{'fraction_non_alphanumeric': 0.0681126018602354, 'fraction_numerical': 0.0323894970779488, 'mean_word_length': 3.7109344707250873, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 28, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L^{\infty}(L^2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L^{\infty}(L^{\infty})-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L^{\infty}(L^2)-norm are derived again under minimal smoothness on initial data.

Early a priori error estimates for Galerkin finite element methods for solving (1.1)-(1.4) (without integral terms) were derived by Dupont [10] using a standard energy argument. These estimates were improved by Baker [2], who used a technique interpreted later in literature as a nonstandard energy arguments. In [17], Rauch discussed the convergence of a continuous time Galerkin approximation to a second order wave equation and proved optimal error estimates in L ∞ (L 2 )-norm using piecewise linear polynomials, when u 0 ∈ H 3 ∩ H 1 0 (with u 1 = 0). The analysis improves upon the earlier results of Baker and Dougalis [3], where optimal error estimates were derived under the assumptions that u 0 ∈ H 5 ∩ H 1 0 and u 1 ∈ H 4 ∩ H 1 0 . In [19], Sinha and Pani extended Rauch's results to hyperbolic integro-differential equation with quadrature and obtained optimal error estimates in L ∞ (L 2 ) norm in the case u 0 ∈ H 3 ∩ H 1 0 and u 1 ∈ H 2 ∩ H 1 0 . For more on Galerkin methods and optimal error estimates for the problem (1.1)-(1.4), see, [6,16].
In the literature, optimal error estimates for mixed finite element approximations to second order hyperbolic equations were proved in [9,13,15,4,8]. In [13], Geveci derived L ∞ -in-time, L 2 -in-space error bounds for the continuous-in-time mixed finite element approximations of velocity and stress. In [9], a priori error estimates were obtained for mixed finite element approximations to both displacement and stress requiring less regularity than needed in [13]. Stability for a family of discrete-in-time schemes was also demonstrated under regularity u 0 ∈ H 4 ∩ H 1 0 and u 1 ∈ H 3 , when r = 2. In [15], a mixed finite element displacement formulation was proposed for the acoustic wave equation under reduced regularity requirement on the displacement variable. Further, for a completely discrete explicit scheme, stability results and error estimates were established.
For the standard mixed methods, the problem (1.1) is often rewritten by introducing a new variable  with u(0) = u 0 and u t (0) = u 1 . To the best of our knowledge, there are few results available on optimal estimates of mixed approximations to problem (1.1)-(1.4) with minimum smoothness on initial data. In the context of parabolic problems, based on mixed methods related to the weak formulation (1.10)-(1.11), Sinha et al. [21] have derived an optimal convergence rate O(ht −1 ) for the velocity σ(t) in L 2 -norm and suboptimal convergence rate O(ht −1/2 ) for the pressure u(t) in L 2 -norm with t ∈ (0, T ], when u 0 ∈ L 2 (Ω). However, optimal rate O(h 2 t −1 ) for u is only established for a class of problems when A = aI and B = b(t, s)I, where a and b are independent of spatial variable x. These results have recently been improved by Goswami et al. [14], who have established optimal convergence rate O(h 2 t −1 ) for u(t) in L 2 -norm and O(ht −1 ) for σ(t) L 2 -norm for all t ∈ (0, T ], when u 0 ∈ L 2 (Ω). Similar results have been obtained for the extended finite element method described below. In both papers only semidiscrete problems have been considered.
