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Numerical approximation of a Bresse-Maxwell-Cattaneo system

  • In this paper, we analyze, from the numerical point of view, a new thermoelastic problem involving the so-called Bresse system. The heat conduction is modeled by using the Maxwell-Cattaneo law, which is of hyperbolic type. An existence and uniqueness result and an energy decay property are recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. First, we prove that the discrete solution is stable, and secondly, we provide an a priori error analysis. This allows us to conclude the linear convergence under suitable additional regularity on the continuous solution. Finally, numerical results are presented to demonstrate the convergence of the scheme and the behavior of the discrete energy.

    Citation: Noelia Bazarra, José R. Fernández, Irea López, María Rodríguez-Damián. Numerical approximation of a Bresse-Maxwell-Cattaneo system[J]. Electronic Research Archive, 2025, 33(6): 3883-3900. doi: 10.3934/era.2025172

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  • In this paper, we analyze, from the numerical point of view, a new thermoelastic problem involving the so-called Bresse system. The heat conduction is modeled by using the Maxwell-Cattaneo law, which is of hyperbolic type. An existence and uniqueness result and an energy decay property are recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. First, we prove that the discrete solution is stable, and secondly, we provide an a priori error analysis. This allows us to conclude the linear convergence under suitable additional regularity on the continuous solution. Finally, numerical results are presented to demonstrate the convergence of the scheme and the behavior of the discrete energy.



    The study of the so-called Bresse system (introduced by J. A. C. Bresse in 1856 (see [1])), and also known as the circular arch problem, includes the mechanical deformation of beams with natural length . This problem can be considered as a generalization of the well-known Timoshenko problem, which is obtained in the particular case where the longitudinal displacement is not modeled and supposing zero initial curvature.

    If we denote by φ, ψ and w the vertical displacement, the rotation angle of the cross-section and the horizontal displacement, respectively, the corresponding system of equations is written as (see, for details, the derivation provided in[2])

    ρ1φttκ(φx+ψ+lw)xlκ0(wxlφ)=0,ρ2ψttbψxx+κ(φx+ψ+lw),ρ1wttκ0(wxlφ)x+lκ(φx+ψ+lw)=0,

    where ρ1 and ρ2 are the mass density and the moment of mass inertia of the beam, κ is the shear modulus of elasticity, l is the initial curvature, κ0=Eh, where E is the Young's modulus and h is the cross-sectional area, and b stands for the rigidity coefficient of the cross section.

    There is a large number of papers dealing with different issues involving this Bresse system. In this way, just as a few examples, we can recall the work of Ma and Nunes [3], where the limit case corresponding to a zero arch curvature (i.e., the Timoshenko system) and the long-time dynamics are shown, the work of Baibeche et al. [4], who introduced the microtemperatures effect with a delay term on a boundary condition, the work of Li et al. [5], who studied the asymptotic behavior of this system with mass diffusion, the work of Muñoz-Rivera and Nunes [6], where the pointwise stabilization on an interior point is assumed to prove the exponential stability, the work of Copetti et al. [7], where a contact problem with the viscoelastic Bresse system is mathematically studied, the work of Bazarra et al. [8], where the so-called dual-phase-lag model is incorporated into the heat conduction, the works of Afilal et al. [9,10], where the second sound effect is also incorporated, and the work of Suzuki and Ueda [11], where the dissipation of a thermoelastic Bresse system is studied with different damping effects. We also recall the works of Alabau Boussouira et al. [12], where the lack of exponential stability is proved if the velocities of wave propagation are not the same, and the work of Almeida Júnior [13], where the locking phenomenon on the shear force is considered for the Timoshenko equation. Moreover, we could cite the recent numerical works of Wang et al. [14] and Yang et al. [15] for the study of some related fourth-order problems.

    In this paper, we want to continue the contributions [16,17], studying a numerical approximation of the thermoelastic Bresse system by using the heat conduction model proposed by Maxwell and Cattaneo [18], where a relaxation parameter is introduced into the constitutive equations to be compatible with the causality principle and to overcome the instantaneous propagation of thermal waves [19].

    Let us denote by (0,) the interval occupied by the curved thermoelastic beam of length >0, and let x[0,] and t[0,T] be the spatial and time variables, respectively, where T>0 denotes the final time.

    The linearized version of the Bresse-Maxwell system by using the hyperbolic heat conduction proposed by Cattaneo is written as follows in (0,)×(0,T) (see [16,17,20]):

    ρ1φttκ(φx+ψ+lw)xlκ0(wxlφ)+lγξ=0,ρ2ψttbψxx+κ(φx+ψ+lw)+γϑx=0,ρ1wttκ0(wxlφ)x+lκ(φx+ψ+lw)+γξx=0,ρ3ϑt+px+γψxt=0,ςϖpt+p+ϖϑx=0,ρ3ξt+qx+γ(wxtlφt)=0,τϖqt+q+ϖξx=0, (1.1)

    where now ϑ is the temperature deviation from a fixed reference temperature in the vertical direction, ξ is the temperature deviation from a fixed reference temperature in a horizontal direction, p denotes the vertical heat flux, and q stands for the horizontal heat flux. The new coefficients γ, ς, ϖ, and ρ3 account for the physical properties of the beam, and τ represents the delay parameter.

    As it is pointed out in [16], these coefficients satisfy the constraints:

    κ0=bρ1ρ2,b>κρ2ρ1.

