A unified error analysis for nonlinear wave-type equations with application to acoustic boundary conditions

In this work we present a unified error analysis for abstract space discretizations of wave-type equations with nonlinear quasi-monotone operators. This yields an error bound in terms of discretization and interpolation errors that can be applied to various equations and space discretizations fitting in the abstract setting. We use the unified error analysis to prove novel convergence rates for a non-conforming finite element space discretization of wave equations with nonlinear acoustic boundary conditions and illustrate the error bound by a numerical experiment.


Introduction
In this paper we present a unified error analysis for abstract non-conforming space discretizations of nonlinear wave-type equations with quasi-monotone operators. The unified error analysis was introduced in [15,16] for linear wave equations and extended in [18] to semilinear problems. It is an abstract framework in which wave equations as well as a variety of spatial discretizations are considered as evolution equations in Hilbert spaces. Using such an abstract framework allows to derive an abstract error bound in terms of approximation properties of the space discretization method. This error bound can then be used to prove convergence rates for all space discretizations of wave-type equations which fit into the abstract setting. This was demonstrated in [16]  The aim of this paper is to extend the unified error analysis to nonlinear evolution equations with quasi-monotone operators. As a specific application, we use this theory to prove error bounds for a non-conforming finite element discretization of the wave equation with nonlinear acoustic boundary conditions. This is a generalization of the results in the thesis [20].
Acoustic boundary conditions were first mentioned in [5]. Since then, many papers studied their properties, wellposedness, and stability, and they are still in the focus of current research, cf. [4,10,11,21,23,26] and references therein.
However, there are only very few numerical papers considering these boundary conditions. We are aware of [16] and [17]. In these papers space discretizations for wave equations with linear acoustic boundary conditions were derived and analyzed in the energy and the L 2 -norm, respectively. In the present paper, we now consider the space discretization of nonlinear acoustic boundary conditions as proposed in [13,14,28], and extend the results from [16] to this case.
Since acoustic boundary conditions include derivatives on the boundary, they are usually posed on domains with smooth boundaries. A common choice to discretize such problems are isoparametric finite elements. Since this involves to approximate the boundary of the domain, the discretization becomes non-conforming. Unfortunately, this makes the error analysis much more involved since the exact and the numerical solution are not defined on the same domain which causes errors in the bilinear form ot the weak form.
We derive an order p isoparametric finite element discretization of the wave equation with nonlinear acoustic boundary conditions and show that it fits into the setting of the unified error analysis. Using the abstract error bound we prove order p convergence of the method in the energy norm, where we tackle the appearing approximation errors stemming from the domain approximation or interpolation by known error bounds from [7,8]. A major difficulty lies in the discretization of the nonlinearities, since this must be done in such a way that it preserves the quasi-monotonicity to ensure the stability of the numerical scheme. Furthermore, the discretization errors of the nonlinearities have to be bounded. While both is straightforward for conforming discretizations, it turns out to be much more involved in the non-conforming case.
We are not aware of any other results in this direction, neither of such a general error analysis for non-conforming space discretizations of nonlinear wave-type equations, nor of results concerning the discretization of wave equations with nonlinear acoustic boundary conditions. Nevertheless, we mention the following works going in the same direction. In [9], a full discretization in an abstract framework similar to the one used in this paper was considered. But only a conforming space discretization was analyzed and no error bounds but only weak convergence of the discretization was shown. For quasilinear equations, a related framework was introduced in [19,22], covering quasilinear wave and Maxwell equations. However, the error analysis in this work relies on properties of quasilinear operators that cannot be used for nonlinear acoustic boundary conditions and in general for equations with maximal quasi-monotone operators.
This paper is structured as follows. In Sect. 2 we introduce the wave equation with nonlinear acoustic boundary conditions with a corresponding finite element space discretization and state an error bound of the spatial discretization. We then present in Sect. 3 the unified error analysis for nonlinear first-order evolution equations and use the results in Sect. 4 to analyze nonlinear second-order wave-type equations. As main results we derive abstract error bounds for the space discretizations. Finally, in Sect. 5 we use these abstract bounds of the unified error analysis to prove the space discretization error bound for the wave equations with nonlinear acoustic boundary conditions and illustrate it with some numerical experiments.