In the first part of this paper, an extended mixed method for (1.1) is proposed and analyzed. Such a method was analysed earlier in [7] and [1] for elliptic problem, [23] for degenerate nonlinear parabolic problem, [14] for parabolic integro-differential equations and references cited in. To motivate this new mixed method, we introduce two variables: Then, the equation (1.1) takes the form u tt − ∇ · σ = 0. Now, the weak mixed formulation of (1.1)-(1.4) which forms a basis of our mixed Galerkin method is to find with u(0) = u 0 and u t (0) = u 1 . The main goal of this paper is to establish optimal convergence rate for the approximate solutions of (1.1), when the initial functions u 0 ∈ H 3 ∩ H 1 0 and u 1 ∈ H 2 ∩ H 1 0 . Our analysis is essentially based on a simple energy technique and a use of a time integration without exploiting the inverse of the associated discrete elliptic operator. Essentially, integrating in time leads to a first order evolution process and hence, is instrumental in reducing the regularity requirements on the solution. Further, due to the presence of the integral term, it is observed that the concept of mixed Ritz-Volterra projections used earlier in [11], [12], [21] and [14] plays a crucial role in our analysis. For the completely discrete scheme, a major difficulty associated with a use of the nonstandard energy formulation by Baker is the presence of a fourth order time derivative of the displacement u in the bounds, see [9]. Moreover, additional difficulty is caused by the quadrature error associated with a second order midpoint quadrature rule which is used to approximate the integral term. Special care is needed to arrest these issues otherwise for optimal convergence rate, we land up with higher regularity assumption on the initial data, namely; u 0 ∈ H 4 ∩ H 1 0 . Therefore, a modification of Baker's approach is adopted and optimal error estimates in ℓ ∞ (L 2 )-norm are derived, when u 0 ∈ H 3 ∩H 1 0 and u 1 ∈ H 2 ∩H 1 0 . This technique is proved to be powerful and can successfully be applied even to the problem in [19]. Finally, the analysis has been extended to the standard mixed method corresponding to the formulation (1.10)-(1.11) and error analysis has been briefly discussed. Now compared to the (1.10)-(1.11), the extended or expanded method corresponding to (1.13)-(1.15) may have introduced one more variable leading to a computation of one more extra variable. However, in Section 2, it is shown that it is possible to eliminate q h , which is an approximation of the gradient vector q. Hence, both these schemes have almost comparable computational cost. Moreover, in the new formulation, we need not invert the coefficient matrix A.
Throughout this article, we denote by C, a generic positive constant which may vary from context to context, and whenever there arises no confusion, we would denote u(t) simply as u for the sake of convenience.
An outline of the paper is as follows. In Section 2, we give some a priori bounds and regularity results for (1.13)-(1.15), and briefly present the finite element approximation of the extended mixed formulation (1.13)-(1.15). In Section 3, the extended mixed Ritz-Volterra projection is introduced and analyzed. In Section 4, error estimates for Galerkin approximations of u, σ and q for the semidiscrete problem are derived. The completely discrete problem is discussed in Section 5 and optimal error estimates in ℓ ∞ (L 2 )-norm are established. Finally, in Section 6, an extention of our analysis to the mixed formulation (1.10)-(1.11) with minimal smoothness assumptions on the initial data is briefly discussed.

Extended Mixed Finite Element Method
For our analysis, we shall use the standard notations for L 2 (Ω), H 1 0 (Ω), H m (Ω) and H(div; Ω) spaces with their norms and seminorms. To be more specific, L 2 (Ω) is equipped with inner product (·, ·) and norm · . The norm on H(div; Ω) is given by Further, the standard Sobolev space H m (Ω) of order m is equipped with its norm · H m (Ω) , which we denote simply by · m . Since the matrix A in (1.1) is positive definite, there exist positive constants a 0 and a 1 such that Further, assume that all coefficients of B and their derivatives are bounded in their respective domain of definitions by the positive constant a 1 . Under our assumptions on the domain and on the coefficient matrix A, we note that the following elliptic regularity result holds: there exists a positive constant C such that for For our subsequent use, we state without proof a priori estimates for u, q and σ satisfying (1.13)-(1.15) under appropriate regularity conditions on the initial data u 0 and u 1 . For more details, we refer to [19] and [20].
