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[ Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, Al-Khod 123, Muscat, Oman    [ Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India
###### Abstract

In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro-differential equations in a two-dimensional convex polygonal domain. The effect of numerical quadrature is also examined. In the semidiscrete case, optimal error estimates in and - norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate in derived in -norm under higher regularity on the data. Based on a second order explicit method in time, a completely discrete scheme is examined and optimal error estimates are established with a mild condition on the space and time discretizing parameters. Finally, some numerical experiments are conducted which confirm the theoretical order of convergence.

finite volume element, hyperbolic integro-differential equation, semidiscrete method, numerical quadrature, Ritz-Volterra projection, completely discrete scheme, optimal error estimates.

2004

FVEM for Hyperbolic PIDE] A priori Error Estimates for Finite Volume Element Approximations to Second Order Linear Hyperbolic Integro-Differential Equations S. Karaa]Samir Karaa

A. K. Pani]Amiya K. Pani

\subjclass

[2000]65N30, 65N15

## 1 Introduction

In this paper, we discuss and analyze a finite volume element method for approximating solutions to the following class of second order linear hyperbolic integro-differential equations:

 utt−∇⋅(A(x)∇u+∫t0B(x,t,s)∇u(s)ds) = f(x,t)in  Ω×J, u(x,t) = 0on ∂Ω×J, (1.1) u(x,0) = u0(x)in Ω, ut(x,0) = u1(x)in Ω,

with given functions and , where is a bounded convex polygonal domain, , and is given function defined on the space-time domain Here, and are matrices with smooth coefficients. Further, assume that is symmetric and uniformly positive definite in . Problems of this kind arise in linear viscoelastic models, specially in the modelling of viscoelastic materials with memory (cf. Renardy et al. [23]).

Earlier, the finite volume difference methods which are based on cell centered grids and approximating the derivatives by difference quotients have been proposed and analyzed, see [15] for a survey. Another approach, which we shall follow in this article was formulated in the framework of Petrov-Galerkin finite element method using two different grids to define the trial space and test space. This is popularly known finite volume element methods (FVEMs). Here and also in literature, the trial space consists of - piecewise linear polynomials on the finite element partition of and the test space is piecewise constants over the control volume to be defined in Section 2. Earlier, the FVEM has been examined by Bank and Rose [3], Cai [4], Chatzipantelidis [8], Li et al. [17], Ewing et al. [12], etc. for elliptic problems, for parabolic and parabolic type problems by Chou et al. [7], Chatzipantelidis et al. [9], Ewing et al. [13], Sinha et al. [25] and for second order wave equations by Kumar et al. [16]. For a recent survey on FVEM, see, a review article by Lin et al. [19].

For linear elliptic problems, Li et al. [17] have established optimal error estimates in and -norms. More precisely, for -norm the following estimate are derived:

 ∥u−uh∥0≤Ch2∥u∥W3,p(Ω),      p>1,

where is the exact solution and is the finite volume element approximation of . Compared to the error analysis of finite element methods, it is observed that this method is optimal in approximation property, but is not optimal with respect to the regularity of the exact solution as for order convergence, the exact solution For convex polygonal domain , it may be difficult to prove -regularity for the solution Therefore, an attempt has been made in [12] to establish optimal error estimate under the assumption that the exact solution and the source term A counter example has also been provided in [12] to show that if , then FVE solution may not have optimal error estimates in norm. The analysis has been extended to parabolic problems in convex polygonal domain in [9] and optimal error estimates have been derived under some compatibility conditions on the initial data. Further, the effect of quadrature, that is, when the inner product is replaced by numerical quadrature has been analyzed. Subsequently, Ewing et al. [13] have employed FVEM for approximating solutions of parabolic integro-differential equations and derived optimal error estimates under regularity for the exact solution and regularity for its time derivative. Then on convex polygonal domain, Sinha et al. [25] have examined semidiscrete FVEM and proved optimal error estimates for smooth and non smooth data. The analysis is further generalized to a second order linear wave equation defined on a convex polygonal domain and a priori error estimates have been established only for semidiscrete case, see, Kumar et al. [13]. Further, the effect of quadrature and maximum norm estimates are proved under some additional conditions on the initial data and the forcing function. In the present article, an attempt has been made to extend the analysis of FVEM to a class of second order linear hyperbolic integro-differential equations in convex polygonal domains with minimal regularity assumptions on the initial data. Moreover, a completely discrete scheme based on a second order explicit method has been analyzed.

