker ) × H (0 , T ; M ) × H (0 , T ; Λ ) be the unique solutionof system (2.2) with given right-hand side g , cf. Lemma 2.3. The introduction of e p := p − p − B − h, e m := m − m, e λ := λ − λ (2.4)leads, together with (2.1), to the system˙ e p + A e p − K ∗ e m + B ∗ e λ = g + C A p − ˙ p − AB − h − B − ˙ h in P ∗ ,ε ˙ e m + K e p + D e m = f − KB − h − ε ˙ m in M ∗ , B e p = 0 in Λ ∗ with initial values e p (0) = p (0) − p (0) − B − h (0) ∈ H ker and e m (0) = m (0) − m (0) ∈ M .Since we have B e p = 0, the solution e p takes values in P ker . Hence, we can reduce the ansatzand test space accordingly, leading to the equivalent system˙ e p + A e p − K ∗ e m = g + C A p − ˙ p − AB − h − B − ˙ h in P ∗ ker , (2.5a) SINGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE ε ˙ e m + K e p + D e m = f − KB − h − ε ˙ m in M ∗ . (2.5b)At this point, we would like to emphasize that an equation stated in P ∗ ker means thatwe only consider test functions in P ker . With the state x := [ √ ε e p, e m ] T , equation (2.5)becomes the abstract Cauchy problem˙ x = √ ε A γ x + F = √ ε A γ x + " √ ε ( g + C A p − ˙ p − AB − h − B − ˙ h ) ε ( f − KB − h ) − ˙ m (2.6a)with initial condition x (0) = (cid:2) √ ε e p (0) , e m (0) (cid:3) T . (2.6b)Here, A γ equals the operator from Lemma 2.4 and γ = √ ε . Since the right-hand sidesatisfies F ∈ L (0 , T ; H ∗ × M ∗ ) and x (0) ∈ H ker × M , the Cauchy problem has a uniquemild solution x ∈ C ([0 , T ] , H ker ×M ). Thus, p = e p + p + B − h ∈ C ([0 , T ] , H ) has a derivativein L (0 , T ; P ∗ ker ) by (2.5a) and m = e m + m is an element of C ([0 , T ] , M ). Finally, λ canbe constructed as in the proof of [EM13, Th 3.3]. (cid:3) Considering higher regularity for the given data, we can show the existence of a classicalsolution. For this, we again analyze the corresponding Cauchy problem.
Proposition 2.6 (Existence of a classical solution) . Let Assumption 2.1 hold and theright-hand sides satisfy f ∈ H (0 , T ; M ∗ ) , g = g + g with g ∈ H (0 , T ; P ∗ ) , g ∈ H (0 , T ; H ∗ ) , and h ∈ H (0 , T ; Λ ∗ ) . Further assume consistent initial data p (0) ∈ P , i. e., B p (0) = h (0) , m (0) ∈ M , and the existence of an element h ∈ H ker with ( h , q ker ) H = hK ∗ m (0) + g (0) , q ker i (2.7) for all q ker ∈ P ker . Then there exists a unique classical solution ( p, m, λ ) of (2.1) with p ∈ C ([0 , T ] , P ) ∩ C ([0 , T ] , H ) , m ∈ C ([0 , T ] , M ) , λ ∈ C ([0 , T ] , Λ ) . Proof.
Consider the auxiliary problem (2.2) with right-hand side g , leading to ( p, m, λ ) ∈ H (0 , T ; P ker ) × H (0 , T ; M ) × H (0 , T ; Λ ), cf. Lemma 2.3. Following the proof of Propo-sition 2.5, we notice that the right-hand side of the Cauchy problem (2.6) is an elementof H (0 , T ; H ∗ ×M ∗ ) by the more regular right-hand sides. For the initial data from (2.6b),we note that x (0) ∈ D ( A √ ε ), since e p (0) ∈ P ker and e m (0) satisfies K ∗ e m (0) = K ∗ m (0) − K ∗ m (0) = K ∗ m (0) + g (0) − A p (0) = h − A p (0) in H ∗ ker due to (2.2a). This means that K ∗ e m (0) has a representation in H ker . The claimed solutionspaces of p and m follow by [Paz83, Ch. 4, Cor. 2.10]. Since the operator B satisfies an inf-sup condition by assumption, there exists a unique (and continuous) multiplier λ , whichsatisfies (2.1a), cf. [Bra07, Lem. III.4.2]. (cid:3) In the upcoming analysis of Section 3, we will apply specific estimates for the mildsolution of (2.1). For this, we first consider the classical solution from Proposition 2.6 and( e p, e m ) as defined in (2.4). Using e p and e m as test functions in (2.5), we obtain by takingthe sum and integrating over time, k e p ( t ) k H + ε k e m ( t ) k M + c D Z t k e m ( s ) k M d s ≤ k e p (0) k H + ε k e m (0) k M + (1 + 2 C A ) Z t k ˜ p ( s ) k H d s INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 7 e e e e e e v v v v v v Figure 2.1.