Then, there is a positive constant C such that the following regularity results hold: . Based on the mixed formulation (1.13)-(1.15) for the problem (1.1)-(1.3), we now introduce the extended mixed finite element Galerkin method. Let T h be a regular triangulation of Ω by triangles of diameter at most h. Let V h × W h denote a pair of finite element spaces satisfying the following conditions: Further, assume that the finite element spaces satisfy the following approximation properties: and on a quasi-uniform mesh, Although, we can have several choices for V h and W h , here we consider only the Raviart-Thomas elements of order one [5]. Note that P h and Π h satisfy Now, the corresponding semidiscrete mixed finite element formulation is to seek a triplet (u h , q h , σ h ) : with initial data u h (0) and u ht (0) to be defined later. Since W h and V h are finite dimensional spaces, the discrete problem (2.6)-(2.8) leads to a linear system consisting of differential, integral and algebraic equations.
Compared to the two field formulation which is based on the mixed weak form (1.10)-(1.11) and is stated in Section 6, the new mixed system (2.6)-(2.8) which depends on three field formulation has one more additional variable to compute. However, one variable, say q h can be easily eliminated with negligible computational cost and hence, it can have comparable computational cost. Further, there is no need of inverting the coefficient matrix A.

Mixed Ritz-Volterra Type Projections
In this section, the extended mixed Ritz-Volterra projections are introduced and analyzed. The projections are defined as follows: Given (u(t), q(t), , for a given triplet {u, q, σ}, leads to a system of linear equations combined with algebraic constraints for {ũ h ,q h ,σ h }. Note that when B = 0, the system has a unique solution, see [7]. Now using theory of linear Volterra equations of second kind and Picard's iteration, it is straightforward to prove that the system (3.1)-(3.3), for a given triplet {u, q, σ}, has a unique solution {ũ h ,q h ,σ h }.
In this section, we discuss estimates of η u , η q and η σ . Using definitions of P h and Π h , we rewrite η u , η q and η σ as Because the estimates of θ u , θ q and θ σ are known, it is sufficient to estimate ρ u , ρ q . Now rewrite (3.1)-(3.3) as Below, we derive estimates of η σ and η q . The analysis of Section 4 of [14] can be suitably modified to prove the following results in Lemmas 3.1 and 3.2, but for completeness, we indicate the proofs here.
Proof. First, observe that Then, a use of the Cauchy-Schwarz inequality with the boundedness of B and the positive definiteness property of A yields Notice that by (1.12), it follows that A substitution of (3.10) in (3.9) with Lemma 2.1 shows that An application of Gronwall's Lemma yields A use of the triangle inequality establishes the estimate (3.7) for j = 0. Now, differentiate (3.5) with respect to time to obtain Again, apply the Cauchy-Schwarz inequality and the boundedness of A and B to arrive at Taking into account approximation properties (2.3) and (1.12), it follows that θ q t ≤ Ch r q t r ≤ Ch r u t r+1 , and θ σ t ≤ Ch r u t r+1 , and hence, by Lemma 2.1, Now, a use of the triangle inequality completes the proof of (3.7) for j = 1. For j = 2, differentiate again (3.5) with respect to t and repeat the above arguments to obtain the estimate. For the second estimate (3.8), we use (3.1) for any v ∈ V to arrive at By taking supremum over all nonzero v ∈ V, we obtain the desired estimate and this concludes the proof. Now, we use a duality argument to estimate η u .
Proof. Consider the auxiliary elliptic problem: ∇ · (A∇ζ) = η u in Ω, (3.13) ζ = 0 on ∂Ω, (3.14) and set p = ∇ζ and ψ = Ap so that ∇ · ψ = η u . Then, from the elliptic regularity result (2.2), it follows that Clearly, the following system of equations is satisfied for all .18), respectively, and then add the resulting equations to arrive at Then substitute in (3.19) and use the Cauchy-Schwarz inequality and (2.3) for r = 1 to obtain Note that for fixed s, t, we can use (2.3) for r = 1 to arrive at A use of estimates of η q from (3.7) with the elliptic regularity (3.15) yields Substituting (3.8) in (3.21), apply Gronwall's lemma to obtain and this concludes the proof of (3.11) for j = 0.