    This system (1.1) is complemented with the following boundary conditions:

    φ(X,t)=ψ(X,t)=w(X,t)=ϑ(X,t)=ξ(X,t)=0 (1.2)

    for X=0,, and the initial conditions:

    φ(x,0)=φ0(x),φt(x,0)=ϕ0(x),ψ(x,0)=ψ0(x),ψt(x,0)=η0(x),w(x,0)=w0(x),wt(x,0)=e0(x),ϑ(x,0)=ϑ0(x),p(x,0)=p0(x),ξ(x,0)=ξ0(x),q(x,0)=q0(x) (1.3)

    for a.e. x(0,).

    The following existence and uniqueness result was proved in [17].

    Theorem 1. The operator that generates the solutions to problem (1.1)(1.3) is the infinitesimal generator of a C0-semigroup of contractions in a suitable Hilbert space. Therefore, problem (1.1)(1.3) has a unique solution with the regularity:

    φ,ψ,wC1([0,T];H10(0,))C([0,T];H2(0,)),p,qC1([0,T];L2(0,))C([0,T];H1(0,)),ϑ,ξC1([0,T];L2(0,))C([0,T];H10(0,)).

    Moreover, even if the stability of the solutions to problem (1.1)–(1.3) has been discussed by some authors, we recall the recent result obtained in [16].

    Theorem 2. Let us define the stability numbers:

    χς=(ςρ3ρ1κ)(bκρ2ρ1)+γ2ς,χτ=(τρ3ρ1κ)(κ0κ)+γ2τ.

    Therefore, the semigroup generated by the solutions to problem (1.1)(1.3) is exponentially stable if and only if χ ς χτ=0. On the contrary, it is (semiuniformly) polynomially stable with optimal decay rate t.

    From [17], we observe that the energy norm given by

    E(t)=12(ρ1φt2L2+ρ2ψt2L2+ρ1wt2L2+bψx2L2+κφx+ψ+lw2L2+κ0wxlφ2L2+ρ3(ϑ2L2+θ2L2)+ςp2L2+τq2L2)

    is equivalent to the usual norm used in the analysis of these thermoelastic problems defined as

    ˜E(t)=12(φt2L2+ψt2L2+wt2L2+ψ2H1+φ2H1+w2H1+ϑ2L2+θ2L2+p2L2+q2L2)

    whenever we assume the general condition lnπ for all nN.

    Now, we will derive the variational formulation of problem (1.1)–(1.3). Let us define the variational spaces V=H10(0,) and E=H1(0,), and denote by (,) the classical inner product in the space Y=L2(0,) with corresponding norm . Therefore, multiplying the equations of system (1.1) by adequate test functions and taking into account the boundary conditions (1.2), we arrive at the following weak formulation written in terms of the variables ϕ=φt, η=ψt, e=wt, ϑ, p, ξ, and q.

    Find the vertical velocity ϕ:[0,T]V, the rate of the rotation angle η:[0,T]V, the horizontal velocity e:[0,T]V, the vertical temperature ϑ:[0,T]V, the horizontal temperature ξ:[0,T]V, the vertical heat flux p:[0,T]E, and the horizontal heat flux q:[0,T]E such that, for a.e. t(0,T) and for all r,v,z,m,χV,s,ζE,

    ρ1(ϕt(t),r)+κ(φx(t),rx)κ(ψx(t),r)(κ+κ0)l(wx(t),r)+l2κ0(φ(t),r)+lγ(ξ(t),r)=0, (1.4)
    ρ2(ηt(t),v)+b(ψx,vx)+κ(φx(t)+lw(t),v)+κ(ψ(t),v)+γ(ϑx(t),v)=0, (1.5)
    ρ1(et(t),z)+κ0(wx(t),zx)+l(κ0+κ)(φx(t),z)+lκ(ψ(t),z)+l2κ(w(t),z)+γ(ξx(t),z)=0, (1.6)
    ρ3(ϑt(t),m)+(px(t),m)+γ(ηx(t),m)=0, (1.7)
    ςϖ(pt(t),s)+(p(t),s)+ϖ(ϑx(t),s)=0, (1.8)
    ρ3(ξt(t),χ)+(qx(t),χ)+γ(ex(t)lϕ(t),χ)=0, (1.9)
    τϖ(qt(t),ζ)+(q(t),ζ)+ϖ(ξx(t),ζ)=0, (1.10)

    where the vertical displacement φ:[0,T]V, the rotation angle ψ:[0,T]V, and the horizontal displacement w:[0,T]V are obtained from the relations

    φ(t)=t0ϕ(s)ds+φ0,ψ(t)=t0η(s)ds+ψ0,w(t)=t0e(s)ds+w0. (1.11)

    The rest of the paper is outlined as follows. In Section 2 we study numerically an approximation of this problem (1.4)–(1.11) by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A discrete stability property and some a priori error estimates are proved, from which the linear convergence is obtained under suitable regularity conditions on the continuous solution. Finally, some numerical simulations are presented in Section 3 to demonstrate the convergence of the approximation, the discrete energy decay, and the behavior of the solution.

    In this section, we will numerically study the variational problem (1.4)–(1.11). Therefore, we will provide first a numerical scheme based on the classical finite element method and the implicit Euler scheme.

    In this subsection, we introduce a fully discrete approximation of the weak problem introduced previously. We will do it in two steps. In order to provide the spatial approximation, let us define a uniform partition of the spatial domain denoted by a0=0<<aM=, and let us construct the finite element spaces:

    Vh={vhC([0,])V;vh[ai,ai+1]P1([ai,ai+1])for i=0,,M1},Wh={whC([0,])E;wh[ai,ai+1]P1([ai,ai+1])for i=0,,M1}.