Example of a network of pipes, represented by a graph withvertices v , . . . v (junctions) and edges e , . . . , e (interconnections).+ Z t (cid:13)(cid:13) ( g + C A p − ˙ p − AB − h − B − ˙ h )( s ) (cid:13)(cid:13) H ∗ + c D (cid:13)(cid:13) ( f − KB − h − ε ˙ m )( s ) (cid:13)(cid:13) M ∗ d s. By the continuity result of Lemma 2.3, the last two integrals are bounded by a multipleof C := k f k L (0 ,T ; M ∗ ) + k g k H (0 ,T ; P ∗ ) + k g k L (0 ,T ; H ∗ ) + k h k H (0 ,T ; Λ ∗ ) . (2.8)An application of Gronwalls lemma hence yields k e p ( t ) k H + ε k e m ( t ) k M + c D Z t k e m ( s ) k M d s . e (1+2 C A ) t (cid:0) k p (0) k H + ε k m (0) k M + C (cid:1) . This estimate is also satisfied for the mild solution ( e p, e m ) of (2.5) under the assumptionsof Proposition 2.5. Here we used the continuity of the semigroup generated by A γ , thedensity of D ( A γ ) in H ker × M , and the density of H ℓ +1 (0 , T ; X ) in H ℓ (0 , T ; X ) for aBanach space X and ℓ ≥
0. Finally, by p = e p + p + B − h and m = e m + m we conclude(again by Lemma 2.3) that k p ( t ) k H + ε k m ( t ) k M + c D Z t k m ( s ) k M d s . e (1+2 C A ) t (cid:0) k p (0) k H + ε k m (0) k M + C (cid:1) . (2.9)We close this section with a particular example from the field of gas dynamics.2.4. Example.
As a demonstrative example, which fits into the framework of Section 2.1and particularly equation (2.1), we consider the (linear) propagation of pressure wavesin a network of pipes [Osi87, BGH11]. The geometry of the underlying network can beencoded by a directed graph. Therein, the edges represent interconnections (the pipes),whereas the vertices model junctions of the gas network. An exemplary illustration ingiven in Figure 2.1.From an analytical point of view, it is sufficient to consider a single pipe [EK18b].Hence, let Ω = (0 ,
1) denote the physical domain modeling a one-dimensional pipe. Ac-cordingly, we define P := H (Ω) and H = M := L (Ω). The unknowns p and m equal thepressure and the mass flux, respectively. If we incorporate the boundary conditions forthe potential p in form of a constraint, then B equals the trace operator (point evaluationat the end points) and hence, Λ = R , P ker = H (Ω), and H ker = H , since H (Ω) is densein L (Ω). Remark . Within this setting, a sufficient condition for the existence of h ∈ H ker inProposition 2.6 is m (0) ∈ H (Ω) and g (0) ∈ L (Ω). In this case, the function h equalsthe spatial derivative of m (0) plus g (0).Under certain simplifying model assumptions this leads to a system of the form (2.1),cf. [EK18a]. In this simple model, the operators A and D denote the multiplication by SINGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE constants and include damping and friction to the model, whereas K : P → M ∗ equalsthe spatial derivative. With the help of the right-hand sides one can incorporate, e. g., theslope of a pipe. Finally, the parameter ε equals the product of the adiabatic coefficientand the square of the Mach number and is of order 10 − , cf. [BGH11].If the full network is considered, then one may define P as the space of globally con-tinuous and piecewise H -functions and M as the space of piecewise L -functions. Inthis setting, typical coupling conditions resemble the Kirchhoff circuit laws, which can beformulated as a right-hand side g . For more details, we refer to [AZ18b]. An alternativeapproach to the network case is considered in [Mug14]. We would like to emphasize, how-ever, that the PDAE model (2.1) is more general and not restricted to the here discussedone-dimensional example. 3. Expansion of the Solution
This section is devoted to an expansion of the solution triple ( p, m, λ ) of (2.1) in termsof the small parameter ε . Assuming ε ≪
1, we consider the expansion p = p + ε p + . . . , m = m + ε m + . . . , λ = λ + ε λ + . . . . (3.1)Note that the triple ( p , m , λ ) is the solution of (2.1) in the limit case ε = 0. The aimof this section is to prove approximation properties of first order for p and m as well asof second order for ˆ p := p + εp and ˆ m := m + εm . Resulting approximation properties of the Lagrange multiplier will then be discussed inSection 3.4. As a starting point, however, we first discuss the solvability of the limit case.3.1.
Parabolic limit case.
