Volume 2012, Issue 1 391372

Research Article

Open Access

A New Positive Definite Expanded Mixed Finite Element Method for Parabolic Integrodifferential Equations

Yang Liu,

Yang Liu

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China imu.edu.cn

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Hong Li,

Corresponding Author

Hong Li

[email protected]

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China imu.edu.cn

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Jinfeng Wang,

Jinfeng Wang

School of Statistics and Mathematics, Inner Mongolia Finance and Economics College, Hohhot 010070, China

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Wei Gao,

Wei Gao

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China imu.edu.cn

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Yang Liu,

Yang Liu

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China imu.edu.cn

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Hong Li,

Corresponding Author

Hong Li

[email protected]

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China imu.edu.cn

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Jinfeng Wang,

Jinfeng Wang

School of Statistics and Mathematics, Inner Mongolia Finance and Economics College, Hohhot 010070, China

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Wei Gao,

Wei Gao

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China imu.edu.cn

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First published: 28 June 2012

https://doi.org/10.1155/2012/391372

Citations: 6

Academic Editor: Jinyun Yuan

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Abstract

A new positive definite expanded mixed finite element method is proposed for parabolic partial integrodifferential equations. Compared to expanded mixed scheme, the new expanded mixed element system is symmetric positive definite and both the gradient equation and the flux equation are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete scheme are proved and error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are provided to confirm our theoretical analysis.

1. Introduction

In this paper, we consider the following initial-boundary value problem of parabolic partial integrodifferential equations:

()

where Ω is a bounded convex polygonal domain in ℝ^d, (d = 1,2, 3) with a smooth boundary ∂Ω, J = (0, T] is the time interval with 0 < T < ∞. The coefficients a = a(x, t), b = b(x, t) are two functions, which satisfy the property that there exist some positive constants a_min, a_max, b_min, and b_max such that 0 < a_min ≤ a(x, t) ≤ a_max and 0 < b_min ≤ b(x, t) ≤ b_max.

Parabolic integrodifferential equations are a class of very important evolution equations which describe many physical phenomena such as heat conduction in material with memory, compression of viscoelastic media, and nuclear reactor dynamics. In recent years, a lot of researchers have studied the numerical methods for parabolic integrodifferential equations, such as finite element methods [1–5], mixed finite element methods [6–9], and finite volume element method [10] and so forth.

In 1994, Chen [11, 12] proposed a new mixed method, which is called a expanded mixed finite element method and proved some mathematical theories for second-order linear elliptic equation. Compared to standard mixed element methods, the expanded mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, its gradient, and its flux (the tensor coefficient times the gradient). In 1997, Arbogast et al. [13] derived and exploited a connection between the expanded mixed method and a certain cell-centered finite difference method. And Chen proved some mathematical theories for second-order quasilinear elliptic equation [14] and fourth-order elliptic problems [15]. With the development of the expanded mixed finite element method, the method was applied to many evolution equations. In [16], some error estimates of the expanded mixed element for a kind of parabolic equation were given. Woodward and Dawson [17] studied the expanded mixed finite element method for nonlinear parabolic equation. Wu and Chen et al. [18–22] studied the two-grid methods for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Song and Yuan [23] proposed the expanded upwind-mixed multistep method for the miscible displacement problem in three dimensions. Guo and Chen [24] developed and analysed an expanded characteristic-mixed finite element method for a convection-dominated transport problem. In 2010, Chen and Wang [25] proposed an H¹-Galerkin expanded mixed method for a nonlinear parabolic equation in porous medium flow, and Liu and Li [26] studied the H¹-Galerkin expanded mixed method for pseudo-hyperbolic equation. Liu [27], studied the H¹-Galerkin expanded mixed method for RLW-Burgers equation and proved semidiscrete and fully discrete optimal error estimates. Jiang and Li [28] studied an expanded mixed semidiscrete scheme for the problem of purely longitudinal motion of a homogeneous bar. In [29, 30], the expanded mixed covolume method was studied for the linear integrodifferential equation of parabolic type and elliptic problems, respectively. In [31], a posteriori error estimator for expanded mixed hybrid methods was studied and analysed.