    In this definition, we have denoted by P1([ai,ai+1]) the space of polynomials with degree less than or equal to one in the subinterval [ai,ai+1] and, as usual, let h=ai+1ai=/M be the spatial parameter.

    Now, we can define the discrete initial conditions as

    φ0h=Ph1φ0,ϕ0h=Ph1ϕ0,ψ0h=Ph1ψ0,η0h=Ph1η0,w0h=Ph1w0,e0h=Ph1e0,ϑ0h=Ph1ϑ0,p0h=Ph2p0,ξ0h=Ph1ξ0,q0h=Ph2q0,

    where Ph1 and Ph2 are the finite element interpolation operators over Vh and Eh, respectively [21].

    In order to obtain the discretization of the time derivatives, we denote by t0=0<<tN=T a uniform partition of the time interval with time step k=t1t0=T/N and nodes tn=nk for n=0,,N. As a usual notation, let wn=w(tn) be the value of a continuous function w(t) at time t=tn, and for a sequence {wn}Nn=0, let δwn=(wnwn1)/k be its divided differences.

    By using the well-known implicit Euler scheme, we have the following fully discrete problem.

    Find the discrete vertical velocity {ϕhkn}Nn=0Vh, the discrete rate of the rotation angle {ηhkn}Nn=0Vh, the discrete horizontal velocity {ehkn}Nn=0Vh, the discrete vertical temperature {ϑhkn}Nn=0Vh, the discrete horizontal temperature {ξhkn}Nn=0Vh, the discrete vertical heat flux {phkn}Nn=0Eh, and the discrete horizontal heat flux {qhkn}Nn=0Eh such that, for n=1,,n and for all rh,vh,zh,mh,χhVh,sh,ζhEh,

    ρ1(δϕhkn,rh)+κ((φhkn)x,rhx)κ((ψhkn)x,rh)(κ+κ0)l((whkn)x,rh)+l2κ0(φhkn,rh)+lγ(ξhkn,rh)=0, (2.1)
    ρ2(δηhkn,vh)+b((ψhkn)x,vhx)+κ((φhkn)x+lwhkn,v)+κ(ψhkn,vh)+γ((ϑhkn)x,vh)=0, (2.2)
    ρ1(δehkn,zh)+κ0((whkn)x,zhx)+l(κ0+κ)((φhkn)x,zh)+lκ(ψhkn,zh)+l2κ(whkn,zh)+γ((ξhkn)x,zh)=0, (2.3)
    ρ3(δϑhkn,mh)+((phkn)x,mh)+γ((ηhkn)x,mh)=0, (2.4)
    ςϖ(δphkn,sh)+(phkn,sh)+ϖ((ϑhkn)x,sh)=0, (2.5)
    ρ3(δξhkn,χh)+((qhkn)x,χh)+γ((ehkn)xlϕhkn,χh)=0, (2.6)
    τϖ(δqhkn,ζh)+(qhkn,ζh)+ϖ((ξhkn)x,ζh)=0, (2.7)

    where the discrete vertical displacement {φhkn}Nn=0Vh, the discrete rotation angle {ψhkn}Nn=0Vh, and the discrete horizontal displacement {whkn}Nn=0Vh are obtained from the relations

    φhkn=knj=1ϕhkj+φ0h,ψhkn=knj=1ηhkj+ψ0h,whkn=knj=1ehkj+w0h. (2.8)

    Remark 3. We note that the above fully discrete problem can be written in terms of the unknowns ϕhkn, ηhkn, ehkn, ϑhkn, phkn, ξhkn, and qhkn by using the definition of the divided differences (we recall the notation δwn=(wnwn1)/k); see the description of the numerical resolution given in Section 3. Therefore, it leads to a linear system written in terms of a product variable Uhkn=(ϕhkn,ηhkn,ehkn,ϑhkn,phkn,ξhkn,qhkn), where the associated matrix is positive definite thanks to the required assumptions on the constitutive coefficients. Therefore, applying Lax-Milgram lemma, it is straightforward to show that the fully discrete problem (2.1)(2.8) has a unique solution.

    In the rest of the section, we will show the numerical analysis of the above problem, providing its discrete stability and a main a priori error estimate result.

    First, the discrete stability is summarized in the following.

    Lemma 4. There exists a positive constant C, which is independent of the discretization parameters h and k, such that, for n=1,,N,

    ϕhkn+φhknV+ηhkn+ψhknV+ehkn+whknV+ϑhkn+phkn+ξhkn+qhknC,

    where X stands for the usual norm in the Hilbert space X.

    Proof.

    First, we obtain the estimates for the discrete vertical velocity. Taking as a test function rh=ϕhkn in the discrete variational equation (2.1), we find that

    ρ1(δϕhkn,ϕhkn)+κ((φhkn)x,(ϕhkn)x)κ((ψhkn)x,ϕhkn)(κ+κ0)l((whkn)x,ϕhkn)+l2κ0(φhkn,ϕhkn)+lγ(ξhkn,ϕhkn)=0.