For the expansion in ε , we need to analyze the limit ofthe PDAE (2.1) for ε = 0. It turns out that this leads to a PDAE of parabolic type.Let ( p , m , λ ) denote the solution of the system˙ p + A p − K ∗ m + B ∗ λ = g in P ∗ , (3.2a) K p + D m = f in M ∗ , (3.2b) B p = h in Λ ∗ . (3.2c)Since there is only a single differential variable (in time) left – recall that ˙ m does notappear anymore – we consider as initial condition p (0) = p (0). As before, we discuss theexistence of solutions for different regularity assumptions. Proposition 3.1 (Existence of a weak solution ( p , m )) . Consider Assumption 2.1 andright-hand sides f ∈ L (0 , T ; M ∗ ) , g ∈ L (0 , T ; P ∗ ) , and h ∈ H (0 , T ; Λ ∗ ) . Furtherassume that the initial data is consistent in the sense that p (0) − B − h (0) ∈ H ker . In thiscase, system (3.2) has a unique weak solution with p ∈ L (0 , T ; P ) ∩ C ([0 , T ] , H ) and m ∈ L (0 , T ; M ) . The Lagrange multiplier λ exists in a distributional sense with ˙ p + B ∗ λ ∈ L (0 , T ; P ∗ ) .Proof. Since equation (3.2b) is stated in M ∗ ∼ = M and D is invertible, we can insert thisequation into (3.2a), which results in˙ p + ( L + A ) p + B ∗ λ = g + K ∗ D − f in P ∗ , (3.3a) B p = h in Λ ∗ (3.3b)with the operator L introduced in Lemma 2.2. Note that the operator L + A satisfies aGarding inequality on P ker . As a result, the existence of a unique partial solution ( p , λ ) INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 9 with the claimed properties follows by [EM13, Th. 3.3]. Finally, with equation (3.2b) theflow variable is given by m = D − ( f − K p ) and therefore unique and an element of L (0 , T ; M ). (cid:3) For the subsequent analysis we also need solutions with higher regularity including acontinuous multiplier λ . This is subject of the following proposition. Proposition 3.2 (Weak solution with higher regularity) . Consider Assumption 2.1 andright-hand sides f ∈ H (0 , T ; M ) , g ∈ W (0 , T ; P ∗ , P ∗ ker ) , and h ∈ H (0 , T ; Λ ∗ ) . Furtherassume consistent initial data p (0) ∈ P with B p (0) = h (0) and the existence of a function p ∈ H ker such that ( p , q ker ) H = (cid:10) g (0) − ( L + A ) p (0) − B − ˙ h (0) + K ∗ D − f (0) , q ker (cid:11) (3.4) for all q ker ∈ P ker . Then the solution of system (3.2) satisfies p ∈ H (0 , T ; P ) ∩ C ([0 , T ] , H ) , m ∈ H (0 , T ; M ) , λ ∈ C ([0 , T ] , Λ ) . Proof.
Due to (3.3) the function e p := p − B − h satisfies B e p = 0 and(3.5) ˙ e p + ( L + A ) e p = g − ( L + A ) B − h − B − ˙ h + K ∗ D − f in P ∗ ker . The given assumptions imply that the right-hand side is an element of H (0 , T ; P ∗ ker )and ˙ e p (0) = p ∈ H ker . By [Wlo92, Th. IV.27.2] we conclude that e p ∈ H (0 , T ; P ker ) ∩ C ([0 , T ] , H ). This, in turn, implies the stated regularity for m = D − ( f − K p ) and theexistence of λ by [Bra07, Lem. III.4.2]. (cid:3) Remark . For the example of Section 2.4, a sufficient condition for the existence of p asin (3.4) is that the difference between f (0) and the spatial derivative of the initial value p (0) has a weak derivative in space and that g (0) is an L -function. Hence, for sufficientlysmooth right-hand sides this reduces to p (0) ∈ H (Ω).We are now in the position to compare the solutions of the two systems (2.1) and (3.2).3.2. First-order approximation.