The wave equation with nonlinear acoustic boundary conditions
In this section we present the analytical framework for the wave equation with acoustic boundary conditions and a suitable finite element space discretization. Additionally, we present a space discretization error bound which we will prove by application of the unified error analysis in Sect. 5.

Problem statement and analytical framework
Let Ω ⊂ R n , n = 2, 3, be a bounded domain with C 2 -boundary Γ and outer normal vector n. We consider the acousitc wave equation with non-local reacting acoustic boundary conditions in the following form: seek u : Here Δ Γ denotes the Laplace-Beltrami operator an Γ .

Remark 2.1
It is possible to include nonlinear forcing terms F Ω (x, u) and F Γ (x, δ) at the right-hand side of (1a) and (1b), respectively. This was considered in [20] for the wave equation with kinetic boundary conditions and such terms can be treated similarly for the acoustic boundary conditions. We omit this here for the sake of a clearer presentation.
We make the following assumptions on the coefficients and nonlinearities in (1).

Assumption 2.2 a) The constants satisfy
b) The function θ ∈ C(R; R) satisfies θ(0) = 0 and is strictly monotonically increasing with for some θ 0 > 0. Further, there exist and a constant C > 0 such that for all ξ ∈ R c) The function η : R → R is globally Lipschitz continuous and satisfies η(0) = 0. We then have thatη defined viaη(ξ ) = η(ξ ) − ρ c Ω ξ is also Lipschitz continuous and denote the Lipschitz constant ofη by L η . d) The inhomogeneities satisfy Weak formulation To prove wellposedness and derive a finite element discretization, we now present a weak formulation of the wave equation with acoustic boundary conditions (1). We make use of the densely embedded Hilbert spaces Note that for k ≥ 3, the spaces H k (Γ ) require more boundary regularity to be welldefined, e.g., Γ ∈ C k , which denotes that Γ is a C k boundary.
By multiplying (1a) and (1b) with test functions defined on Ω and Γ , respectively, applying integration by parts and inserting the nonlinear coupling (1c), we obtain the the weak formulation of (1): for t ≥ 0 and all ϕ ∈ V , where Note that m is an inner product on H andã:=a + m is an inner product on V . (4) is globally wellposed, we comment on this in Sect. 4.1, cf.Corollary 4.4, and Sect. 5.

Finite element space discretization
For the space discretization of (1) we consider the bulk-surface finite element method from [7] which was also used in [16] to discretize the wave equation with linear acoustic boundary conditions. We give a brief introduction of the finite element spaces and refer to [7] for further details on the bulk-surface finite element method.
The bulk-surface finite element method Let Γ ∈ C p+1 for some p ≥ 1 and let T Ω h be a consistent and quasi-uniform mesh consisting of isoparametric elements K of degree p which discretizes Ω. By h we denote the maximal mesh width of T Ω h . The discretized domain is then given by and its boundary by Γ h = ∂Ω h . The bulk and the surface finite element space of order p are then defined by respectively. Here, K denotes the reference triangle with corresponding polynomial space P p ( K ) of order p, and F K is the transformation from K to K . Note that by , the discretization is non-conforming. Hence, to relate functions in V h with functions in V , in [7], was constructed. By I h,Ω : C(Ω) → V Ω h, p and I h,Γ : C(Γ ) → V Γ h, p we denote the order p nodal interpolation operators in Ω and on Γ , respectively, and set for v = The spatially discretized equation We now state the finite element discretization of (1). For this, let be an elementwise defined quadrature formula that approximates the integral Γ h · ds. We require that the quadrature formula has positive weights and is of order greater than 2 p, s.t.polynomials up to degree 2 p are integrated exactly and we have for all For Then, the spatial discretization of (1) is given by

Remark 2.4
The use of the quadrature formulas instead of the interpolation in the definition of the discretized nonlinearity D h is required to prove that D h is quasimonotone, cf.Lemma 5.3.
To prove an error bound of the discretization we pose the following assumptions on the exact solution and the data: Assumption 2.5 a) Let T > 0. For the inhomogeneities and the nonlinearities in (1) we assume the additional regularity Furthermore, we assume that the strong solution u, δ of (1) satisfies on [0, T ] b) Let the discrete initial values satisfy As main theorem, we state the following error bound for the finite element discretization of the wave equation with nonlinear acoustic boundary conditions.

and C is a constant independent of h.
In the next two sections we will now present a general theory for the error analysis of non-conforming space discretizations which we then use to proof Theorem 2.6 in Sect. 5.