    Taking into account that ϕhkn=δφhkn and the estimates:

    ρ1(δϕhkn,ϕhkn)ρ12k[ϕhkn2ϕhkn12],κ((φhkn)x,(ϕhkn)x)κ2k[(φhkn)x2(φhkn1)x2],(κ+κ0)l(φhkn,ϕhkn)(κ+κ0)l2k[φhkn2φhkn12],

    using Cauchy-Schwarz inequality and Cauchy's inequality

    abϵa2+14ϵb2for a,b,ϵR with ϵ>0, (2.9)

    we have

    12k[ϕhkn2ϕhkn12]+12k[(φhkn)x2(φhkn1)x2]+12k[φhkn2φhkn12]C((ψhkn)x2+ϕhkn2+(whkn)x2+ξhkn2),

    where, here and in what follows, C>0 is a positive constant that does not depend on the discretization parameters h and k.

    In the same way, we can obtain the estimates for the discrete rate of the rotation angle and the discrete horizontal velocity (we omit here the details for the sake of clarity in the presentation). So, we find that

    12k[ηhkn2ηhkn12]+12k[(ψhkn)x2(ψhkn1)x2]+12k[ψhkn2ψhkn12]+((ϑhkn)x,ηhkn)C((φhkn)x2+ηhkn2+whkn2),12k[ehkn2ehkn12]+12k[(whkn)x2(whkn1)x2]+12k[whkn2whkn12]+((ξhkn)x,ehkn)C((φhkn)x2+ehkn2+ψhkn2).

    Now, we derive the estimates for the discrete vertical temperature and the discrete vertical heat flux. By using the test functions mh=ϑhkn and sh=phkn in discrete variational equations (2.4) and (2.5), respectively, we find that

    ρ3(δϑhkn,ϑhkn))+((phkn)x,ϑhkn)+γ((ηhkn)x,ϑhkn)=0,ςϖ(δphkn,phkn)+(phkn,phkn)+ϖ((ϑhkn)x,phkn)=0.

    Keeping in mind the estimates

    (δϑhkn,ϑhkn)12k[ϑhkn2ϑhkn12],(δphkn,phkn)12k[phkn2phkn12],(phkn,phkn)0,((phkn)x,ϑhkn)=(phkn,(ϑhkn)x),((ηhkn)x,ϑhkn)=(ηhkn,(ϑhkn)x),

    using several times Cauchy-Schwarz inequality and Cauchy inequality (2.9), we have

    12k[ϑhkn2ϑhkn12]+12k[phkn2phkn12](ηhkn,(ϑhkn)x)0.

    Proceeding in an analogous way with the discrete horizontal temperature and the discrete horizontal heat flux, it follows that

    12k[ξhkn2ξhkn12]+12k[qhkn2qhkn12](ehkn,(ξhkn)x)C(ϕhkn2+ξhkn2).

    Combining all these estimates, we find that

    12k[ϕhkn2ϕhkn12]+12k[(φhkn)x2(φhkn1)x2]+12k[φhkn2φhkn12]+12k[ηhkn2ηhkn12]+12k[(ψhkn)x2(ψhkn1)x2]+12k[ψhkn2ψhkn12]+12k[ehkn2ehkn12]+12k[(whkn)x2(whkn1)x2]+12k[whkn2whkn12]+12k[ϑhkn2ϑhkn12]+12k[phkn2phkn12]+12k[ξhkn2ξhkn12]+12k[qhkn2qhkn12]C((ψhkn)x2+ϕhkn2+(whkn)x2+ξhkn2+ηhkn2+whkn2+(φhkn)x2+ehkn2+ψhkn2).

    Multiplying these estimates by k and summing up to n, we have

    ϕhkn2+(φhkn)x2+φhkn2+ηhkn2+(ψhkn)x2+ψhkn2+ehkn2+(whkn)x2+whkn2+ϑhkn2+phkn2+ξhkn2+qhkn2Cknj=1((ψhkj)x2+ϕhkj2+(whkj)x2+ξhkj2+ηhkj2+whkj2+(φhkj)x2+ehkj2+ψhkj2)+C(ϕ0h2+φ0h2V+η0h2+ψ0h2V+e0h2+w0h2+ϑ0h2+p0h2+ξ0h2+q0h2).

    Finally, applying a discrete version of Gronwall's lemma (see [22] for details), we obtain the desired stability estimates.

    Now, we will obtain some a priori error estimates. First, subtracting the variational equation (1.4) at time t=tn and for a test function r=rhVhV and discrete variational equation (2.1), it follows that

    ρ1(ϕtnδϕhkn,rh)+κ((φnφhkn)x,rhx)κ((ψnψhkn)x,rh)(κ+κ0)l((wnwhkn)x,rh)+l2κ0(φnφhkn,rh)+lγ(ξnξhkn,rh)=0,

    and so, we have, for all rhVh,

    ρ1(ϕtnδϕhkn,ϕnϕhkn)+κ((φnφhkn)x,(ϕnϕhkn)x)κ((ψnψhkn)x,ϕnϕhkn)(κ+κ0)l((wnwhkn)x,ϕnϕhkn)+l2κ0(φnφhkn,ϕnϕhkn)+lγ(ξnξhkn,ϕnϕhkn)=ρ1(ϕtnδϕhkn,ϕnrh)+κ((φnφhkn)x,(ϕnrh)x)κ((ψnψhkn)x,ϕnrh)(κ+κ0)l((wnwhkn)x,ϕnrh)+l2κ0(φnφhkn,ϕnrh)+lγ(ξnξhkn,ϕnrh).