We first discuss the approximation property of thepair ( p , m ), coming from system (3.2). For the special case discussed in Section 2.4, itwas shown in [EK18a, Th. 1] that this approximation is of order √ ε and – under certainassumptions on the initial data – of order ε . We will rediscover this result in the moregeneral setting, using an alternative technique of proof. For this, we consider the differenceof the two systems (2.1) and (3.2), which leads to dd t ( p − p ) + A ( p − p ) − K ∗ ( m − m ) + B ∗ ( λ − λ ) = 0 in P ∗ , (3.6a) K ( p − p ) + D ( m − m ) = − ε ˙ m in M ∗ , (3.6b) B ( p − p ) = 0 in Λ ∗ . (3.6c)The corresponding initial condition reads ( p − p )(0) = 0. Due to (3.6c) we expect thedifference p − p to take values in P ker . Because of this, we will often restrict the test spacein (3.6a) to P ker . In this case, the Lagrange multipliers vanish and we have the equation dd t ( p − p ) + A ( p − p ) − K ∗ ( m − m ) = 0 in P ∗ ker . (3.7)In the following, we consider two approaches to derive error estimates: First, we under-stand (3.6) as a parabolic system with a right-hand side ε ˙ m . Second, we will subtract ε ˙ m from (3.6b) and consider the result as a hyperbolic system with a right-hand side − ε ˙ m . Theorem 3.4 (First-order approximation I) . Suppose that all assumptions of Proposi-tion 2.6 are satisfied such that the PDAE (2.1) has a classical solution. Then the differenceof ( p, m ) and ( p , m ) is bounded for ≤ t ≤ T by k ( p − p )( t ) k H + c L Z t k ( p − p )( s ) k P d s + c D Z t k ( m − m )( s ) k M d s . ε e (1+4 C A ) t k f (0) − K p (0) − D m (0) k M ∗ + ε e (1+4 C A ) t e C data with a constant e C data only depending on k p (0) k H , k h k H , and the right-hand sides k f k H (0 ,T ; M ∗ ) , k g k H (0 ,T ; P ∗ ) , k g k H (0 ,T ; H ∗ ) , and k h k H (0 ,T ; Λ ∗ ) .Proof. We test (3.7) and (3.6b) by p − p and m − m , respectively. Adding and integratingthe resulting equations, we obtain by Young’s and Gronwall’s inequality(3.8) k ( p − p )( t ) k H + c D Z t k ( m − m )( s ) k M d s ≤ ε c D e C A t Z t k ˙ m ( s ) k M d s. On the other hand, considering test functions ( D − ) ∗ K ( p − p ) in place of m − m , weconclude by the continuity of the operators that(3.9) k ( p − p )( t ) k H + c L Z t k ( p − p )( s ) k P d s ≤ ε C K c L c D e C A t Z t k ˙ m ( s ) k M d s. Note that we have used that ˙ m ( t ) ∈ M , which is guaranteed by Proposition 2.6. Further,we note that the derivative of the classical solution ( m, p, λ ) is again a mild solution of (2.1)where we replace the right-hand sides by their temporal derivatives. Hence, we can applyestimate (2.9), which yields c D Z t k ˙ m ( s ) k M d s . e (1+2 C A ) t (cid:0) k ˙ p (0) k H + ε k ˙ m (0) k M + ˙ C (cid:1) with the constant ˙ C defined accordingly to (2.8), namely˙ C := k f k H (0 ,T ; M ∗ ) + k g k H (0 ,T ; P ∗ ) + k g k H (0 ,T ; H ∗ ) + k h k H (0 ,T ; Λ ∗ ) . It remains to bound the initial values of ˙ p and ˙ m . For ˙ p (0) we use (2.1a) and the factthat K ∗ m (0) + g (0) has a representation in H ker by (2.7), k ˙ p (0) k H = h g (0) − A p (0) + K ∗ m (0) , ˙ p (0) i = h h + g (0) − A p (0) , ˙ p (0) i . Thus, we have k ˙ p (0) k H . k h k H + k g k H (0 ,T ; H ∗ ) + k p (0) k H . For an estimate of ˙ m (0), wesimply apply (2.1b) to obtain ε k ˙ m (0) k M = ε − k f (0) − K p (0) − D m (0) k M ∗ . In total, this gives ε Z t k ˙ m ( s ) k M d s . e (1+2 C A ) t h ε k f (0) −K p (0) −D m (0) k M ∗ + ε (cid:0) k p (0) k H + k h k H + ˙ C (cid:1)i , which completes the proof. (cid:3) As mentioned above, the second approach considers system (3.6) as a hyperbolic system.For this, we need to assume the existence of ˙ m (cf. Proposition 3.2), which then appearson the right-hand side. INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 11