In order to estimate η ut , consider again the elliptic problem (3.13)-(3.14) with replacing η u on the right hand side of (3.13) by η ut . By setting p = ∇ζ, ψ = Ap, From the standard regularity results, it follows that Now, differentiate with respect to time the three equations in (3.1)-(3.1) to obtain Following the previous steps for deriving the estimate of η u , a use of (3.22)-(3.25) at the appropriate steps leads to where P h is the L 2 -projection onto the conforming finite element space consisting of C 0 -piecewise linear elements which is a subspace of H 1 0 (Ω). Since for all other terms except the last three terms on the right hand side of (3.26), it is easy to derive the estimates, it is enough to estimate the last three terms. An application of integration by parts shows that For the last two terms on the right hand side of (3.26), we observe that and a use of integration by parts yields All together, we obtain using the Cauchy-Schwarz inequality, Lemmas 2.1, 3.1 and 3.2, the approximation properties (2.3) of projections P h and Π h , with elliptic regularity result (3.22) in (3.26) the following estimate Finally, by differentiating (3.23)-(3.25) with respect to time and following the previous steps we establish the estimate for j = 2.
In order to show the estimate (3.12), we write (3.1) as Using Lemma 2.1 in [22], it follows that for some constant C independent of h. A use of (2.4) and (3.7) completes the proof.

Semidiscrete Error Estimates
In this section, error estimates for the semidiscrete problem are derived. Using the mixed Ritz-Volterra projections defined in section 3, we rewrite where, (u, q, σ) and (u h , q h , σ h ) are solutions of (1.13)-(1.15) and (2.6)-(2.8), respectively. Note that, (e u , e q , e σ ) satisfy the following equations Since estimates of η u , η q and η σ are known from Lemmas 3.1 and 3.2, it is sufficient to estimate ξ u , ξ q and ξ σ . Using (3.1)-(3.3), we rewrite (4.1)-(4.3) as Below, one of the main results for the semidiscrete problem is proved. and Proof. First differentiate (4.4) with respect to time and set v h = ξ σ in the resulting equation, z h = −ξ qt in (4.5) and w h = ξ ut in (4.6). Then, add the resulting equations to arrive at (4.9) 1 2 Apply integration by parts to the integral term on the right hand side of (4.9) to find that Substitute (4.10) in (4.9), then integrate the resulting equation from 0 to t. A use of the Cauchy-Schwarz inequality with the boundedness of B and the positive definite property of A yields ξ ut (t) 2 + ||A 1/2 ξ q (t)|| 2 ≤ ξ ut (0) 2 + ||A 1/2 ξ q (0)|| 2 + 2 t 0 ||η utt || ||ξ ut || ds (4.11) Then, at t = t * , (4.11) becomes and hence, Now an application of Gronwall's lemma yields To estimate ξ σ , choose z h = ξ σ in (4.5). Then, use the Cauchy-Schwarz inequality to arrive at From the triangle inequality, we obtain e ut ≤ ξ ut + η ut .
Apply Lemmas 3.1 and 3.2 with the choices u h (0) = P u 0 , u ht (0) = P h u 1 and q h (0) = Π h (∇u 0 ) to arrive at the estimate (4.7). In a similar way, we can establish (4.8) and this completes the rest of the proof. As a consequence of Theorem 4.1, we have the following L ∞ (L ∞ ) estimate.
we easily derive the following estimate of u − u h : Since from (4.14), we obtain an optimal error estimate of u − u h L ∞ (L 2 (Ω)) , when u 0 ∈ H 4 ∩ H 1 0 and u 1 ∈ H 3 ∩ H 1 0 , we now use a variant of Baker's nonstandard formulation (see [2]) to provide a proof of L ∞ (L 2 ) estimate of u − u h under reduced regularity conditions on u 0 and u 1 . More precisely, we shall obtain optimal L ∞ (L 2 ) estimate for u − u h , when u 0 ∈ H 3 ∩ H 1 0 and u 1 ∈ H 2 ∩ H 1 0 . In the rest of this section, we make use of the following notation forφ:φ After integrating (4.5) and (4.6) with respect to t, the following system of equations is derived Note that in (4.17), we have used the fact that u ht (0) = P h u 1 , that is, (e ht (0), w h ) = 0 for all w h ∈ W h .  Integrate from 0 to t to deduce Let I denote the last term on the right hand side of (4.19). Integration by parts yields For I 1 , we again integrate by parts in time so that Similarly for I 2 , we note that Using the Cauchy-Schwarz inequality and the bounds for I 1 and I 2 , we obtain for some t * ∈ [0, t]. Then, at t = t * , we note that and therefore, An application of Gronwall's lemma yields Finally, a use of the triangle inequality with Lemma 3.2 concludes the proof of Theorem 4.2.