    Now, keeping in mind that

    ρ1(δϕnδϕhkn,ϕnϕhkn)ρ12k[ϕnϕhkn2ϕn1ϕhkn12],κ((φnφhkn)x,(δφnδφhkn)x)κ2k[(φnφhkn)x2(φn1φhkn1)x2],(κ+κ0)l(φnφhkn,δφnδφhkn)(κ+κ0)l2k[φnφhkn2φn1φhkn12],

    we obtain the estimates:

    12k[ϕnϕhkn2ϕnϕhkn12]+12k[(φnφhkn)x2(φn1φhkn1)x2]+12k[φnφhkn2φn1φhkn12]C(ϕtnδϕn2+φtnδφn2V+ϕnrh2V+(φnφhkn)x2+φnφhkn2+(ψnψhkn)x2+ϕnϕhkn2+(wnwhkn)x2+ξnξhkn2+(δϕnδϕhkn,ϕnrh))rhVh,

    where C>0 is again a positive constant that does not depend on the discretization parameters h and k, but it depends now on the continuous solution.

    Similarly, we derive the following estimates for the rate of the rotation angle and the horizontal velocity. As we pointed out in the proof of the discrete stability, we omit the details for the sake of clarity in the presentation. So, we obtain

    12k[ηnηhkn2ηn1ηhkn12]+12k[(ψnψhkn)x2(ψn1ψhkn1)x2]+12k[ψnψhkn2ψn1ψhkn12]+((ϑnϑhkn)x,ηnηhkn)C(ηtnδηn2+ψtnδψn2V+(ψnψhkn)x2+ηnηhkn2+wnwhkn2+ηnvh2V+(φnφhkn)x2+ϑnϑhkn2+(δηnδηhkn,ηnvh))vhVh,12k[enehkn2en1ehkn12]+12k[(wnwhkn)x2(wn1whkn1)x2]+12k[wnwhkn2wn1whkn12]+((ξnξhkn)x,enehkn)C(etnδen2+wtnδwn2V+(φnφhkn)x2+enehkn2+enzh2V+wnwhkn2+(wnwhkn)x2+ξnξhkn2+ψnψhkn2+(δenδehkn,enzh))zhVh.

    Secondly, we obtain the estimates for the vertical temperature and the vertical heat flux. Thus, we subtract the variational equations (1.7) and (1.8) at time t=tn and for test functions m=mhVhV and s=shEhE, respectively, and the discrete variational equations (2.4) and (2.5) to obtain

    ρ3(ϑtnδϑhkn,mh)+((pnphkn)x,mh)+γ((ηnηhkn)x,mh)=0,ςϖ(ptnδphkn,sh)+(pnphkn,sh)+ϖ((ϑnϑhkn)x,sh)=0,

    and so, we conclude that, for all mhVh and shEh,

    ρ3(ϑtnδϑhkn,ϑnϑhkn)+((pnphkn)x,ϑnϑhkn)+γ((ηnηhkn)x,ϑnϑhkn)=ρ3(ϑtnδϑhkn,ϑnmh)+((pnphkn)x,ϑnmh)+γ((ηnηhkn)x,ϑnmh),ςϖ(ptnδphkn,pnphkn)+(pnphkn,pnphkn)+ϖ((ϑnϑhkn)x,pnphkn)=ςϖ(ptnδphkn,pnsh)+(pnphkn,pnsh)+ϖ((ϑnϑhkn)x,pnsh).

    By using the following estimates:

    (δϑnδϑhkn,ϑnϑhkn)12k[ϑnϑhkn2ϑn1ϑhkn12],(δpnδphkn,pnphkn)12k[pnphkn2pn1phkn12],(pnphkn,pnphkn)0,((pnphkn)x,ϑnϑhkn)=(pnphkn,(ϑnϑhkn)x),((ηnηhkn)x,ϑnϑhkn)=(ηnηhkn,(ϑnϑhkn)x),((ηnηhkn)x,ϑnmh)=(ηnηhkn,(ϑnmh)x),((ϑnϑhkn)x,pnsh)=(ϑnϑhkn,(pnsh)x),

    applying several times Cauchy-Schwarz inequality and Cauchy's inequality (2.9), it follows that, for all mhVh and shEh,

    12k[ϑnϑhkn2ϑn1ϑhkn12]+12k[pnphkn2pn1phkn12](ηnηhkn,(ϑnϑhkn)x)C(ϑtnδϑn2+ptnδpn2+ϑnmh2V+pnsh2E+pnphkn2+ϑnϑhkn2+ηnηhkn2+(δpnδphkn,pnsh)+(δϑnδϑhkn,ϑnmh)).

    Proceeding in an analogous way with the discrete horizontal temperature and the discrete horizontal heat flux, we obtain, for all χhVh and ζhEh,

    12k[ξnξhkn2ξn1ξhkn12]+12k[qnqhkn2qn1qhkn12](enehkn,(ξnξhkn)x)C(ξtnδξn2+qtnδqn2+ξnχh2V+qnζh2E+qnqhkn2+ξnξhkn2+ηnηhkn2+(δqnδqhkn,qnζh)+(δξnδξhkn,ξnχh)+ϕnϕhkn2).