Theorem 3.5 (First-order approximation II) . Under the assumptions of Proposition 3.2,the difference of the solutions of (2.1) and (3.2) satisfy for ≤ t ≤ T the estimate k ( p − p )( t ) k H + ε k ( m − m )( t ) k M + c D Z t k ( m − m )( s ) k M d s . ε e C A t k m (0) − m (0) k M + ε e C A t e C data with a constant e C data only depending on the initial data, k p k H ker , and the right-hand sides k f k H (0 ,T ; M ) , k g k W (0 ,T ; P ∗ , P ∗ ker ) , and k h k H (0 ,T ; Λ ∗ ) .Proof. Adding − ε ˙ m to (3.6b), we obtain the to (3.6) equivalent system dd t ( p − p ) + A ( p − p ) − K ∗ ( m − m ) + B ∗ ( λ − λ ) = 0 in P ∗ ,ε dd t ( m − m ) + K ( p − p ) + D ( m − m ) = − ε ˙ m in M ∗ , B ( p − p ) = 0 in Λ ∗ . We would like to emphasize that this system has the same structure as the originalPDAE (2.1). By Proposition 3.2 we conclude that ˙ m ∈ L (0 , T ; M ). Hence, we canapply the estimate for mild solutions (2.9) with right-hand sides g = 0, f = − ε ˙ m ,and h = 0. With p (0) = p (0) we conclude k ( p − p )( t ) k H + ε k ( m − m )( t ) k M + c D Z t k ( m − m )( s ) k M d s . e C A t (cid:0) ε k m (0) − m (0) k M + ε k ˙ m k L (0 ,T ; M ∗ ) (cid:1) . The second term of the right-hand side is bounded by ε e C data , since D ˙ m = ˙ f − K ˙ p =˙ f − KB − ˙ h − K ˙ e p , where ˙ e p solves the formal derivative of (3.5) with initial value p ; seealso [Wlo92, Th. IV.27.2]. (cid:3) Remark . In the case ε ˙ m (0) = f (0) −K p (0) −D m (0) = 0, which is equivalent to m (0) = m (0), Theorems 3.4 and 3.5 state that p is a first-order approximation of p in terms of ε ,measured in L ∞ (0 , T ; H ) and L (0 , T ; P ). Further, m is a first-order approximation of m in L (0 , T ; M ) and, under the conditions of Theorem 3.5, an approximation of order in C ([0 , T ] , M ). Remark . In the finite-dimensional case, the obtained results for m − m also matchwith [KKO99, Ch. 2.5, Th. 5.1]. For p − p , however, one has k p − p k L ∞ (0 ,T ) = O ( ε ) inthe finite-dimensional setting, independent of the initial data. A careful analysis showsthat the here derived estimates are optimal [AZ18b, App. C]. This is also numericallyconfirmed in Section 4. Remark . If the operator A is non-negative in addition to Assumption 2.1, then theconstant C A in the estimates of Theorems 3.4 and 3.5 can be set to zero. If A is elliptic,then the bounds can be further improved such that the exponential function therein isstrictly monotonic decreasing.3.3. Second-order approximation.
So far, we have shown that ( p , m ) provides afirst-order approximation of ( p, m ) if the initial value of m is chosen appropriately. Wenow include the second term of the expansion (3.1). To be precise, we analyze the ap-proximation properties of ˆ p = p + ε p and ˆ m = m + ε m . We are especially interestedin the needed regularity assumptions to gain an additional half or full order in terms of ε .As a first step, we note that the tuple ( p , m , λ ) introduced in (3.1) solves the PDAE˙ p + A p − K ∗ m + B ∗ λ = 0 in P ∗ , (3.10a) K p + D m = − ˙ m in M ∗ , (3.10b) B p = 0 in Λ ∗ (3.10c)with the initial condition p (0) = 0. For the solvability of system (3.10) we note that it hasthe same structure as (3.2). Hence, we only need to analyze the regularity of the right-handside, i. e., of ˙ m . The weak differentiability of m has been discussed in Proposition 3.2such that an application of Proposition 3.1 leads to the following result. Proposition 3.9 (Existence of a weak solution ( p , m )) . Given the assumptions of Propo-sition 3.2, system (3.10) is uniquely solvable with p ∈ L (0 , T ; P ker ) ∩ C ([0 , T ] , H ) and m ∈ L (0 , T ; M ) . Furthermore, λ exists in a distributional sense. In the previous subsection, we have observed that the initial data may cause a reductionin the approximation order, cf. Remark 3.6. To focus on the improvements resulting fromthe incorporation of p and m , we assume in the following that f (0) − K p (0) − D m (0) = 0,i. e., m (0) = m (0). Theorem 3.10 (Second-order approximation) . Consider the assumptions of Proposi-tion 2.6 with additional regularity of the form f ∈ H (0 , T ; M ∗ ) , g = g + g with g ∈ H (0 , T ; P ∗ ) , g ∈ H (0 , T ; H ∗ ) , and h ∈ H (0 , T ; Λ ∗ ) . Further assume the