Abstract space discretizations of first-order evolution equations with monotone operators
In this section we present the unified error analysis for abstract space discretizations of first-order evolution equations with maximal monotone operators. This generalizes the results from [16] and [18] for linear and semilinear equations, respectively. The results of this section are part of the dissertation [20]. We first present the continuous equation and the corresponding abstract space discretization, before we prove an error bound.

Analytical setting
Let X be a Hilbert space with scalar product ·, · X in which we consider the evolution equation In the following, we omit the t arguments in evolution equations. We pose the following classical assumptions to ensure that (10) is wellposed.

Theorem 3.2 Let Assumption 3.1 hold true. Then, the evolution equation
We further state the following stability result which is essential for the latter error analysis.
Proof The result can be derived with energy estimates similar to [25, Theorem IV.4.1A].

Abstract space discretization
We now present an abstract space discretization of the evolution equation (10). Let (X h ) h be a family of finite dimensional vector spaces with scalar products ·, · X h , where h is a discretization parameter, e.g., the maximal mesh width of a finite element discretization. For all X h ∈ (X h ) h we seek an approximations x h ∈ X h to the solution x of (10). Therefore, let S h and g h be approximations of S and g, respectively, which satisfy the following assumptions similar to Assumption 3.1.

Assumption 3.4
a) The nonlinear operator S h : The discretized evolution equation is then given by Since these assumptions are similar to the continuous case, we obtain by Theorem 3.2 that (12) is globally wellposed. In the following we introduce a framework for the error analysis of the abstract space discretization that is similar to the linear case presented in [16]. To cover nonconforming space discretizations where X h X , as they appear in Sect. 2, we make the following assumptions to relate the discrete and the continuous problem.
for some constant C X > 0 independent of h. The adjoint of the lift operator As the term L h J h −I appears in the error bound in Theorem 3.7, the reference operator should be chosen such that the approximation L h J h z ≈ z (z ∈ Z ) is of the order of convergence which one wants to proof for the spatial discretization. It could, e.g., be an interpolation or a projection operator. One possible example for a lift operator is the lift defined in (6). We will consider this in Sect. 5. The space discretization error bound is given in terms of the following terms: a) The remainder of the nonlinear monotone operator is given by b) We define the error term We now can state and prove an error bound of the abstract space discretization, cf.
is the discrete error. The full error can thus be bounded by and we further investigate the discrete error. By applying the adjoint lift to (10a) we obtain , and g h on both sides yields where Under Assumption 3.4, the stability estimate from Theorem 3.3 holds also true in the discrete case with c qm instead of c qm . Hence, we obtain by Theorem 3.3 applied to (12) and (18) the following bound for the discrete error where we used (19) and (14). Together with (17), we finally obtain (16).
In the following section we will use this result to derive error bounds for secondorder nonlinear wave-type equations.

Abstract space discretizations of second-order evolution equations with nonlinear damping
In this section we apply the theory of Sect. 3 to second-order evolution equations. As in the previous section, we first introduce the continuous problem and then present and analyze the abstract space discretization. This is a generalization of the linear unified error analysis introduced in [16] and also an extension of the framework considered in the dissertation [20] which does not cover the acoustic boundary conditions with nonlinear coupling from Sect. 2, cf.Remark 4.2 and Sect. 5.

Analytical setting
Let V , H be Hilbert spaces es and let V be densely embedded in H . We consider the following variational equation, which is typical for a weak formulation of a secondorder partial differential equation. Seek To ensure the wellposedness of (21) we pose the following assumptions. :=a + c G m is a scalar product on V with induced complete norm · ã . From now on, we equip V withã.
Formulation as evolution equation We identify H with its dual space H * to obtain the Gelfand triple with dense embeddings. We thus have for all To reformulate (21) as an evolution equation, we define the operator Then, we can rewrite (21) equivalently as an evolution equation in V * : Note that (25) implicitly contains the condition

