Abstract
The expanded mixed covolume Element (EMCVE) method is studied for the two-dimensional integro-differential equation of Sobolev type. We use a piecewise constant function space and the lowest order Raviart-Thomas (\(\mathit{RT}_{0}\)) space as the trial function spaces of the scalar unknown u and its gradient σ and flux λ, respectively. The semi-discrete and backward Euler fully-discrete EMCVE schemes are constructed, and the optimal a priori error estimates are derived. Moreover, numerical results are given to verify the theoretical analysis.
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1 Introduction
We consider the linear integro-differential equation of Sobolev type
for \((x,t)\in\Omega\times J\), with boundary and initial conditions
where Ω is a convex and bounded polygonal domain in \(R^{2}\) with boundary denoted by ∂Ω, \(J=(0,T]\) with \(0< T<\infty\), the initial function \(u_{0}(x)\), the source function \(f(x,t)\), and coefficients \(k(x,t,\tau)\), \(a(x)\), \(b(x)\) and \(c(x)\) are given bounded and smooth functions, and there exist some constants \(a_{0}\), \(a_{1}\), \(b_{0}\), \(b_{1}\), \(c_{0}\) and \(c_{1}\) such that
Partial integro-differential equations are often used to describe various physical processes such as heat conduction behavior in memory material, nuclear reactor dynamics, compression of viscoelastic media and the propagation of sound in viscous media. Various numerical studies have been reported based on the finite element methods [1–3], finite volume element methods [4, 5], mixed finite element methods [6–9], discontinuous mixed covolume methods [10] etc. Numerical solutions for the integro-differential equation of Sobolev type have been given by Cui [11] who constructed a finite element scheme and obtained optimal error estimate by introducing Sobolev-Volterra projection; Che et al. [12] who considered \(H^{1}\)-Galerkin expanded mixed finite element method; and Guezane-Lakoud et al. [13] who developed Rothe’s method for one-dimensional problem with integral conditions.
Mixed covolume element (MCVE) method was first introduced by Russell [14] to solve the mixed formulation of linear elliptic problems. Subsequently, Chou et al. [15, 16] considered the MCVE method for the elliptic boundary value problems by using the \(\mathit{RT}_{0}\) space on the triangular grids and rectangular grids, respectively. This method not only can calculate several different physical quantities (such as pressure and Darcy velocity in [15]) but also maintains the mass local conservation law, and this is very important in fluid numerical computations. The satisfactory numerical simulation results on different test problems were obtained in [15–17]. The MCVE methods have been used to solve quasi-linear second order elliptic equations [18], parabolic equations [19, 20], and so on.
This article proposes an EMCVE scheme to solve the 2D linear integro-differential equation of Sobolev type. We introduce the variables \(\boldsymbol{\sigma}(x,t)=-\nabla u(x,t)\) and \(\boldsymbol{\lambda}(x,t)=-(a(x)\nabla u(x,t)+b(x)\nabla u_{t}+\int _{0}^{t}k(x,t,\tau)\nabla u(x,\tau)\,\mathrm{d}\tau)\) and write problem (1) as the system of first order PDEs
The EMCVE scheme is obtained by integrating these equations on local covolume directly and using the Green’s formula when proper. And then, the local conservation law with the discrete solution holds. This method skillfully combines finite volume element methods [21, 22] with expanded mixed finite element methods [23, 24], can use the advantage of finite volume element methods to calculate more different physical quantities simultaneously. Rui and Lu [25] applied the EMCVE method to solve the elliptic problem on rectangular grids in the rectangular area. In this article, we propose a semi-discrete and backward Euler fully-discrete EMCVE scheme based on triangular grids and obtain the optimal order error estimates by introducing a Volterra-type generalized EMCVE projection. Moreover, we give numerical results for a model equation to verify the feasibility and effectiveness of the scheme.
The expanded mixed weak formulation of (3) is to solve \((u,\boldsymbol{\sigma},\boldsymbol{\lambda})\in L^{2}(\Omega)\times \mathbf{H}(\operatorname{div},\Omega)\times\mathbf{H}(\operatorname{div},\Omega)\) satisfying
where \(\mathbf{H}(\operatorname{div},\Omega)=\{\mathbf{z}\in(L^{2}(\Omega))^{2}: \operatorname{div}\mathbf{z} \in L^{2}(\Omega)\}\).