In 2001, Yang [32] proposed a new mixed finite element method called the splitting positive definite mixed finite element procedure to treat the pressure equation of parabolic type in a nonlinear parabolic system describing a model for compressible flow displacement in a porous medium. Compared to standard mixed methods whose numerical solutions have been quite difficult because of losing positive definite properties, the proposed one does not lead to some saddle point problems. From then on, the method was applied to the hyperbolic equations [33] and pseudo-hyperbolic equations [34].

In this paper, our purpose is to propose and analyse a new expanded mixed method based on the positive definite system [32–34] for parabolic integrodifferential equations. Compared to expanded mixed methods, the proposed mixed element system is symmetric positive definite and avoids some saddle point problems. What is more, both the gradient equation and the flux equation are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete scheme are proved and error estimates are derived for both semidiscrete and fully discrete schemes.

The layout of the paper is as follows. In Section 2, the positive definite expanded mixed weak formulation and semidiscrete mixed scheme are formulated, and the proof of the existence and uniqueness of the discrete solutions is given. Error estimates are derived for both semidiscrete and fully discrete schemes for problems, respectively, in Sections 3 and 4. In Section 5, some numerical results are provided to illustrate the efficiency of our method. Finally in Section 6, we will give some concluding remarks about the positive definite expanded mixed finite element method.

Throughout this paper, C will denote a generic positive constant which does not depend on the spatial mesh parameters h_u, h_σ and time-discretization parameter δ and may be different at their occurrences. Usual definitions, notations, and norms of the Sobolev spaces as in [35–37] are used. We denote the natural inner product in L²(Ω) or [L²(Ω)] ^d by (·, ·) with norm or and introduce the function space W = H(div ; Ω) = {ω ∈ [L²(Ω)] ^d; ∇·ω ∈ L²(Ω)}.

2. A New Expanded Mixed Variational Formulation

Introducing the auxiliary variables:

()

Then we obtain the equivalent system of parabolic partial integrodifferential equations for the problem (1.1):

()

with the initial values λ(x, 0) = ∇u₀(x), σ(x, 0) = a∇u₀(x), and u(x, 0) = u₀(x).

Then, the following expanded mixed weak formulation of (2.2) can be given by:

()

From (2.3)(b) we derive

()

Taking v = ∇·w in (2.3)(a) for w ∈ W and then substituting it into (2.4), we derive a new equivalent expanded mixed weak formulation of the system (2.3):

()

Let and be two families of quasi-regular partitions of the domain Ω, which may be the same one or not, such that the elements in the partitions have the diameters bounded by h_u and h_σ, respectively. Let and be finite element spaces defined on the partitions and .

Now the semidiscrete positive definite expanded mixed finite element method for (2.5) consists in determining

such that

()

with given an initial approximation

Remark 2.1. Compared to expanded mixed weak formulation (2.3), the new expanded mixed element system (2.6) is symmetric positive definite, that is to say the gradient function and the flux function system (2.6)(a,b) is symmetric positive definite. And both the gradient equation and the flux equation are separated from its scalar unknown equation (2.6)(c).

Theorem 2.2. There exists a unique discrete solution to the system (2.6).

Proof. Let and be bases of and , respectively. Then, we have the following expressions:

()

Substituting these expressions into (2.6) and choosing v_h = ψ_m, w_h = z_h = φ_l, then the problems (2.6) can be written in vector matrix form as: find {U(t), Λ(t), Σ(t)} such that, for all t ∈ (0, T]

()

where

()

It is easy to see that both A and D are symmetric positive definite. From (2.8), the problems can be written as follows:

()

Thus, by the theory of differential equations [38, 39], (2.10) has a unique solution, and equivalently (2.6) has a unique solution.

Remark 2.3. It is easy to see that the coefficient matrixes A, B, C, and H of system (2.8) are symmetric positive definite. In view of this, the new expanded mixed element system (2.6) is symmetric positive definite.

3. Semidiscrete Error Estimates

Let

and

be finite dimensional subspaces of L²(Ω) and W, respectively, with the inverse property (see [36]) and the following approximation properties (see [40–44]): for 0 ≤ p ≤ +∞ and r, r^*, k positive integers

()

where r^* = r + 1 for the Brezzi-Douglas-Fortin-Marini spaces [43] and the Raviart-Thomas spaces [42] and r^* = r for the Brezzi-Douglas-Marini spaces [40, 43].