In order to put the present investigation into a proper perspective visa-vis earlier results, we discuss, below, the literature for the second order hyperbolic equations. Li et al. [17] have proved an optimal order of convergence in -norm without quadrature using elliptic projection, but the regularity of the exact solution assumed to be higher than the regularity assumed in our results even when for the problem (1). On a related finite element analysis for the second order hyperbolic equations without quadrature, we refer to Baker [1] and with quadrature, see, Baker and Dougalis [2] and Dupont [11]. Baker and Dougalis [2] have proved optimal order of convergence in for the semidiscrete finite element scheme, provided the initial displacement and the initial velocity Subsequently, Rauch [22] has derived the convergence analysis for the Galerkin finite element methods when applied to a second order wave equation by using piecewise linear polynomials and established optimal estimate with and which are turned out to be the minimal regularity conditions for the second order wave equation. Subsequently, Pani et al. [26] have examined the effect of numerical quadrature on finite element method for hyperbolic integro-differential equations with minimal regularity assumptions on the initial data, that is, and . On a related article on a linear second order wave equation, we refer to Sinha [24] and on hyperbolic PIDE, see, [6]. When FVEM is combined with quadrature for approximating solution of (1), we have, in this article, proved optimal estimate with minimal regularity assumptions on the initial data.

The organization of the present paper is as follows: Section deals with some notations, weak formulation and the regularity results for the exact solution. Section is devoted to the primary and dual meshes for finite volume element method and semidiscrete FVE approximation to the problem (1). Section focuses on a priori error estimates for the semidiscrete FVE approximations and optimal order of convergence in and norms are established under minimal regularity assumptions on the initial data. Further, quasi-optimal order of convergence in maximum norm has also been derived. Section is on completely discrete scheme which is based on a second order explicit scheme in time and a priori error estimates are established. Section deals with the effect of numerical quadrature and the related error estimates are derived again with minimal regularity assumption on the initial data. Finally in Section , some numerical experiments are conducted which confirm our theoretical order of convergence.

Through out this paper, is a generic positive constant independent of discretising parameters and

## 2 Notation and Preliminaries.

This section is devoted to some notations and preliminary results related to the weak solution of (1).

Let denote the standard Sobolev space with the norm

 ∥u∥m,p,Ω=⎛⎝∑|α|≤m∥Dαu∥pLp(Ω)⎞⎠1/p    for 1≤p<∞,

and for ,

 ∥u∥m,∞,Ω=sup|α|≤m∥Dαu∥L∞(Ω).

When there is no confusion, we denote by For , we simply write as and denote its norm by For a Banach space with norm and let be defined by

 Wm,p(0,T;X):={v:(0,T)⟶X|∥Djtv∥X∈Lp(0,T),0≤j≤m}.

with its norm

 ∥v∥Wm,p(0,T;X)=∥u∥Wm,p(X):=m∑j=0(∫T0∥Djtv∥pXdt)1/p,

with the standard modification for , see [14]. For , is simply the space . Finally, let and denote, respectively, the inner product and its induced norm on

With define the bilinear forms and on by

 A(u,v)=∫ΩA(x)∇u⋅∇vdx,

and

 B(t,s;u(s),v)=∫ΩB(x,t,s)∇u(s)⋅∇vdx.

Then, the weak formulation for (1) is to seek such that

 (utt,v)+A(u,v)+∫t0B(t,s;u(s),v)ds=(f,v)∀v∈H10(Ω) (2.1)

with and

Since is symmetric and uniformly positive definite in , the bilinear form satisfies the following condition: there exist positive constants and with such that

 Λ∥v∥21≥A(v,v)≥α∥v∥21∀v∈H10(Ω). (2.2)

For our subsequent use, we state without proof a priori estimates of the solution of the problem (1) under appropriate regularity conditions and compatibility conditions on , and . Its proof can be easily obtained by appropriately modified arguments in the proof of Theorem 3.1 of [26]. For similar estimates for second order linear hyperbolic equations, see Lemma 2.1 of [16].