We also use the general notations and definitions of the Sobolev spaces as in [26]. Let \((\cdot,\cdot)\) be the inner product in \(L^{2}(\Omega)\) and \((L^{2}(\Omega))^{2}\), that is, \((\psi,\phi)=\int_{\Omega}\psi\phi\,\mathrm{d}x\) (if \(\psi,\phi \in L^{2}(\Omega)\)) and \((\mathbf{z},\mathbf{w})=\int_{\Omega}\mathbf{z}\cdot\mathbf{w}\,\mathrm{d}x\) (if \(\mathbf{z},\mathbf{w}\in(L^{2}(\Omega))^{2}\)), and either \(\|\cdot\|_{L^{2}(\Omega)}\) or \(\|\cdot\|_{(L^{2}(\Omega))^{2}}\) is denoted as \(\|\cdot\|\). We also use the norm \(\|\mathbf{z}\|_{\mathbf{H}(\operatorname{div},\Omega)}=(\| \mathbf{z}\|^{2}+\|\operatorname{div}\mathbf{z}\|^{2})^{\frac{1}{2}}\) of the space \(\mathbf{H}(\operatorname{div},\Omega)\). Throughout this paper, the constant \(C>0\) does not depend on the spatial and time mesh parameters h and Δt.
2 Expanded mixed covolume element formulation
In order to describe the EMCVE scheme for system (1), we construct the partition \(\mathcal{T}_{h}\) of the domain Ω. As in [15], let \(\mathcal{T}_{h}=\{K_{B}\}\) be a quasi-uniform triangulation partition, where \(K_{B}\) is the triangle with barycenter point B, and \(h=\max\{h_{K_{B}}\}\), \(h_{K_{B}}\) stands for the diameter of triangle \(K_{B}\). We define the nodes to be the midpoints on the edges of every triangular element, where \(P_{1}, P_{2}, \ldots, P_{N_{\tau}}\) stand for interior nodes, and \(P_{N_{\tau}+1}, \ldots, P_{N}\) stand for boundary nodes.
We use the \(\mathit{RT}_{0}\) space as the trial function space \(\mathbf {H}_{h}\) for variables σ and λ, where
and use \(L_{h}\) as a trial space for variable u, where
Now the dual partition \(\mathcal{T}_{h}^{*}\) is constructed by a union of interior quadrilaterals and border triangle. Referring now to Figure 1, and the quadrilateral \(A_{1}B_{1}A_{3}B_{2}\) is the dual element \(K_{P_{3}}^{*}\) with interior node \(P_{3}\), which contains two elements \(K_{L}\) (the triangle \(\triangle A_{1}B_{1}A_{3}\)) and \(K_{R}\) (the triangle \(\triangle A_{1}A_{3}B_{2}\)); the triangle \(\triangle A_{4}A_{5}B_{3}\) is the dual element \(K_{P_{6}}^{*}\) with boundary node \(P_{6}\), which contains one element \(K_{I}\) (the triangle \(\triangle A_{5}B_{3}A_{4}\)).
Integrate (3) on these primal and dual elements to obtain
Similar to [15, 27], we define a transfer operator \(\gamma_{h}: \mathbf{H}_{h}\rightarrow(L^{2}(\Omega))^{2}\) by
where \(\chi_{ K}\) means the characteristic function of a set K. Then we choose the range of \(\gamma_{h}\) as the test function space \(\mathbf{Y}_{h}\). By using the transfer operator \(\gamma_{h}\), we can rewrite equations (a) and (b) in (7) as
Applying Green’s integral formula, we have
for \(\forall\mathbf{w}_{h}\in\mathbf{H}_{h}\), where n stands for the unit out-normal direction.