For our subsequent error analysis, we introduce two operators. It is well known that, in any one of the classical mixed finite element spaces, there exists an operator R_h from H(div ; Ω) onto

, see [40–44], such that, for 1 ≤ p ≤ +∞,

()

We also define the L²-project operator P_h from L²(Ω) onto

such that

()

Using the definitions of the operators R_h and P_h, we can easily obtain the following lemma.

Lemma 3.1. Assume that the solution of system (2.5) has the regular properties that u_t ∈ L²(H^k+1(Ω)), λ_t, λ_tt, σ_t ∈ L²(H^r+1(Ω)), then one has the following estimates:

()

Let

()

Subtracting (2.6) from (2.5) and using projections (3.2) and (3.3), one obtains

()

Theorem 3.2. Assume that the approximate properties (3.1) hold, and the solution of system (2.5) has regular properties that u_t, u_tt ∈ L²(H^k+1(Ω)), λ_t, λ_tt, σ_t, σ_tt ∈ L²(H^r+1(Ω)). Then one has the error estimates

()

Proof . Choose w_h = θ in (3.6)(a) and z_h = −ξ_t in (3.6)(b), and add the two equations to obtain

()

Integrate with respect to time from 0 to t and apply the Cauchy-Schwarz′s inequality and the Young′s inequality to obtain

()

Choose z_h = θ in (3.6)(b) to get

()

Combining (3.9) and (3.10), we obtain

()

Using the fact that ξ(0) = 0 and Gronwall′s lemma, we obtain

()

Differentiating (3.6)(b) and taking z_h = ξ_t, we obtain

()

Choosing w_h = θ_t in (3.6)(a), we obtain

()

Add (3.13) and (3.14) to obtain

()

Integrate (3.15) with respect to time from 0 to t to obtain

()

Substitute (3.12) into (3.16) to have

()

Choosing v_h = ς in (3.6)(b) and applying Cauchy-Schwarz′s inequality, we obtain

()

Using Lemma 3.1, (3.17), and Gronwall′s lemma, we get

()

Using (3.12), (3.17), (3.19), (3.2), (3.3), and Lemma 3.1, we apply the triangle inequality to complete the proof.

4. Fully Discrete Error Estimates

In this section, we get the error estimates of fully discrete schemes. For the backward Euler procedure, let 0 = t₀ < t₁ < t₂ < ⋯<t_M = T be a given partition of the time interval [0, T] with step length δ = T/M, for some positive integer M. For a smooth function ϕ on [0, T], define ϕⁿ = ϕ(t_n) and ∂_tϕⁿ = (ϕⁿ − ϕⁿ⁻¹)/δ. For approximating the integrals, we use the composite left rectangle rule

()

Note that ϕ ∈ C¹ [0, T], the quadrature error satisfies

()

Equation (2.5) has the following equivalent formulation:

()

where

()

Now we can formulate a fully discrete scheme based on (4.3).

Fully discrete scheme: find

such that

()

with given an initial approximation

For fully discrete error estimates, we now split the errors

()

From (4.3) to (4.5), we then obtain

()

Lemma 4.1. Assume that the solution of system (2.5) has regular properties that u_t ∈ L²(H^k+1(Ω)),λ_t, σ_t ∈ L²(H^r+1(Ω)). Then one has the estimates

()

Theorem 4.2. Assume that ∂²u/∂t², ∂u/∂t ∈ L²(H^k+1(Ω)),∂λ/∂t, ∂²λ/∂t², ∂σ/∂t, ∂²σ/∂t² ∈ L²(H^r+1(Ω)), u ∈ L^∞(H^k+1(Ω)), and λ, σ ∈ L^∞(H^r+1(Ω)), then there exists a constant C such that

()

Proof . Set w_h = θⁿ in (4.7)(a) and z_h = −∂_tξⁿ in (4.7)(b) and add the two equations to obtain

()

Note that

()

So, we get

()

Note that

()

Substitute (4.12)-(4.13) into (4.10) to get

()

Summing from 1 to n, we find that

()

Choose z_h = θⁿ in (3.6)(b) to get

()

Substitute (4.16) into (4.15) and note that ξ⁰ = 0 to get

()

Using Gronwall′s lemma, we obtain

()

Note that

()

Therefore, substituting the above estimates into (4.18) and choosing δ₀ in such a way that for 0 < δ ≤ δ₀, (1 − Cδ) > 0, we obtain

()

By (4.7)(b), we obtain

()