ex-istence of an element p ∈ P ker satisfying (3.4) , f (0) − K p (0) − D m (0) = 0 , and that ˙ g (0) = 0 . In this setting, the difference of the solution of (2.1) and (ˆ p, ˆ m ) is bounded by k ( p − ˆ p )( t ) k H + c L Z t k ( p − ˆ p )( s ) k P d s + c D Z t k ( m − ˆ m )( s ) k M d s . ε e (1+4 C A ) t k ˙ f (0) − K ˙ p (0) k M ∗ + ε e (1+4 C A ) t ˆ C data for ≤ t ≤ T and with ˆ C data depending on the initial data, k p k H ker , and the right-handsides k f k H (0 ,T ; M ∗ ) , k g k H (0 ,T ; P ∗ ) , k g k H (0 ,T ; H ∗ ) , and k h k H (0 ,T ; Λ ∗ ) .Proof. The difference of the solution ( p, m, λ ) of (2.1) and (ˆ p, ˆ m, ˆ λ ) solves dd t ( p − ˆ p ) + A ( p − ˆ p ) − K ∗ ( m − ˆ m ) + B ∗ ( λ − ˆ λ ) = 0 in P ∗ , K ( p − ˆ p ) + D ( m − ˆ m ) = − ε ( ˙ m − ˙ m ) in M ∗ , B ( p − ˆ p ) = 0 in Λ ∗ with initial condition ( p − ˆ p )(0) = 0. Considering p − ˆ p as test function in the first equation,the Lagrange multipliers vanish. Thus, following the proof of Theorem 3.4, we obtain k ( p − ˆ p )( t ) k H + Z t k ( p − ˆ p )( s ) k P + k ( m − ˆ m )( s ) k M d s . ε e C A t Z t k ( ˙ m − ˙ m )( s ) k M d s and it remains to find an estimate of the integral of the error ˙ m − ˙ m . For this, we considerthe formal derivative of system (3.6). Similar to the estimate (3.8), we can show that Z t k ( ˙ m − ˙ m )( s ) k M d s . k ( ˙ p − ˙ p )(0) k H + ε Z t k ¨ m ( s ) k M d s. As mentioned in Remark 3.6, the assumption f (0) − K p (0) − D m (0) = 0 implies that m (0) = m (0). Equation (3.6a) thus implies ˙ p (0) = ˙ p (0) in H . For an estimate of ¨ m , weconsider the formal derivative of (2.1) and obtain by estimate (2.9) that Z t k ¨ m ( s ) k M d s . e (1+2 C A ) t (cid:0) k ¨ p (0) k H + ε k ¨ m (0) k M + ¨ C (cid:1) . INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 13
Here, ¨ C denotes the constant similarly defined as in (2.8) but with two additionalderivatives on each occurring function. For an estimate of ¨ m (0) we note that (2.1b)implies together with ˙ m (0) = 0 that ε k ¨ m (0) k M = ε k ˙ f (0) − K ˙ p (0) − D ˙ m (0) k M ∗ = ε k ˙ f (0) − K ˙ p (0) k M ∗ . Note that the right-hand side is bounded due to f ∈ H (0 , T ; M ∗ ) and ˙ p (0) = p (0) + p + B − h (0) ∈ P .In order to find a bound of ¨ p (0), we apply the decomposition p = p + e p + B − h with p being the solution of (2.2) with right-hand side g and e p the function introduced in (2.4).By Lemma 2.3 we know that ¨ p (0) is bounded by the H (0 , T ; P ∗ )-norm of g . Further, B − ¨ h is bounded by the H (0 , T ; Λ ∗ )-norm of h . Finally, (2.5a) implies that (cid:13)(cid:13) ¨ e p (0) (cid:13)(cid:13) H = (cid:13)(cid:13) ˙ g (0) − A ˙ e p (0) − AB − ˙ h (0) − ¨ p (0) − B − ¨ h (0) (cid:13)(cid:13) H ∗ . Here, we used that ˙ e m (0) = ˙ m (0) + ˙ m (0) vanishes by the assumptions ε ˙ m (0) = f (0) −K p (0) − D m (0) = 0 and ˙ g (0) = 0. Finally, since p (0) = p (0), m (0) = m (0), and˙ m (0) = 0 imply that ˙ p (0) = ˙ p (0) by (2.5) and (3.5), we have ˙ e p (0) = p − ˙ p (0) = p ∈ H ker and thus the term A ˙ e p (0) is bounded in H ∗ . (cid:3) Remark . In Theorem 3.10 we have assumed ˙ g (0) = 0 to show that ˙ e m (0) vanishesand, hence, K ˙ e m (0) is bounded in H ∗ ker . To conclude this boundedness, however, it issufficient (and necessary) that ˙ g (0) ∈ H ∗ ker . Remark . Under the additional assumption 0 = ˙ f (0) − K ˙ p (0) = ε ¨ m (0), which isequivalent to m (0) = ˆ m (0) and ˙ m (0) = ˙ m (0), Theorem 3.10 states that ˆ p is a second-orderapproximation of p in terms of ε , measured in L ∞ (0 , T ; H ) and L (0 , T ; P ), respectively.Further, ˆ m is a second-order approximation of m measured in L (0 , T ; M ). Remark . A comparison of Theorem 3.10 with the corresponding finite-dimensionalcase shows once more that the results for m − ˆ m coincide but that the difference p − ˆ p converges with one ε -order less for general initial data, cf. [KKO99, Ch. 2.5, Th. 5.2]. Fora numerical validation of this result, we refer to Section 4.3.4. Estimate of the Lagrange multiplier.