By calculation, it is easy to get the equality \(b(\gamma_{h}\mathbf{w}_{h},v_{h})=-(\operatorname{div}\mathbf{w}_{h},v_{h})\), \(\forall\mathbf{w}_{h}\in\mathbf{H}_{h}\), \(\forall v_{h}\in L_{h}\). Then we can obtain the semi-discrete EMCVE scheme to find \((u_{h},{\boldsymbol{\sigma}}_{h},{\boldsymbol{\lambda}}_{h})\in L_{h} \times\mathbf{H}_{h}\times\mathbf{H}_{h}\) such that
and the initial values \(u_{h}(0)\) and \(\boldsymbol{\sigma}_{h}(0)\) will be defined in Theorems 4.1 and 4.2.
3 Some lemmas
For \(\forall\mathbf{z}_{h}=(z_{h}^{1},z_{h}^{2})\in\mathbf{H}_{h}\), the discrete norms are defined as follows:
Lemma 3.1
[15]
The operator \(\gamma_{h}\) is bounded
and satisfies
Lemma 3.2
[20]
The following symmetry relation
holds, and there is a constant \(\mu_{0}>0\) independent of h such that
For \(\forall x\in K_{B}\), we define \(\bar{a}(x)=a(B)\), \(\bar{b}(x)=b(B)\), \(\bar{k}(x,t,\tau)=k(B,t,\tau)\).
Lemma 3.3
[20]
The following symmetry relation
holds, and there are constants \(\mu_{1}>0\), \(\mu_{2}>0\) independent of h such that
Lemma 3.4
[20]
The following estimates hold:
The Raviart-Thomas projection \(\Pi_{h}: \mathbf{H}(\operatorname{div},\Omega )\rightarrow\mathbf{H}_{h}\) is defined in [29] such that
and the \(L^{2}\) projection \(R_{h}: L^{2}(\Omega)\rightarrow L_{h}\) is defined by
Then the properties of \(\Pi_{h}\) and \(R_{h}\) are known from [28–30]
where \(\mathbf{H}^{1}(\operatorname{div},\Omega)=\{\mathbf{w}\in(L^{2}(\Omega))^{2}: \operatorname{div}\mathbf{w}\in H^{1}(\Omega)\}\).
Lemma 3.5
[20]
The following estimate holds:
Lemma 3.6
The following symmetry relation
holds, and we have
Proof
Let \(K=\triangle A_{1}A_{2}A_{3}\), \(\triangle_{j}=\triangle A_{j+1}BA_{j+2}\) (\(j=1,2,3\)), and \(A_{4}=A_{1}\) (see Figure 1). Denote \(\mathbf{w}_{h}=(w_{h}^{1},w_{h}^{2})\) and \(\mathbf{z}_{h}=(z_{h}^{1},z_{h}^{2})\), then
By applying the numerical quadrature formula, we get
Similarly, we get \(\sum_{j=1}^{3}M_{j2}=0\). Summing over all j and K, then we complete the proof of (15).
To prove (16), using (15), we have
Noting that \(k(x,t,\tau)\) is Lipschitz continuous with variable x, we get the desired conclusion. □
Lemma 3.7
For \(\forall\mathbf{z}_{h}, \mathbf{w}_{h}\in\mathbf{H}_{h}, \forall\mathbf {z}\in(H^{1}(\Omega))^{2}\), we have
Proof
To prove (17), we obtain
By using Lemmas 3.1 and 3.6, we complete the proof of (17).
Next we prove (18). Using (12), we have
This ends the proof of Lemma 3.7. □
Now, we introduce the Volterra-type generalized EMCVE projection. Define \((\tilde{u}_{h},\tilde{\boldsymbol{\sigma}}_{h}, \tilde{\boldsymbol {\lambda}}_{h}): [0,T]\rightarrow L_{h}\times\mathbf{H}_{h}\times\mathbf {H}_{h}\) such that
Theorem 3.1
Suppose \((\tilde{u}_{h},\tilde{\boldsymbol{\sigma}}_{h},\tilde{\boldsymbol {\lambda}}_{h})\) satisfies (19a)-(19c), then there is a constant \(C>0\) independent of h and t such that
Proof
Noting that \(\tilde{\boldsymbol{\lambda}}_{h}=\Pi_{h}{\boldsymbol{\lambda }}\), we have estimates (20) and (21).