    Combining all the previous estimates, we have, for all rh,vh,zh,mh,χhVh,sh,ζhEh,

    12k[ϕnϕhkn2ϕnϕhkn12]+12k[(φnφhkn)x2(φn1φhkn1)x2]+12k[φnφhkn2φn1φhkn12]+12k[ηnηhkn2ηn1ηhkn12]+12k[(ψnψhkn)x2(ψn1ψhkn1)x2]+12k[ψnψhkn2ψn1ψhkn12]+12k[enehkn2en1ehkn12]+12k[(wnwhkn)x2(wn1whkn1)x2]+12k[wnwhkn2wn1whkn12]+12k[pnphkn2pn1phkn12]+12k[ϑnϑhkn2ϑn1ϑhkn12]+12k[ξnξhkn2ξn1ξhkn12]+12k[qnqhkn2qn1qhkn12]C(ϕtnδϕn2+φtnδφn2V+ϕnrh2V+(φnφhkn)x2+φnφhkn2+(ψnψhkn)x2+ϕnϕhkn2+(wnwhkn)x2+ξnξhkn2+(δϕnδϕhkn,ϕnrh)+ηtnδηn2+ψtnδψn2V+ηnηhkn2+wnwhkn2+ηnvh2V+(δηnδηhkn,ηnvh)+etnδen2+wtnδwn2V+enehkn2+enzh2V+(δenδehkn,enzh)+ϑtnδϑn2+ptnδpn2+ϑnmh2V+pnsh2E+pnphkn2+ϑnϑhkn2+(δpnδphkn,pnsh)+(δϑnδϑhkn,ϑnmh)+ξtnδξn2+qtnδqn2+ξnχh2V+qnζh2E+qnqhkn2+(δqnδqhkn,qnζh)+(δξnδξhkn,ξnχh)).

    Multiplying the above estimates by k and summing up to n, we find that, for all {rhj}nj=1,{vhj}nj=1,{zhj}nj=1,{mhj}nj=1,{χhj}nj=1Vh,{shj}nj=1,{ζhj}nj=1Eh,

    ϕnϕhkn2+(φnφhkn)x2+φnφhkn2+ηnηhkn2+(ψnψhkn)x2+ψnψhkn2+enehkn2+(wnwhkn)x2+wnwhkn2+pnphkn2+ϑnϑhkn2+ξnξhkn2+qnqhkn2Cknj=1(ϕtjδϕj2+φtjδφj2V+ϕjrhj2V+(φjφhkj)x2+φjφhkj2+(ψjψhkj)x2+ϕjϕhkj2+(wjwhkj)x2+ξjξhkj2+(δϕjδϕhkj,ϕjrhj)+ηtjδηj2+ψtjδψj2V+ηjηhkj2+wjwhkj2+ηjvhj2V+(δηjδηhkj,ηjvhj)+etjδej2+wtjδwj2V+ejehkj2+ejzhj2V+(δejδehkj,ejzhj)+ϑtjδϑj2+ptjδpj2+ϑjmhj2V+pjshj2E+pjphkj2+ϑjϑhkj2+(δpjδphkj,pjshj)+(δϑjδϑhkj,ϑjmhj)+ξtjδξj2+qtjδqj2+ξjχhj2V+qjζhj2E+qjqhkj2+(δqjδqhkj,qjζhj)+(δξjδξhkj,ξjχhj))+C(ϕ0ϕ0h2+φ0φ0h2V+η0η0h2+ψ0ψ0h2V+e0e0h2+w0w0h2V+ϑ0ϑ0h2+p0p0h2+ξ0ξ0h2+q0q0h2).

    Now, we observe that

    knj=1(δϕjδϕhkj,ϕjrhj)=(ϕnϕhkn,ϕnrhn)+(ϕ0hϕ0,ϕ1rh1)+n1j=1(ϕjϕhkj,ϕjrhj(ϕj+1rhj+1)),

    where similar relations can be found for the remaining variables.

    Finally, applying a discrete version of Gronwall's lemma (see again [22]), we conclude the following a priori error estimates result.

    Theorem 5. Under the assumptions imposed on the constitutive coefficients, let (ϕ,φ,η,ψ,e,w,ϑ,p,ξ,q) and (ϕhk,φhk,ηhk,ψhk,ehk,whk,ϑhk,phk,ξhk,qhk) be the solutions to problems (1.4)–(1.11) and (2.1)–(2.8), respectively. Therefore, it follows that, for all {rhn}Nn=0,{vhn}Nn=0,{zhn}Nn=0,{mhn}Nn=0,{χhn}Nn=0Vh,{shn}Nn=0,{ζhn}Nn=0Eh,

    max0nN{ϕnϕhkn2+φnφhkn2V+ηnηhkn2+ψnψhkn2V+enehkn2+wnwhkn2V+pnphkn2+ϑnϑhkn2+ξnξhkn2+qnqhkn2}CkNj=1(ϕtjδϕj2+φtjδφj2V+ϕjrhj2V+ηtjδηj2+ψtjδψj2V+ηjvhj2V+etjδej2+wtjδwj2V+ejzhj2V+ϑtjδϑj2+ptjδpj2+ϑjmhj2V+pjshj2E+ξtjδξj2+qtjδqj2+ξjχhj2V+qjζhj2E)+Ck1N1j=1[ϕjrhj(ϕj+1rhj+1)2+ηjvhj(ηj+1vhj+1)2+ejzhj(ej+1zhj+1)2+ϑjmhj(ϑj+1mhj+1)2+pjshj(pj+1shj+1)2+ξjχhj(ξj+1χhj+1)2+qjζhj(qj+1ζhj+1)2]+Cmax0nN{ϕnrhn2+ηnvhn2+enzhn2+ϑnmhn2+pnshn2+ξnχhn2+qnζhn2}+C(ϕ0ϕ0h2+φ0φ0h2V+η0η0h2+ψ0ψ0h2V+e0e0h2+w0w0h2V+ϑ0ϑ0h2+p0p0h2+ξ0ξ0h2+q0q0h2),

    where C is a positive constant that depends on the continuous solution but it is independent of the discretization parameters.