Closing this section, we would like to takea closer look at the Lagrange multiplier λ and its approximation λ . Since an estimateof λ − λ depends on the derivatives ˙ p and ˙ p , the regularity assumptions of the first-order estimates in Section 3.2 are not sufficient. Hence, we consider the assumptions ofTheorem 3.10.By Proposition 2.6 we know that λ ∈ C ([0 , T ] , Λ ), whereas λ ∈ C ([0 , T ] , Λ ) was shownin Proposition 3.2. By the inf-sup stability of B we have k λ ( t ) − λ ( t ) k Λ ≤ β kB ∗ ( λ ( t ) − λ ( t )) k P ∗ = 1 β sup q ∈P hB ∗ ( λ ( t ) − λ ( t )) , q ik q k P and thus, by equation (3.6a), k λ ( t ) − λ ( t ) k Λ . k ˙ p ( t ) − ˙ p ( t ) k P ∗ + k p ( t ) − p ( t ) k H + k m ( t ) − m ( t ) k M . Hence, the L (0 , T ; Λ )-error of λ − λ can be bounded by Z t k λ ( s ) − λ ( s ) k Λ d s . Z t k ˙ p ( s ) − ˙ p ( s ) k P ∗ + k p ( s ) − p ( s ) k H + k m ( s ) − m ( s ) k M d s . Z t k ˙ p ( s ) − ˙ p ( s ) k H d s + ε e C A t e C data ≤ Z t ε − / k ˙ p ( s ) − ˙ p ( s ) k P ∗ ker + 12 ε / k ˙ p ( s ) − ˙ p ( s ) k P ker + ε e C A t e C data . ε / e (1+4 C A ) t k ˙ f (0) − K ˙ p (0) k M ∗ + ( ε / + ε ) e C A t e C data + ε / e (1+4 C A ) t ˆ C data with the constants e C data and ˆ C data from Theorems 3.4 and 3.10, respectively. Here, we usedon the one hand that ˙ p − ˙ p can be estimated in P ∗ ker analogously to the lines of Theorem 3.4.On the other hand, ˙ p − ˙ p and ˙ m − ˙ m satisfy the formal derivative of the parabolicsystem (3.6) with the initial condition ˙ p (0) − ˙ p (0) = 0. Similarly to estimate (3.9) wethen have R t k ˙ p − ˙ p k P d s . ε R t k ¨ m k M d s , where the right-hand side can be bounded inthe same manner as in the proof of Theorem 3.10. Remark . If the interpolation space [ P ker , P ∗ ker ] θ with θ > can be embedded in P ∗ ,then the estimate of R t k λ ( s ) − λ ( s ) k Λ d s can be improved to the order O ( ε θ ). Formore details on this, we refer the reader to [Zim21, Rem. 8.39].4. Numerical Validation
This final section is devoted to the numerical confirmation of the convergence results ofSection 3, including the differences in the approximation orders in the finite- and infinite-dimensional setting. For this, we consider the propagation of gas in a single pipe of unitlength as already discussed in Section 2.4. The associated PDE in its strong form is givenby ˙ p + ∂ x m = 0 in (0 , , (4.1a) ε ˙ m + ∂ x p + m = 0 in (0 , . (4.1b)Moreover, the pressure p satisfies homogeneous Dirichlet boundary conditions. In thecorresponding weak formulation, these boundary conditions are included explicitly in formof a constraint. With the trace operator denoted by B , A := 0, and D := id, this then leadsto the PDAE (2.1) with vanishing right-hand sides. In this particular case, the associatedLagrange multiplier λ equals the trace of the mass flux m (if m is sufficiently smooth).The numerical experiments of this section illustrate the transition of the approximationorder in terms of ε from the finite- to the infinite-dimensional setting, cf. Remarks 3.7and 3.13. Recall that Theorems 3.4 and 3.5 imply the estimate k p − p k C (0 ,T ; L (0 , + k m − m k L (0 ,T ; L (0 , ≤ C cons √ ε + O ( ε )with C cons = 0 if the initial data satisfies ∂ x p (0) = − m (0). In finite dimensions, oneshows k p − p k C (0 ,T ; R n ) = O ( ε ) independent of the initial data; see [KKO99, Ch. 2.5,Th. 5.1]. We would like to emphasize that this is an asymptotic result, whereas thebounds in Theorem 3.4 and 3.5 are valid for all ε >
0. Similar statements hold true forTheorem 3.10 and its finite-dimensional counterpart.For the numerical validation of the approximation orders obtained in Section 3, weconsider system (4.1) for different initial values. These are given by p ( x,
0) = ∞ X k =1 sin( πkx ) k . ∈ H (0 ,
1) and m ( x,
0) = 0 ∈ H (0 , INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 15 .
51 discretization parameter n a pp r o x i m a t i o n o r d e r α p − p in L ∞ ( L ) p − p in L ( H ) m − m in L ( L ) λ − λ in L ( R ) Figure 4.1.