Splitting \(\boldsymbol{\sigma}-\tilde{\boldsymbol{\sigma}}_{h}=\boldsymbol {\sigma}-\Pi_{h}\boldsymbol{\sigma}+\Pi_{h}\boldsymbol{\sigma}-\tilde {\boldsymbol{\sigma}}_{h}\) in (19c) yields
Choose \(\mathbf{z}_{h}=\Pi_{h}\boldsymbol{\sigma}-\tilde{\boldsymbol{\sigma }}_{h}\) in (24) and use the Cauchy-Schwarz inequality to get
Using (12) and (20), applying Gronwall’s inequality, we obtain estimate (22).
Noting that \(\operatorname{div}(\mathbf{H}_{h})=L_{h}\), we have \((\operatorname{div}\mathbf{w}_{h},u-R_{h} u)=0,\forall\mathbf{w}_{h}\in\mathbf{H}_{h}\), and rewrite (19b) as
Next we introduce an auxiliary elliptic problem. Given \(\varphi\in L^{2}(\Omega)\), let ψ satisfy the following elliptic problem:
And we have the following elliptic regularity result:
Using the projection \(\Pi_{h}\) and \(R_{h}\), and (26)-(28), we have
Noting that
using (12), (28) and Lemma 3.5, we have
and
Using (30)-(32) in (29) yields
Apply the triangle inequality with (14) and (33) to obtain (23). □
Differentiating (19a)-(19c) with respect to time variable t, we can also obtain the following projection estimates.
Theorem 3.2
Suppose \((\tilde{u}_{h},\tilde{\boldsymbol{\sigma}}_{h},\tilde{\boldsymbol {\lambda}}_{h})\) satisfies (19a)-(19c), then there is a constant \(C>0\) independent of h and t such that
4 The error estimates of semi-discrete expanded mixed covolume element formulation
In this section, we first discuss the existence and uniqueness of solution for the semi-discrete EMCVE scheme (11).
Theorem 4.1
Set \(u_{h}(0)=\tilde{u}_{h}(0)\), \(\boldsymbol{\sigma}_{h}(0)=\tilde {\boldsymbol{\sigma}}_{h}(0)\), then there is a unique solution for system (11).
Proof
Let \(\{\chi_{j}\}_{j=1}^{N_{1}}\) and \(\{\varphi_{j}\}_{j=1}^{N}\) be the basis of \(L_{h}\) and \(\mathbf{H}_{h}\), respectively. Then \(\boldsymbol{\sigma}_{h},\boldsymbol{\lambda}_{h},\tilde{\boldsymbol {\sigma}}_{h}(0)\in\mathbf{H}_{h}\), \(\tilde{u}_{h}(0),u_{h}\in L_{h}\) can be expressed as
Substitute the above expressions into system (11) and set \(\mathbf{w}_{h}, \mathbf{z}_{h}=\phi_{i}\) (\(i=1,2,\ldots,N\)), \(v_{h}=\chi_{i}\) (\(i=1,2,\ldots,N_{1}\)), then we write system (11) as the following matrix form:
where
It is easy to see that A and C are symmetric positive definite matrixes, and \(A_{1}\) and \(A_{2}\) are invertible matrixes. We rewrite equation (c) in (37) as
where \(G=A^{-1}BC^{-1}B^{T}A^{-1}\).