    The a priori error estimates shown above can be used to derive the convergence order. As an example, if we assume that the continuous solution has the regularity

    φ,ψ,wH3(0,T;Y)C2([0,T];V)C1([0,T];H2(0,)),p,qH2(0,T;Y)C([0,T];H2(0,)E),ξ,ϑH2(0,T;Y)C([0,T];H2(0,)V),

    then we approximations provided by the fully discrete problem (2.1)–(2.8) converge linearly; that is, there exists a positive constant C>0, independent of the discretization parameters h and k, such that

    max0nN{ϕnϕhkn+φnφhknV+ηnηhkn+ψnψhknV+enehkn+wnwhknV+pnphkn+ϑnϑhkn+ξnξhkn+qnqhkn}C(h+k).

    In this final section, we describe the numerical scheme implemented in MATLAB for solving the fully discrete problem (2.1)–(2.8), and we show some numerical examples to demonstrate the accuracy of the approximations and the decay of the discrete energy.

    As a first step, given the solution φhkn1,ψhkn1,whkn1,ϑhkn1,phkn1,ξhkn1 and qhkn1 at time tn1, variables ϕhkn,ηhkn,ehkn,ϑhkn,phkn,ξhkn and qhkn are obtained by solving the discrete linear system, for all rh,vh,zh,mh,χhVh,sh,ζhEh,

    ρ1(ϕhknk,rh)+κk((ϕhkn)x,rhx)κk((ηhkn)x,rh)(κ+κ0)kl((ehkn)x,rh)+l2κ0k(ϕhkn,rh)+lγ(ξhkn,rh)=ρ1(ϕhkn1k,rh)κ((φhkn1)x,rhx)+κ((ψhkn1)x,rh)+(κ+κ0)l((whkn1)x,rh)l2κ0(φhkn1,rh)+(F1n,rh),ρ2(ηhknk,vh)+bk((ηhkn)x,vhx)+κk((ϕhkn)x+lk(ehkn),vh)+κk(ηhkn,vh)+γ((ϑhkn)x,vh)=ρ2(ηhkn1k,vh)b((ψhkn1)x,vhx)κ((φhkn1)x+l(whkn1),vh)κ(ψhkn1,vh)+(F2n,vh),ρ1(ehknk,zh)+κ0k((ehkn)x,zhx)+l(κ0+κ)k((ϕhkn)x,zh)+lκk(ηhkn,zh)+l2κk(ehkn,zh)+γ((ξhkn)x,zh)=ρ1(ehkn1k,zh)κ0((whkn1)x,zhx)l(κ0+κ)((φhkn1)x,zh)lκ(ψhkn1,zh)l2κ(whkn1,zh)+(F3n,zh),ρ3(ϑhknk,mh)+((phkn)x,mh)+γ((ηhkn)x,mh)=ρ3(ϑhkn1k,mh)+(F4n,mh),ςϖ(phknk,sh)+(phkn,sh)+ϖ((ϑhkn)x,sh)=ςϖ(phkn1k,sh)+(F5n,sh),ρ3(ξhknk,χh)+((qhkn)x,χh)+γ((ehkn)xlϕhkn,χh)=ρ3(ξhkn1k,χh)+(F6n,χh),τϖ(qhknk,ζh)+(qhkn,ζh)+ϖ((ξhkn)x,ζh)=τϖ(qhkn1k,ζh)+(F7n,ζh),

    where Fi, for i=1,,7, represents artificial supply terms that are introduced in order to make the system more general.

    The numerical scheme was implemented on a 2.9 GHz PC using MATLAB, and a typical run (using parameters h=k=0.01) took about 0.12 sec of CPU time.

    As an academic example, in order to show the accuracy of the approximations, we have solved an example with the following data:

    T=1,=1,ρ1=1,ρ2=1,ρ3=1,κ=1,l=1,γ=1,ς=1,ϖ=1,τ=1,b=2,κ0=2.

    By using the initial conditions, for all x(0,1),

    φ0(x)=ϕ0(x)=x(x1),ψ0(x)=η0(x)=x(x1),e0(x)=w0(x)=x(x1),ϑ0(x)=p0(x)=ξ0(x)=q0(x)=x(x1),

    considering homogeneous Dirichlet boundary conditions for the variables φ,ψ,w,ϑ,ξ and the (artificial) supply terms, for all (x,t)[0,1]×[0,1],

    F1(x,t)=et(4x212x+2),F2(x,t)=et(3x2+x6),F3(x,t)=et(3x2+5x8),F4(x,t)=et(x2+3x2),F5(x,t)=F7(x,t)=et(2x21),F6(x,t)=et(4x2),

    the exact solution to the above problem can be easily calculated, and it has the form, for (x,t)[0,1]×[0,1]:

    φ(x,t)=ψ(x,t)=w(x,t)=p(x,t)=q(x,t)=ϑ(x,t)=ξ(x,t)=etx(x1).

    Here, if we approximate the numerical errors by

    max0nN{ϕnϕhkn+φnφhknV+ηnηhkn+ψnψhknV+enehkn+wnwhknV+pnphkn+ϑnϑhkn+ξnξhkn+qnqhkn},

    they are depicted in Table 1 using different values of h and k. Furthermore, in Figure 1 we show the evolution of the above numerical errors depending on the parameter h+k. It is clear that the convergence is found, and the linear convergence (that is, first order), stated in the above section, is achieved.