Estimate of the approximation order α corresponding to The-orem 3.4 with the initial values from (4.2).considering Theorem 3.4 and p ( x,
0) = 0 ∈ H (0 ,
1) and m ( x,
0) = π ∞ X k =1 cos( πkx ) k . ∈ L (0 , ∂ x p (0) = − m (0), i. e., C cons = 0.For the spatial discretization, we consider enriched spectral finite elements given by P n := span { e − x , e x , sin( πx ) , . . . , sin( nπx ) } ⊂ P = H (0 ,
1) for the pressure variable and M n := span { , cos( πx ) , . . . , cos( nπx ) } ⊂ M = L (0 ,
1) for the mass flux, n ∈ N . Sincethe space Λ for the Lagrange multiplier is simply R , there is no need for an additionaldiscretization. To identify the approximation rate as a function of the discretizationparameter n , we calculate for fixed n the differences p n ( · ; ε ) − p ,n , m n ( · ; ε ) − m ,n ,and λ n ( · ; ε ) − λ ,n for ε = 1 / (8 √ j ), j = 1 , . . . ,
30. For each error measure err( n, ε ) – weconsider different variables in different norms – we make the ansatz err( n, ε ) = C ( n ) ε α ( n ) .With this, we obtain an approximation of the rate by the median of the slopes of thelogarithmic errors between two successive values of ε .For the initial data mentioned above, Figures 4.1 and 4.2 illustrate the estimated ap-proximation orders as a function of the discretization parameter n . For the pressure p , weobserve a fading from order one (as expected for the finite-dimensional case) to a valueclose to 0 .
5. A rigorous analysis shows that the asymptotic limit is 0 . m has a constant rate of 0 . . λ . Recall that wehave no theoretical predictions for these particular regularity assumptions. For small n ,it is close to 0 .
5, since λ n − λ ,n = ( B n M − n B Tn ) − B n M − n K Tn ( m n − m ,n )with the mass matrix M n , the discrete partial derivative K n , and the discrete trace op-erator B n . For increasing n , however, the approximation orders decrease. Due to thestructure of system (4.1), one can prove that k ∂ x ( m − m ) k L (0 ,T ; L (0 , = O (1) for the .
51 discretization parameter n a pp r o x i m a t i o n o r d e r α p − p in L ∞ ( L ) √ ε ( m − m ) in L ∞ ( L ) m − m in L ( L ) λ − λ in L ( R ) Figure 4.2.
Estimate of the approximation order α corresponding to The-orem 3.5 with the initial values from (4.3).10 . . . n a pp r o x i m a t i o n o r d e r α p − p in L ∞ ( L ) p − p in L ( H ) √ ε ( m − m ) in L ∞ ( L ) m − m in L ( L ) λ − λ in L ( R ) Figure 4.3.
Estimate of the approximation order α corresponding to The-orems 3.4 and 3.5 with consistent initial values given in (4.4), i. e., with m (0) = m (0).example considered in Figure 4.1. Since[ L (0 , , H − (0 , / − δ = [ H / − δ (0 , ∗ ֒ → [ H (0 , ∗ for every δ ∈ (0 , . .
25; see also Section 3.4. For the second example, the rate α analytically tends to zero.In Section 3.2 we also proved that the convergence orders improve by half an order if m (0) = m (0) or, equivalently, ∂ x p (0) = − m (0) is satisfied. This is numerically confirmedin Figure 4.3, where the associated initial values are given by p ( x,
0) = ∞ X k =1 sin( πkx ) k . ∈ H (0 ,
1) and m ( x,
0) = − π ∞ X k =1 cos( πkx ) k . ∈ H (0 , . (4.4)At this point, we would like to emphasize that this also improves the rate of the Lagrangemultiplier λ to 0 .
78 for larger n . With similar arguments as made for the first example,we would expect a rate 0 . INGULAR PERTURBATION RESULTS FOR LINEAR PDAES OF HYPERBOLIC TYPE 17 . . .
82 discretization parameter n a pp r o x i m a t i o n o r d e r α p − ˆ p in L ∞ ( L ) p − ˆ p in L ( H ) m − ˆ m in L ( L ) Figure 4.4.
Estimate of the approximation order α corresponding to The-orem 3.10 with the initial values from (4.5).Finally, to verity the results of Theorem 3.10, i.e., considering ˆ p and ˆ m , we set as initialdata p ( x,
0) = ∞ X k =1 sin( πkx ) k . ∈ H (0 ,
1) and m ( x,
0) = − π ∞ X k =1 cos( πkx ) k . ∈ H (0 , . (4.5)These values are obviously consistent, i. e., ∂ x p (0) = − m (0), and satisfy the smoothnessrequirements of Theorem 3.10. The resulting approximations of the rates are displayed inFigure 4.4. Here, the approximated order for the pressure fades from two to around 1 . . ε -rates forthe pressure and the Lagrange multiplier show a decreasing asymptotic behavior in thediscretization parameter n . The approximate orders for the mass flux m , on the otherhand, are more or less independent of n as predicted in Section 3.5. Conclusion
In this paper, we have considered linear PDAEs of hyperbolic type with a small pa-rameter ε >
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