Using quadratic form theory, we can know that \((A_{2}+G^{-1})\) is an invertible matrix, and problem (38) has a unique solution by the theory of differential equations. Thus, systems (37) and (11) have a unique solution. □
Now we write the errors as
where \((\tilde{u}_{h},\tilde{\boldsymbol{\sigma}}_{h},\tilde{\boldsymbol {\lambda}}_{h})\) is the Volterra-type generalized EMCVE projection of \((u,\boldsymbol{\sigma},\boldsymbol {\lambda})\). Using (11) and (4), we have the error equations
Theorem 4.2
Let \((u,\boldsymbol{\sigma},\boldsymbol{\lambda})\), \((u_{h},\boldsymbol {\sigma}_{h},\boldsymbol{\lambda}_{h})\) be the solutions of (4) and (11), respectively, and set that \(u_{h}(0)=\tilde{u}_{h}(0)\), \(\boldsymbol{\sigma}_{h}(0)=\tilde {\boldsymbol{\sigma}}_{h}(0)\). Then there is a constant \(C>0\) independent of h and t such that
where
Proof
Differentiating (39a) with respect to variable t, we have
Setting \(v_{h}=\phi_{t}\) in (39c), \(\mathbf{w}_{h}=\zeta\) in (45), and \(\mathbf{z}_{h}=\xi_{t}\) in (39b), we have
Noting that
and \((\bar{a}\xi,\gamma_{h}\xi_{t})=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}(\bar{a}\xi ,\gamma_{h}\xi)\), using Lemmas 3.3-3.5 and Lemma 3.7, we get
Integrating the above inequality from 0 to t, we get
Noting that \(\xi(0)=0\), \((\bar{a}\xi,\gamma_{h}\xi)\geq\mu_{1}\|\xi\|^{2}\), applying Gronwall’s inequality, we have
Now, we set \(v_{h}=\phi\) in (39c), \(\mathbf{w}_{h}=\zeta\) in (39a), and \(\mathbf{z}_{h}=\xi\) in (39b) to obtain
Using Lemmas 3.3, 3.4 and 3.7, we get
Integrating (50) from 0 to t yields
Noting that \(\phi(0)=0\), and substituting (48) into (51), we get that
Using Gronwall’s inequality yields
Next, using Lemmas 3.3 and 3.4 in (46), we get
Substituting (48) and (52) into (53) yields
To estimate \(\|\boldsymbol{\lambda}-\boldsymbol{\lambda}_{h}\|\) and \(\| \boldsymbol{\lambda}-\boldsymbol{\lambda}_{h}\|_{\mathbf{H}(\operatorname{div},\Omega)}\), we choose \(\mathbf{z}_{h}=\zeta\) in (39b) to see that
Using Lemmas 3.2 and 3.4, we get
Substituting (48), (52) and (54) into (55), we have that
Choosing \(v_{h}=\operatorname{div}\zeta\) in (39c) yields
And we have
Thus, combine (48), (52), (54) and (57), apply the triangle inequality to complete the proof. □
5 The fully-discrete expanded mixed covolume element formulation
Let Δt be the time step length, and \(t_{n}=n\Delta t\) (\(n=0,1,2,\ldots,M\)) for some positive integer M. Define \(\varphi^{n}=\varphi(t_{n})\) and \(\partial_{t}\varphi^{n}=\frac{\varphi^{n}-\varphi^{n-1}}{\Delta t}\) for a function φ. To approximate the integral term, we select the left rectangle quadrature formula
and the quadrature error \(\varepsilon^{n}(\varphi)=\int_{0}^{t_{n}}\varphi(s)\, \mathrm{d}s-\Delta t\sum_{j=0}^{n-1}\varphi(t_{j})\) satisfies
Now, we define the backward Euler fully-discrete scheme: find \((u_{h}^{n},\boldsymbol{\sigma}_{h}^{n},\boldsymbol{\lambda}_{h}^{n})\in L_{h}\times \mathbf{H}_{h}\times\mathbf{H}_{h}\), \(n=0,1,\ldots,N\), such that
where \(k^{n,j}=k(x,t_{n},t_{j})\).
The above calculation of \(\{u_{h}^{n},\boldsymbol{\sigma}_{h}^{n},\boldsymbol {\lambda}_{h}^{n}\}\) (\(n=1,2,\ldots,M\)) only involves the inverse operation of stiffness matrix with the spaces \(\mathbf{H}_{h}\) and \(L_{h}\). \(u_{h}^{0}\) and \(\boldsymbol{\sigma}_{h}^{0}\) are calculated by solving (58a) and (58b). The calculation proceeds by solving (58c), (58d) and (58e) equations for \(\{\boldsymbol{\sigma}_{h}^{n},\boldsymbol{\lambda}_{h}^{n},u_{h}^{n}\}\) with using already calculated \(\{\boldsymbol{\sigma}_{h}^{n-1},u_{h}^{n-1}\}\). It is easy to get that there is a unique solution for the fully-discrete scheme (58a)-(58e).