###### Lemma 2.1

Let be a weak solution of . Then, there is a positive constant such that the following estimates

 ∥Dj+2tu(t)∥0+∥Dj+1tu(t)∥1+∥Djtu(t)∥2 ≤ C(∥u0∥j+2+∥u1∥j+1+ j∑k=0∥Dktf∥L1(Hj−k)+∥Dj+1tf∥L1(L2)),

hold for where

We shall have occasion to use the following identity for where is a Banach space

 ϕ(t)=ϕ(0)+∫t0ϕt(s)ds. (2.3)

## 3 Finite Volume Element Method

This section deals with primary and dual meshes on the domain , construction of finite dimensional spaces, finite volume element formulation and some preliminary results.

Let be a family of regular triangulations of the closed, convex polygonal domain into closed triangles and let where denotes the diameter of Let be set of nodes or vertices, that is, and let be the set of interior nodes in with cardinality . Further, let be the dual mesh associated with the primary mesh which is defined as follows. With as an interior node of the triangulation let be its adjacent nodes (see, FIGURE 1 with ). Let denote the midpoints of and let be the barycenters of the triangle with . The control volume is constructed by joining successively . With as the nodes of let be the set of all dual nodes . For a boundary node , the control volume is shown in the FIGURE 1. Note that the union of the control volumes forms a partition of .

Assume that the partitions and are quasi-uniform in the sense that there exist positive constants and independent of such that

 C1h2≤|KQi|≤C2h2     ∀Qi∈N∗h, (3.1)
 C1h2≤|K∗Pi|≤C2h2     ∀Pi∈Nh, (3.2)

where

We consider a finite volume element discretization of (1) in the standard -conforming piecewise linear finite element space on the primary mesh , which is defined by

 Uh={vh∈C0(¯¯¯¯Ω):vh|Kis linear for % all K∈Thandvh|∂Ω=0},

and the dual volume element space on the dual mesh given by

 U∗h={vh∈L2(Ω):vh|K∗P0is % constant for allK∗P0∈T∗handvh|∂Ω=0}.

Now, and , where ’s are the standard nodal basis functions associated with nodes and ’s are the characteristic basis functions corresponding to the control volume given by

 χi(x)={1,if x∈K∗Pi0,elsewhere.

The semidiscrete finite volume element formulation for (1) is to seek such that

 (uh,tt,vh)+Ah(uh,vh)+∫t0Bh(t,s;uh(s),vh)ds=(f,vh)∀vh∈U∗h, (3.3)

with given initial data and in to be defined later. Here, the bilinear forms and are defined, respectively, by

 Ah(uh,vh)=−∑Pi∈N0hvh(Pi)∫∂K∗PiA(x)∇uh⋅nds,

and

 Bh(t,s;uh,vh)=−∑Pi∈N0hvh(Pi)∫∂K∗PiB(x,t,s)∇uh⋅nds

for all , with denoting the outward unit normal to the boundary of the control volume . Notice that by taking the inner product of (1) with and then integrating, we obtain a similar equation for as

 (utt,vh)+Ah(u,vh)+∫t0Bh(t,s;u(s),vh)ds=(f,vh)∀vh∈U∗h. (3.4)

For the error analysis, we first introduce two interpolation operators. Let be the piecewise linear interpolation operator and be the piecewise constant interpolation operator. These interpolation operators are defined, respectively, by

 Πhu=∑Pi∈N0hu(Pi)ϕi(x) and Π∗hu=∑Pi∈N0hu(Pi)χi(x). (3.5)

Now for , has the following approximation property, (see, Ciarlet [10]):

 ∥ψ−Πhψ∥0≤Ch2∥ψ∥2. (3.6)

Further, we introduce the following discrete norms

 ∥vh∥0,h=⎛⎝∑K∈Th|vh|20,h,K⎞⎠1/2and∥vh∥1,h=(∥vh∥20,h+|vh|21,h)1/2,

where the seminorm , and for ,

 |vh|0,h,K={13(vh(P1)2+vh(P2)2+vh(P3)2)|K|}1/2
 |vh|1,h,K={(|∂vh∂x|2+|∂vh∂y|2)|K|}1/2.