Error Estimates for a Completely Discrete Scheme
In this section, we introduce further notations and formulate a completely discrete scheme by applying an implicit finite difference method to discretize the time variable of the semidiscrete system (2.6)-(2.8). Then, we discuss optimal error estimates. Let k (0 < k < 1) be the time step, k = T /N for some positive integer N , and t n = nk. For any function φ of time, let φ n denote φ(t n ). We shall use this notation for functions defined for continuous in time as well as those defined for discrete in time. Set and define the following terms for the difference quotients: and The discrete-in-time scheme is based on a symmetric difference approximation around the nodal points, and integral terms are computed by using the second order quadrature formula ǫ n (φ) = k n−1 j=0 g(t j+1/2 ) ≈ tn 0 g(s) ds, with t j+1/2 = (j + 1/2)k.
The quadrature error E n (g) is defined by Using Peano's kernel theorem, see [16], it can be written as Then, the discrete-in-time scheme for the problem (1. 13 This choice of the time discretization leads to a second order accuracy in k. For φ ∈ V h , we define a linear functional E n B (φ) representing the error in the quadrature formula by Notice that E 0 B (φ) = 0. In our analysis, we shall use the following lemma, which can be found in [16].
In order to derive a priori error estimates for the completely discrete scheme, we rewrite Since estimates for η U , η Q and η Z are known from Lemmas 3.1 and 3.2, it is sufficient to estimate ξ U , ξ Q and ξ Z . From (5.5)-(5.8), we obtain the following system 2 k Below, one of the main theorems of this section is proved.
With Ψ n = (ξ n U , ξ n Q ), define Multiply (5.16) by 2k and sum from n = 2 to m to obtain Define for some m ⋆ with 0 ≤ m ⋆ ≤ m, A use of the Cauchy-Schwarz inequality yields we now estimate one of the term of I n 3 . Note that =I n 31 + I n 32 . For the second term I n 32 , we use the fact that and hence after summing up from n = 2 to m and multiplying by k All together, we obtain We can now estimate ǫ n−1/2 (B n−1/2 (ξ Q , δ t ξ n Q )) in a similar way, but without having the term A 1/2 ξ m+1/2 Q 2 on the right hand side and thus, we arrive at In a similar way, using (5.18), the term I n 2 can be estimated as follows Notice that, since E 0 B = 0, it follows that and hence, we obtain It remains now to bound the term |||Ψ 3/2 ||| on the right hand side of (5.17). Take (5.16) at n = 1 to obtain Substitution of estimates involving I n 1 , · · · , I n 3 and (5.19) in (5.17) yields Choose k 0 > 0 such that for 0 < k ≤ k 0 , (1 − a1 a 2 0 k) > 0. Then replace m by m ⋆ in (5.20) and obtain after cancellation of |||Ψ m ⋆ +1/2 ||| from the both sides Then an application of the discrete Gronwall's lemma yields To estimate Ψ 1/2 on the right hand side of this inequality, we first note that, ξ 0 U = 0 since U 0 =ũ h (0), and hence, ξ (5.11), and w h = −ξ 1/2 Q in (5.12). Adding the resulting equations, and taking into account that ǫ 0 (φ) = 0 and E 0 B (φ) = 0, we arrive at For 0 < k ≤ k 0 , 1 − (a 1 k)/(2a 0 ) > 0 and hence, we obtain Thus, To estimate the first two terms on the right hand side of (5.22), it is observed that Finally, a use of the triangle inequality with Lemmas 3.1 and 3.2 completes the rest of the proof. Below, we again recall a variant of Baker's nonstandard energy formulation to prove ℓ ∞ (L 2 )-estimate for the error e n U under reduced regularity