We now rewrite the errors as
where \((\tilde{u}_{h},\tilde{\boldsymbol{\sigma}}_{h},\tilde{\boldsymbol {\lambda}}_{h})\) is the Volterra-type generalized EMCVE projection of \((u,\boldsymbol{\sigma},\boldsymbol {\lambda})\).
Using (19a)-(19c), we obtain the following error equations:
where
Theorem 5.1
Let \((u_{h}^{n},\boldsymbol{\sigma}_{h}^{n},\boldsymbol{\lambda}_{h}^{n})\) be the solution of scheme (58a)-(58e), and suppose that the solution \((u,\boldsymbol{\sigma},\boldsymbol {\lambda})\) of system (4) has properties that \(\boldsymbol{\sigma},\boldsymbol{\lambda}\in L^{\infty}((H^{1}(\Omega))^{2})\), \(\boldsymbol{\sigma}_{t},\boldsymbol{\lambda}_{t}\in L^{2}((H^{1}(\Omega))^{2})\), \(u\in L^{\infty}(H^{1}(\Omega))\), \(u_{t}\in L^{2}(H^{1}(\Omega))\), \(\boldsymbol{\sigma}_{t},\boldsymbol{\sigma}_{tt}\in L^{2}((L^{2}(\Omega))^{2})\), \(u_{tt}\in L^{2}(L^{2}(\Omega))\), then there is a constant \(C>0\) independent of h and Δt such that
Proof
Using (19a)-(19c), we rewrite (59b) as
Then using (59c) and (60), we have
Choosing \(v_{h}=\partial_{t}\phi^{n}\) in (59e), \(\mathbf{w}_{h}=\zeta^{n}\) in (61), and \(\mathbf{z}_{h}=\partial_{t}\xi^{n}\) in (59d), we have
Noting the fact that \((a\xi^{n},\gamma_{h}\partial_{t}\xi^{n})=(\bar{a}\xi^{n},\gamma_{h}\partial_{t}\xi^{n}) +[(a\xi^{n},\gamma_{h}\partial_{t}\xi^{n})-(\bar{a}\xi^{n},\gamma_{h}\partial_{t}\xi^{n})]\), and \((\bar{a}\xi^{n},\gamma_{h}\partial_{t}\xi^{n})\geq\frac{1}{2\Delta t}[(\bar {a}\xi^{n},\gamma_{h}\xi^{n})-(\bar{a}\xi^{n-1},\gamma_{h}\xi^{n-1})]\), we have
Summing from \(n=1\) to m and multiplying (63) by \(2\Delta t\), we have
Note that \((\bar{a}\xi^{m},\gamma_{h}\xi^{m})\geq\mu_{1}\|\xi^{m}\|^{2}\), choose Δt in (64) to satisfy \(C\Delta t<\frac{\mu_{1}}{2}\), and use Gronwall’s inequality to get
Now, by Lemma 3.3, it follows from (62) that
Substituting (65) into (66), we have that
To estimate \(\|\boldsymbol{\lambda}(t_{n})-\mathbf{Z}^{n}\|+\|\boldsymbol {\lambda}(t_{n})-\mathbf{Z}^{n}\|_{\mathbf{H}(\operatorname{div},\Omega)}\), we set \(\mathbf{z}_{h}=\zeta^{n}\) in (59d) and get that
Substituting (65) and (67) into (68), we have
Choose \(v_{h}=\operatorname{div}\zeta^{n}\) in (59e) to obtain
Substituting (67) into (70), we get
Finally, we estimate \(\|u(t_{m})-U^{m}\|\). Setting \(v_{h}=\phi^{n}\) in (59e), \(\mathbf{w}_{h}=\zeta^{n}\) in (59c), and \(\mathbf{z}_{h}=\xi^{n}\) in (59d), we get
Noting that \((c\partial_{t}\phi^{n},\phi^{n})\geq\frac{1}{2\Delta t} (\|c^{\frac{1}{2}}\phi ^{n}\|^{2}-\|c^{\frac{1}{2}}\phi^{n-1}\|^{2})\), and using Lemmas 3.3 and 3.4, we obtain
Summing from \(n=1\) to m, multiplying (73) by \(2\Delta t\), and using (67), we get
Choose Δt in (74) to satisfy \(C\Delta t<\frac{c_{0}}{2}\), and use Gronwall’s inequality to get
Now, we note that
and we have
Further, using (59a) and (59b), we get
Finally, apply the triangle inequality to obtain the error estimates. □
6 Numerical example
For confirming the above theoretical analysis, we give a numerical example and consider the spatial and temporal domain \(\Omega=(0,1)\times(0,1)\), \(J=(0,1]\), the coefficients \(a(x)=1+2x_{1}^{2}+x_{2}^{2}\), \(b(x)=1+x_{1}^{2}+2x_{2}^{2}\), \(c(x)=1\), \(k(x,t,\tau)=(1+x_{1}^{2}+x_{2}^{2}+t^{2})\tau\), and the initial function
The exact solution is
and the source function \(f(x,t)\), auxiliary variables \(\boldsymbol {\sigma}(x,t)=-\nabla u(x,t)\) and \(\boldsymbol{\lambda}(x,t)= -(a(x)\nabla u(x,t)+b(x)\nabla u_{t}(x,t)+\int _{0}^{t}k(x,t,\tau)\nabla u(x,\tau)\,\mathrm{d}\tau)\) are determined by the above functions.