In the following Lemma, a relation between discrete norms and standard Sobolev norms is stated without proof. For a proof, see, [17, pp. 124] and [4].

###### Lemma 3.1

For and are identical; and are equivalent to and , respectively, that is, there exist positive constants and , independent of h, such that

 C3∥vh∥0,h≤∥vh∥0≤C4∥vh∥0,h∀vh∈Uh (3.7)

and

 C3||vh||1,h≤||vh||1≤C4||vh||1,h∀vh∈Uh. (3.8)

Note that Below, we state without proof the properties of the interpolation operator For a proof, we refer to [17, pp. 192].

###### Lemma 3.2

The following statements hold true.

 (ϕh,Π∗hvh)=(vh,Π∗hϕh)∀ϕh, vh∈Uh. (3.9)

(ii)  With , the norms and are equivalent on , that is, there exist positive constants and , independent of , such that

 ceq||ϕh||0≤∥|ϕh∥|≤Ceq||ϕh||0∀ϕh∈Uh. (3.10)

## 4 A Priori Error Estimates

This section is devoted to a priori error estimates of the approximation to the spatial semidiscrete scheme (3.3).

For the derivation of optimal error estimates, we split as

 e:=(u−Vhu)+(Vhu−uh)=:ρ+θ,

where is the Ritz-Volterra projection defined by

 A(u−Vhu,χh)+∫t0B(t,s;u−Vhu,χh)ds=0∀χh∈Uh. (4.1)

With some abuse of notations, we will denote by the Ritz projection of onto defined by

 A(u0−Vhu0,χh)=0∀χh∈Uh.

For our subsequent analysis, we state without proof following error estimates for the Ritz-Volterra projection. For a proof, see, [26], [5], [18], [20] and [21].

###### Lemma 4.1

There exist positive constants , independent of , such that for , and the following estimates hold:

 ∥Djtρ(t)∥0+h∥Djtρ(t)∥1≤Chr[j∑l=0∥Dltu(t)∥r+∫t0∥u(s)∥rds], (4.2)

and

 ∥ρ(t)∥0,∞≤Ch2(log1h)(∥u(t)∥2,∞+∫t0||u(s)||2,∞ds). (4.3)

Now, define

 ϵh(f,χ)=(f,χ)−(f,Π∗hχ)∀χ∈Uh,
 ϵA(ψ,χ)=A(ψ,χ)−Ah(ψ,Π∗hχ)∀ψ, χ∈Uh,

and

 ϵB(t,s;ψ,χ)=B(t,s;ψ,χ)−Bh(t,s;ψ,Π∗hχ)∀ψ, χ∈Uh.

Then, the following lemma will be of frequent use in our analysis and the proof of which can be found in [8].

###### Lemma 4.2

Assume that the coefficient matrices for Then, there exist positive constant independent of , such that the following estimates hold for and for

 |ϵh(f,χ)|≤Chi+j∥f∥Hi∥χ∥Hj∀f∈Hi, (4.4)

and for

 |ϵA(Vhu,χ)|≤Chi+j(∥u∥H1+i+∫t0∥u(s)∥H1+ids)∥χ∥Hj. (4.5)

Moreover,

 |ϵA(wh,χ)|≤Ch∥wh∥H1∥χ∥H1∀wh∈Uh. (4.6)

The estimates (4.5) and (4.6) are also valid if is replaced by

Now, for and for each introduce a linear functional defined on by

 G(ψ)(χ)=ϵA(ψ,χ)+∫t0ϵB(t,s;ψ(s),χ)ds,χ∈Uh.

Notice that, by using the definition of , (2.1) and (3.4), there follows that

 G(ρ)(χ) = A(u,χ)+∫t0B(t,s;u(s),χ)ds (4.7) −Ah(u,Π∗hχ)−∫t0Bh(t,s;u(s),Π∗hχ)ds−G(Vhu)(χ) = (f−utt,χ)−(f−utt,Π∗hχ)−G(Vhu)(χ) = ϵh(f−utt,χ)−G(Vhu)(χ).

From (3.3) and (3.4), we obtain the equation in for as

 (θtt,vh)+Ah(θ,vh)+∫t0Bh(t,s;θ(s),vh)ds=−Ah(ρ,vh)−∫t0Bh(t,s;ρ,χ)ds−(ρtt,vh).