Remark 4.2
In [20], the stricter assumption D ∈ C(V ; H ) was posed. However, this does not cover the acoustic boundary conditions with nonlinear coupling (1c) as we will see in Sect. 5, cf.Remark 5.2.
First-order formulation We rewrite (25) into a first-order formulation in the framework of Sect. 3.1. For this let u = v and we define with Then, (25) is equivalent to the first-order evolution equation (10).
In the following we show that the assumptions of Sect. 3.1 are satisfied. The subsequent lemma is a slight extension of [20, Lemma 2.14].
In the next step we prove the maximality and proceed similar as in the proof of [27,Theorem 4.1]. We have to show that there exists a λ > 0 such that for every By solving (28a) for v and plugging it into (28b) we obtain We thus investigate the operator T = λ + 1 λ A + D ∈ C(V ; V * ) which can be decomposed via T = T 1 + T 2 with For λ > max{c qm , 2c G , 2β qm } we then have that T is monotone as the sum of monotone operators. Further, we have where we used that T 1 is coercive due to the choice of λ, and T 2 is monotone with We apply [3, Corollary 2.3] stating that continuous, monotone, and coercive operators from a reflexive Banach space to its dual space are surjective. This yields the existence of a solution v ∈ V of (29) and thus also of a solution x = [v, w] ∈ V × V of (28). We further obtain by (28b) x ∈ D(S) since  By Theorem 3.2 we then directly obtain the wellposedness of (21).

Space discretization
We consider a family (V h ) h of finite dimensional vector spaces related to a discretization parameter h and the following discretized version of (21) Here, m h , a h , D h , and f h are approximations of the corresponding continuous counterparts. We pose the following assumptions similar to Assumption 4.1.
The operator A h ∈ L(V h ; V h ) related to a h is defined via We then can reformulate (30) as an evolution equation in V h : Analogously to the continuous equation we rewrite (32) in a first-order formulation and therefore define X h = V h × H h . With (32) is then of the form (12).
Proof Since the setting in the discrete case from Assumption 4.6 is similar to the continuous one from Assumption 4.1 with constants independent of h, the proof of Lemma 4.3 transfers directly to the discrete case.
Similar to the first-order case, we require the existence of suitable operators to relate continuous and discrete functions of the abstract non-conforming space discretization.

Assumption 4.8 a) There exists a lift operator L
with a constant C I h > 0 independent of h.
To apply the results of Sect. 3.2, we now define the first-order reference and lift operator.

Lemma 4.10
The first-order lift and reference operators from Definition 4.9 satisfy Assumption 3.5 with C X = max{ C V , C H } and C J h = max{ C V , C I h }.
Proof This is a direct consequence of Assumption 4.8.
In the following we now bound the first-order remainder term which is for z = [v, w] ∈ D(S) ∩ Z given by To do so, we use the following error terms in the scalar products, which are for i.e., against errors in the scalar products, interpolation errors, and the discretization error of the nonlinear operator.
Proof The proof works similar to the proof of [16,Lemma 4.7] and relies on the identity where ·, · X h is the scalar product on X h . Thus, let and we bound the first two summands separately. To bound the first one, we use (39), (34), and ϕ h ã h ≤ 1 to obtain By using the definitions ofã,ã h , ψ h m h ≤ 1 and (39), (22), (34), (31), we bound the second summand in (41) via Similar to (42), we further estimate We finally obtain the assertion by collecting all terms.
We are now in the position to prove the following error bound which is a generalization of [20,Thm. 2.24], cf. Remark 4.2. It is applicable to all equations and space discretizations fitting in the abstract framework of this section. In the dissertation [20], it was used to prove novel convergence rates for the wave equation with nonlinear kinetic boundary conditions and in this paper, we use it to prove Theorem 2.6.
with a constant C that is independent of h and t. The other constants are given by and the abstract space discretization errors Proof By Corollaries 4.4, 4.7, and Lemma 4.10, we have that the first-order formulations of (25) and (32) satisfy all assumptions of Theorem 3.7.
By applying Theorem 3.7 and employing the error bound (16), we obtain In the remaining proof, we bound the different terms against E h,i , i = 1, . . . , 4. For the remainder term we apply the bound (40) and obtain for all t ∈ [0, T ] By the definitions of J h and L * h we further have for the discretization errors of the initial values and the inhomogeneity The reference error can be decomposed for all t ∈ [0, T ] via where we have similar to (42) In the same way, we finally bound Having this abstract theory at hand, we can now return to the wave equation with nonlinear acoustic boundary conditions from Sect. 2 and give the proof of Theorem 2.6 in the next section.