conditions on the initial data. Now introduce the following notations for proving the next theorem. Definê and similarly, Multiplying (5.4) by k, summing over n, and taking into account (5.1), we obtain the new equation The key idea here, which differs from Baker's approach, is that we compare the above equation with which is derived by integrating (1.15) with respect to time, whereσ(t) = t 0 σ(s)ds. By taking the average of (5.30) at t n+1 and t n and using (5.29) we arrive at Using the definitions ofũ h andσ h , it follows that To complete the system of error equations, we multiply (5.12) by k, sum over n, and take the average of the resulting equations. Including (5.12), we end up with following system Theorem 5.2 Let Ω be a bounded convex polygonal domain in IR 2 , and let (u, q, σ) be the solution of (1.13)-(1.15). Further, let (U n , Q n , Z n ) ∈ W h × V h × V h be the solution of (5.1)-(5.4). With U 0 = P h u 0 and Q 0 = P h ∇u 0 , there exists a positive constant C, independent of h and k, such that for small k with k = O(h), the following estimate holds: Proof. Choose w h =ξ n+1/2 Z in (5.31), z h = −ξ n+1/2 Q in (5.32), and v h = ξ n+1/2 U in (5.33). Adding the resulting equations, we find that For convenience, we use the following notations: B s r = B(t s , t r ),∂ i φ i+1/2 = (φ i+1/2 − φ i−1/2 )/k, and denote by ψ · φ the L 2 inner product of ψ and φ. Further, let |||(ξ n U ,ξ n Q )||| 2 = ||ξ n U || 2 + ||A 1/2ξ n Q || 2 and for some m ⋆ ∈ [0; m + 1], define To estimate the first term I m 1 , we observe that by (5.28) we now obtain Next, we estimate the terms Θ n+1 and Υ n+1 . Using (5.27) we have and hence, using the definition of Θ n , it follows that Similarly, On substitution of (5.37) and (5.38) in (5.36), a use of Cauchy-Schwarz inequality with ||D t,s B(t, s)|| ≤ a 1 , yields Since, similar bounds can be obtained for other terms in k m n=0ǫ n+1/2 (ξ Q , ξ n+1/2 Q ), we finally conclude that Since, the terms in k m n=0Ê ) have a similar form, we deduce that For I m 3 and I m 4 , we have To estimate the last term I m 5 , we first notice that (∇ · σ, ξ Thus, it follows that On substituting the above estimates in (5.35) and following steps in previous theorems, we arrive at The first two terms on the right hand side can be bounded as follows: For the third term, we note thatq h = −(q −q h ) + q = −η Q + q, and hence For the last term on the right hand side of (5.39), use Lemmas 5.  Use integration by parts to find that the boundary terms become zero and therefore, we arrive at Following similar line of proof, we can easily show that the last two terms in (5.38) are also bounded by C(T )(kh + k 2 )( u 0 3 + u 1 2 ). Finally, by using the triangle inequality and the estimates in Lemmas 3.1 and 3.2 we complete the proof of the theorem.

Error Estimates for the Standard Mixed Method
Now, we extend our analysis to discuss optimal error estimates for u and σ, satisfying the standard mixed method (1.10)-(1.11) with minimal regularity of the initial data. We first recall the following regularity results.  With W h and V h defined as in Section 2, we define the corresponding semidiscrete mixed finite element approximation to (1.10)-(1.11) as: Find a pair (u h , σ h ) ∈ W h × V h such that with u h (0) = P h u 0 , and u ht (0) = P h u 1 .
Below, we present the main theorem of this section.
The following lemma can be easily obtained by combining Theorem 2.6 of [11] and Lemma 6.1.