Choosing and using the definition of and (4.1), we find that

 (θtt,Π∗hχ)+A(θ,χ)ds + ∫t0B(t,s;θ(s),χ)ds=G(ρ)(χ) (4.8) + G(θ)(χ)−(ρtt,Π∗hχ)∀χ∈Uh.

For any continuous function in , define by

 ^ϕ(t)=∫t0ϕ(s)ds.

Notice that and . Then, integrate (4.8) from to to obtain

 (θt,Π∗hχ)+A(^θ,χ) = ^G(ρ)(χ)+^G(θ)(χ)+(−ρt,Π∗hχ)+(et(0),Π∗hχ) (4.9) −∫t0B(s,s;^θ(s),χ)ds+∫t0∫s0Bτ(s,τ;^θ(τ),χ)dτds,

where

 ^G(ϕ)(χ)=ϵA(^ϕ,χ)+∫t0ϵB(s,s;^ϕ(s),χ)ds−∫t0∫s0ϵBτ(s,τ;^ϕ(τ),χ)dτds.

For a linear functional defined on , set

 ∥F∥−1,h=sup0≠χ∈Uh|F(χ)|∥χ∥1.

We shall need the following lemmas in our subsequent analysis.

###### Lemma 4.3

With and as above, there exists a positive constant such that the following estimates

 ∥DjtG(Vhu)∥−1,h≤Ch2(j∑ℓ=0∥Dℓtu(t)∥2+∫t0∥u(s)∥2ds), (4.10)

and

 ∥Djt^G(Vhu)∥−1,h≤Ch2(j∑ℓ=0∥Djt^u(t)∥2+∫t0∥^u(s)∥2ds), (4.11)

hold for .

Proof. Using (4.5) and the estimates in Lemma 2.1, we obtain

 |G(Vhu)(χ)| ≤ |ϵA(Vhu,χ)|+∫t0|ϵB(t,s;Vhu(s),χ)|ds ≤ Ch2(∥u∥2+∫t0∥u(s)∥2ds)∥χ∥1,

and

 |Gt(Vhu)(χ)| ≤ Ch2(∥ut∥2+∥u∥2+∫t0∥u(s)∥2ds)∥χ∥1 ≤ Ch2(∥ut∥2+∥u∥2+∫t0∥u(s)∥2ds)∥χ∥1

In a similar manner, we derive the second estimate (4.11) and this completes the rest of the proof.

In the error analysis, we shall frequently use the following inverse assumption:

 ∥χ∥1≤Cinvh−1∥χ∥0,χ∈Uh. (4.12)

### 4.1 H1- error estimate

###### Theorem 4.1

Let and be the solutions of and respectively, and assume that , and . Further, assume that and , where is the interpolation operator defined in . Then, there exists a positive constant , independent of , such that for the following estimate

 ∥u(t)−uh(t)∥1≤Ch(∥u0∥3+∥u1∥2+∫t0(∥f∥1+∥ft∥0+∥ftt∥0)ds)

holds.

Proof. Since and estimates of are known from the Lemma 4.1, it is sufficient to estimate Choose in (4.8) and use (4.7) to obtain

 (θtt,Π∗hθt)+A(θ,θt)+∫t0B(t,s;θ(s),θt)ds = ϵh(f−utt,θt)−G(Vhu)(θt) +G(θ)(θt)−(ρtt,Π∗hθt).

Now use (3.9) and symmetry of the bilinear form to arrive at

 12ddt[(θt,Π∗hθt)+A(θ,θ)] = ϵh(f−utt,θt)−G(Vhu)(θt)+G(θ)(θt)−(ρtt,Π∗hθt) −∫t0B(t,s;θ(s),θt(t))ds.

Integration from to yields

 12(∥|θt∥|2+A(θ,θ)) = 12(∥|θt(0)∥|2+A(θ(0),θ(0)))+∫t0ϵh(f−utt,θt)ds (4.13) −∫t0G(Vhu)(θt)ds+∫t0G(θ)(θt)ds+∫t0(−ρtt,Π∗hθt)ds −∫t0∫s0B(s,τ;θ(τ),θt(s))dτds = J1+J2