Numerical analysis of wave equations with nonlinear acoustic boundary conditions
In this section we will use the unified error analysis for second-order equations from Sect. 4 to prove the error bound from Theorem 2.6. We start by verifying that all assumptions are satisfied. Proof We clearly have that m is a scalar product on H and thatã:=a + m is a scalar product on V . Further, Assumption 2.2 d) directly implies Assumption 4.1 d).
Thus it remains to prove Assumption 4.1 c). By Assumption 2.2 a), b), c) and (5a), (5c) we obtain This proves the the quasi-monotonicity of D.
In the next step we show D ∈ C(V ; V * ). We emphasize that the trace inequality holds true for q = ζ + 1 with ζ from the growth condition (3) We hence obtain By the trace inequality (45), the growth condition (3), the relation ζ = q − 1, and [12,

Remark 5.2
It is not possible to prove the stronger condition D ∈ C(V , H ). This is due to the fact that the calculation (46) strongly relies on ϕ ∈ H 1 (Ω) and is not possible for a test function ϕ ∈ L 2 (Ω).  It remains to prove Assumption 4.6 c). To show the quasi-monotonicity, we proceed analogously to the proof in the continuous case from Lemma 5.1 and obtain by (8c) where we used that the quadrature formula has positive weights and satisfies (7). Finally, D h is continuous, since V h is a finite dimensional space and, thus, convergence in V h implies uniform pointwise convergence and especially convergence in all quadrature nodes.
To prove an error bound for the semidiscretization, we apply the theory of Sect. 4.2 and therefore have to specify the operators from Assumption 4.8.

Definition 5.4
a) The lift operator Our error analysis relies on the following properties of the lift and the interpolation operators.
First of all, there exist element-wise norm equivalences related to the lift, which were shown in [8,Lemmas 5.3 and 7.3].
h and for all h < h 0 sufficiently small we have where K Ω = G h (K Ω ), K Γ = G h (K Γ ). By construction, the lift additionally preserves the L ∞ norm, i.e., Further, we have the following bounds of the geometric errors stemming from the domain approximation (cf. [ The nodal interpolation satisfy the following error bounds, which follow from [ with a constant C independent of h. b) Locally, on each element K Ω ∈ T Ω h , K Γ ∈ T Γ h , the interpolation operators satisfy for all 0 ≤ r ≤ k and all v ∈ H k+1 (K Ω ), ϑ ∈ H k+1 (K Γ ), the error bounds ϑ − (I h,Γ ϑ) with a constant C independent of h.
hold true with a constant C independent of h.
The following lemma is a direct consequence of Lemmas 5.6 and 5.7.

Lemma 5.8
The operators defined in Definition 5.4 satisfy Assumption 4.8 with where C Ω,Ω h and C Γ ,Γ h are given in (47).
We are now in the position to prove the error bound of the space discretization.
For the first summand on the right-hand side of (53) we have by the continuity of the lift operator and the interpolation error (49b) The second summand can be bounded using the geometric error estimate (48c) by To bound the third summand on the right-hand side of (53) we use that for the nodal interpolation we have I h,Γ θ(v) = I h,Γ θ((I h,Γ v) ) ∈ V Γ h, p and that the order of quadrature formula is greater than 2 p to obtain

Numerical experiment
In this section we illustrate Theorem 2.6 with a numerical experiment.
We implemented the experiments in the C++ finite element library deal.ii, cf. [1,2]. The code which was used for the numerical is available at https://doi.org/10.5445/ IR/1000150159. For the time integration we use the implicit midpoint rule with time step size ≈ 2·10 −4 and solve the arising nonlinear systems with the simplified Newton method. For the spatial discretization we use the bulk-surface finite element methods of orders p = 1, 2, 3.
We consider the error instead of the error from Theorem 2.6 since the computation of the lift is quite laborious. We evaluated the integrals with a quadrature rule of degree 2 p, so that the Fig. 1 Error from (54) at t = 0.7 for the test example. The dashed plots are straight lines of slope 1, 2 and 3, respectively quadrature error is negligible. The restriction of u to Ω h is possible for this example since we have Ω h ⊂ Ω. In Fig. 1 the error E(t) is plotted against the maximal mesh width h. We chose t = 0.7 since it keeps distance to the roots of sin(2π t). We observe that the error converges with order p as predicted by Theorem 2.6 which indicates that our proven convergence rates are optimal. Note that for order p = 3 and small h, the plot approaches the plateau of the time discretization error.