    Table 1.  Example 1: Numerical errors for some values of h and k.
    hk 0.01 0.005 0.002 0.001 0.0005 0.0002 0.0001
    1/23 0.568372 0.567958 0.567767 0.567713 0.567688 0.567674 0.567669
    1/24 0.284072 0.283581 0.283402 0.283363 0.283347 0.283339 0.283336
    1/25 0.142712 0.141915 0.141666 0.141622 0.141608 0.141603 0.141601
    1/26 0.072787 0.071325 0.070886 0.070818 0.070800 0.070794 0.070792
    1/27 0.039020 0.036392 0.035564 0.035439 0.035406 0.035397 0.035395
    1/28 0.023655 0.019530 0.018021 0.017780 0.017718 0.017701 0.017698
    1/29 0.017225 0.011865 0.009460 0.009010 0.008890 0.008855 0.008850
    1/210 0.014732 0.008661 0.005478 0.004730 0.004505 0.004437 0.004428
    1/211 0.013842 0.007420 0.003769 0.002740 0.002365 0.002238 0.002219
    1/212 0.013553 0.006976 0.003091 0.001886 0.001370 0.001156 0.001119
    1/213 0.013469 0.006832 0.002842 0.001548 0.000944 0.000643 0.000578
    1/214 0.013447 0.006790 0.002759 0.001424 0.000775 0.000417 0.000322

     | Show Table
    DownLoad: CSV
    Figure 1.  Example 1: Asymptotic constant error.

    In this last example, we are going to study the energy decay of the discrete problem. Therefore, we assume that there are no supply terms, and we use the following data:

    T=30,=1,ρ1=1,ρ2=1,ρ3=1,κ=1,l=1,γ=3,ς=0.1,ϖ=0.5,τ=0.1,b=2,κ0=2,

    and the initial conditions, for all x[0,1],

    φ0(x)=ϕ0(x)=p0(x)=q0(x)=e0(x)=w0(x)=0,ψ0(x)=η0(x)=ϑ0(x)=ξ0(x)=x(x1).

    With this data, we note that the stability numbers defined in Theorem 2 are both equal to zero. Thus, the semigroup generated by the solutions of the problem is exponentially stable because the condition χ ς χτ=0 is satisfied. Taking the discretization parameters h=0.001 and k=0.0001, the evolution in time of the discrete energy of the problem, defined as

    Ehkn=12(ρ1ϕhkn2Y+ρ2ηhkn2Y+ρ1ehkn2Y+b(ψhkn)x2Y+κ(φhkn)x+ψ+lwhkn2Y+κ0(whkn)xlφhkn2Y+γm(ϑhkn2Y+θhkn2Y)+γκ1m(ςphkn2Y+τqhkn2Y))

    is plotted in Figure 2 (in both natural and semi-log scales).

    Figure 2.  Example 2: Evolution in time of the discrete energy (natural and semi-log scales).

    As can be seen, it converges to zero, and the exponential decay seems to be achieved.

    Taking the data

    T=30,=1,ρ1=1,ρ2=1,ρ3=1,κ=1,l=1,γ=3,ς=1.1,ϖ=0.5,τ=1.1,b=2,κ0=2,

    and the same initial conditions as before, for all x[0,1]. With the above data, the stability numbers defined in Theorem 2 are both non equal to zero (χ ς =χτ=10), and so the semigroup generated by the solutions to this problem is not exponentially stable. In fact, the energy decay is (semiuniformly) polynomially stable with optimal decay rate t. Taking the discretization parameters h=103 and k=105, the evolution in time of the discrete energy of the problem is plotted in Figure 3 (in both natural and semi-log scales).

    Figure 3.  Example 2: Evolution in time of the discrete energy (natural and semi-log scales).

    Finally, in order to verify if the theoretical polynomial energy decay is achieved, we compare the above discrete energy with the function C(logt)/2, where C is a positive constant to be fixed. After an adjustment procedure, we have found the value C=4, and so in Figure 4 the evolution of the discrete energy (in semi-log scale) is plotted against the evolution of the function 2logt. As can be seen, a similar behavior of both graphs is obtained.

    Figure 4.  Example 2: Evolution in time of the discrete energy (semi-log scale) in comparison with the function 2logt.

    In this work, we have studied, from the numerical point of view, a new thermoelastic Bresse system where the heat conduction has been modeled by using the Cattaneo-Maxwell law. This leads to a hyperbolic law, and it was considered, from the analytical point of view, by Dell'Oro [16] and Lima and Fernández-Sare [17]. Here, we have focused on a fully discrete approximation based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. The stability of the discrete solution has been proved, and an a priori error analysis has been provided. As a particular case, the linear convergence of the approximations has been deduced assuming some regularity conditions on the continuous solution. Finally, we have shown some numerical results, including an example to demonstrate the numerical convergence (Example 1) and another one to analyze the discrete energy decay (Example 2). In the latter one, we have seen how it depends on the condition proved in [17] for the so-called stability numbers χ ς  and χτ.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This paper is part of the project "Qualitative and numerical analyses of some thermomechanical problems (ACUANUTER)" (Ref. PID2024-156827NB-I00), which is currently under evaluation by the Spanish Ministry of Science, Innovation and Universities.

    The authors would like to thank the anonymous reviewers for their helpful and interesting comments, which have improved the quality of this work.

    The authors declare there are no conflicts of interest.



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