Set z_h = ∂_tξⁿ in (4.21) to obtain

()

Set w_h = ∂_tθⁿ in (4.7)(a) to obtain

()

Substitute (4.22) into (4.23) to get

()

Take z_h = ∂_tθⁿ in (4.21) to obtain

()

Add (4.24) and (4.25) to get

()

Apply Cauchy-Schwarz′s inequality and Young′s inequality to obtain

()

Using (4.19) and (4.20) and summing from 1 to n, we obtain

()

Choosing v_h = ςⁿ in (4.7)(c) and applying Cauchy-Schwarz′s inequality, Young′s inequality, and (4.28), we have

()

Summing from 1 to n and using the Gronwall lemma, we obtain

()

Note that

()

Substituting (4.31) into (4.30) and using (3.2), (4.20), and the triangle inequality, we get

()

Combining (3.2), (3.3), (4.20), (4.28), (4.32), and Lemma 4.1, we apply the triangle inequality to complete the proof.

5. Numerical Example

In this section, we analyse some numerical results to illustrate the efficiency of the proposed method. We consider the following 2D parabolic partial integrodifferential equations with initial-boundary value condition:

()

where Ω = [0,1]×[0,1], J = (0,1],

, x = (x₁, x₂), and f(x, t) is chosen so that the exact solution for the scalar unknown function is

()

The corresponding exact gradient is

()

and its exact flux is

()

Dividing the domain Ω into the triangulations of mesh size h_u = h_σ = h uniformly, considering the piecewise constant space with index k = 0 for the scalar unknown function u and the lowest-order Raviart-Thomas triangular space [42, 45] for the gradient λ and the flux σ and using the backward Euler procedure with uniform time step length δ = 1/M, we obtain some convergence results for , , and with in Table 1. With time t = 1, , the exact solution u, λ, σ is shown in Figures 1, 3, and 5, respectively, and the corresponding numerical solution u_h, λ_h, σ_h is shown in Figures 2, 4, and 6, respectively.

Table 1. The errors and convergence order.

(h, δ)		Order
	1.4527e − 002		1.3532e − 001
	6.5250e − 003	1.15	6.9202e − 002
	3.0841e − 003	1.08	3.5363e − 002

(h, δ)		Order

	3.3512e − 001		3.8892e − 001
	1.6734e − 001	1.00	1.7291e − 001
	8.3746e − 002	1.00	8.4431e − 002

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

The exact solution u.

We can see from Table 1 that the convergence rate is order 1 which confirms the theoretical results of Theorem 4.2 for the above chosen spaces and . The numerical results in Table 1 and Figures 1–6 show that new positive definite expanded mixed scheme is efficient.

6. Concluding Remarks

In the paper, a new expanded mixed finite element method based on a positive definite system is proposed for parabolic partial integrodifferential equation. Compared to expanded mixed method and standard mixed methods, the new expanded mixed element system is symmetric positive definite and both the gradient equation and the flux equation are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete scheme are proved and error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are provided to confirm our theoretical analysis. In the near further, we will study the others evolution equations such as hyperbolic wave equation, and miscible displacement of compressible flow in porous media.

Acknowledgments

The authors would like to thank the Editor and the referees for their valuable suggestions and comments. This work is supported by the National Natural Science Fund of China (no. 11061021), the Scientific Research Projection of Higher Schools of Inner Mongolia (no. NJ10006, no. NJ10016, no. NJZZ12011), the Program of Higher-level talents of Inner Mongolia University (SPH-IMU, no. Z200901004, no. 125119), and the YSF of Inner Mongolia University (no. ND0702).

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A New Positive Definite Expanded Mixed Finite Element Method for Parabolic Integrodifferential Equations

Abstract

1. Introduction

2. A New Expanded Mixed Variational Formulation

3. Semidiscrete Error Estimates

4. Fully Discrete Error Estimates

5. Numerical Example

6. Concluding Remarks

Acknowledgments

References

Citing Literature

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A New Positive Definite Expanded Mixed Finite Element Method for Parabolic Integrodifferential Equations

Abstract

1. Introduction

2. A New Expanded Mixed Variational Formulation

3. Semidiscrete Error Estimates

4. Fully Discrete Error Estimates

5. Numerical Example

6. Concluding Remarks

Acknowledgments

References

Citing Literature

Figures

References

Related

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