We use the fifth order Gauss quadrature rule to calculate the errors \(\|u-u_{h}\|_{L^{\infty}(L^{2}(\Omega))}\), \(\|\boldsymbol{\sigma}-\boldsymbol{\sigma}_{h}\|_{L^{\infty}((L^{2}(\Omega))^{2})}\), \(\|\boldsymbol{\lambda}-\boldsymbol{\lambda}\|_{L^{\infty}(H(\operatorname{div},\Omega))}\) and \(\|\boldsymbol{\lambda}-\boldsymbol{\lambda}_{h}\|_{L^{\infty}((L^{2}(\Omega))^{2})}\). The simulation results for the backward Euler fully-discrete scheme are given in Table 1 by using \(\mathit{RT}_{0}\) space with different mesh sizes \(h=\sqrt{2}\Delta t=\frac{\sqrt{2}}{8},\frac{\sqrt{2}}{16},\frac{\sqrt {2}}{32},\frac{\sqrt{2}}{64}\). Based on the error results and convergence rates, we can verify the theoretical analysis.
The graphs of exact solutions for u, σ and λ at \(t=1\) are drawn on Figures 2, 3 and 4, respectively. The graphs of the corresponding discrete solutions for \(u_{h}^{n}\), \(\boldsymbol{\sigma}_{h}^{n}\) and \(\boldsymbol{\lambda}_{h}^{n}\) with the mesh \(h=\frac{\sqrt{2}}{32}\) and \(\Delta t=\frac{1}{32}\) are drawn on Figures 5, 6 and 7, respectively. The numerical results and figures show that the EMCVE scheme is feasible and efficient.
7 Conclusions
We present the EMCVE method for the 2D linear integro-differential equation of Sobolev type. We introduce the transfer operator \(\gamma_{h}\) and construct the semi-discrete, backward Euler fully-discrete EMCVE schemes. We obtain the optimal order error estimates for the scalar unknown u (in \(L^{2}(\Omega)\)-norm), gradient σ (in \((L^{2}(\Omega))^{2}\)-norm) and flux λ (in \((L^{2}(\Omega))^{2}\)-norm and \(\mathbf {H}(\operatorname{div},\Omega)\)-norm) by introducing the Volterra-type generalized EMCVE projection. Moreover, we give the numerical experiment to verify the theoretical analysis.
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Acknowledgements
This work was supported by the National Natural Science Fund of China (11661058, 11361035, 11501311), the Natural Science Fund of Inner Mongolia Autonomous Region (2016BS0105, 2016MS0102, 2017MS0107), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZY14013), the Program of Higher-Level Talents of Inner Mongolia University (30105-135127).
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Fang, Z., Li, H., Liu, Y. et al. An expanded mixed covolume element method for integro-differential equation of Sobolev type on triangular grids. Adv Differ Equ 2017, 143 (2017). https://doi.org/10.1186/s13662-017-1201-7
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DOI: https://doi.org/10.1186/s13662-017-1201-7