A New Expanded Mixed Element Method for Convection-Dominated Sobolev Equation

We propose and analyze a new expanded mixed element method, whose gradient belongs to the simple square integrable space instead of the classical H(div; Ω) space of Chen's expanded mixed element method. We study the new expanded mixed element method for convection-dominated Sobolev equation, prove the existence and uniqueness for finite element solution, and introduce a new expanded mixed projection. We derive the optimal a priori error estimates in L 2-norm for the scalar unknown u and a priori error estimates in (L 2)2-norm for its gradient λ and its flux σ. Moreover, we obtain the optimal a priori error estimates in H 1-norm for the scalar unknown u. Finally, we obtained some numerical results to illustrate efficiency of the new method.

Sobolev equations are a class of important evolution partial differential equations and have a lot of applications in many physical problems, such as the porous theories concerned with percolation into rocks with cracks, the heat conduction problems in different mediums, and the transport problems of humidity in soil. In [1], the finite element method for nonlinear Sobolev equation with nonlinear boundary conditions was studied. In [2], a discontinuous Galerkin method for Sobolev equation was studied. In [3][4][5][6][7], some mixed finite element methods for Sobolev equations are studied and analyzed.
In 1994, Chen [8,9] developed and studied an expanded mixed 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. From then on, the expanded mixed element method has been applied to solving other partial differential equations [10]. At the same time, many researchers proposed and studied some new numerical methods based on Chen's expanded mixed method, such as expanded mixed hybrid methods [11], two-grid expanded mixed finite element method [12][13][14], expanded characteristic-mixed element method [15], expanded mixed covolume method 2 The Scientific World Journal [16,17], and expanded positive definite mixed method [18].
In 2011, we developed and analyzed a new expanded mixed finite element method [19] for elliptic equations based on the mixed schemes [20,21] which have been studied for some partial differential equations [4,[22][23][24][25]. Compared to Chen's expanded mixed method, the gradient for the new expanded mixed method belongs to the simple square integrable space instead of the classical (div; Ω) space. In this paper, we will study the new expanded mixed element method for convection-dominated Sobolev equation. We will give the proof for the existence and uniqueness of the solution for semidiscrete scheme and a new expanded mixed projection and the proof of its uniqueness. We will prove the optimal a priori error estimates in 2 -norm for the scalar unknown and a priori error estimates in ( 2 ) 2 -norm for its gradient and its flux . In particular, we obtained the optimal a priori error estimates in 1 -norm for the scalar unknown . Finally, we obtained some numerical results to confirm our theoretical analysis.
Throughout this paper, will denote a generic positive constant which is free of the space-time parameters ℎ and Δ . At the same time, we denote the natural inner product in 2 (Ω) or ( 2 (Ω)) 2 by (⋅, ⋅) with the corresponding norm ‖ ⋅ ‖. The other notations and definitions of Sobolev spaces as in [26] are used.
Remark 1. Compared to Chen's expanded mixed weak formulation (4), the gradient in the scheme (5) belongs to the simple square integrable space ( 2 (Ω)) 2 instead of the classical H(div; Ω) space. Obviously, the regularity requirements on the solution = ∇ reduced.

Theorem 2.
There exists a unique discrete solution to semidiscrete scheme (6).

Error Estimates for Semidiscrete Scheme
In order to analyze the convergence of the method, we first introduce the new expanded mixed elliptic projection associated with our equations.
Proof. Noting that mixed elliptic projection system (14) is linear, it suffices to prove the associated homogeneous system has the trivial solution.
Integrate (17) with respect to time from 0 to and use Cauchy-Schwarz inequality and Young inequality to obtain Taking w ℎ = ∇̃ℎ in (15) and using Poincaré inequality, we obtaiñℎ Substitute (19) into (18) and use Gronwall lemma to obtain 4 The Scientific World Journal From (20), we havẽℎ Combining (19), (21), and (15)(c), we get Using (21) and (22), we get In the following discussion, we will give some important lemmas based on new mixed scheme.
Using the definition of Π ℎ and ℎ , we rewrite , , and as Since estimates of , , and are known, it is enough to estimate , , and . Using Lemmas 4-6, we rewrite (14) as We discuss the following approximation properties for system (29).
For a priori error estimates, we decompose the errors as − ℎ = −̃ℎ +̃ℎ − ℎ = + ; The Scientific World Journal Using (5)- (6) and (14), we can get the error equations We will prove the error estimates for semidiscrete scheme.
Proof. Choose Adding the above three equations, and using Cauchy-Schwarz inequality and Young inequality, we have Integrate with respect to time from 0 to to obtain Using Gronwall lemma, we obtain Differentiating (51)(b) and taking w ℎ = , we obtain So, we have Choosing z ℎ = in (51)(c) and using (56) and (61), we have Combining Lemmas 7 and 8, (56), (57), (61), (62), and the triangle inequality, we obtain the error estimate for Theorem 9.

Fully Discrete Scheme and Error Estimates
In this section, we get the error estimates of fully discrete schemes. For the backward Euler procedure, let 0 = 0 < 1 < 2 < ⋅ ⋅ ⋅ < = be a given partition of the time interval The Scientific World Journal (64) Now we formulate a completely discrete procedure. Find For the fully discrete error estimates, we now split the errors We will prove the theorem for the fully discrete error estimates.
It is easy to see that we obtained the optimal error estimates for in 2 -norm, 1 -norm, and the error estimates for and in ( 2 ) 2 -norm, which confirm the theoretical results in this paper, in Table 1. The numerical results in Table 1 and Figures 1-6 show that new expanded mixed

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The Scientific World Journal scheme for 2D Sobolev equation with convection term is efficient.

Concluding Remarks
In this paper, a new expanded mixed finite element method is proposed and studied for Sobolev equation with convectionterm. The proof for the existence and uniqueness of the solution for semidiscrete scheme, the new expanded mixed projection, and the proof of its uniqueness are given. The optimal a priori error estimates in 2 for the scalar unknown and the a priori error estimates in ( 2 ) 2 -norm for its gradient and its flux are proved. Especially, the optimal a priori error estimates in 1 -norm for the scalar unknown are derived. Finally, some numerical results are provided to confirm our theoretical analysis.
In the near future, the new expanded mixed method will be applied to other evolution equations such as evolution integrodifferential equations, hyperbolic wave equations, and nonlinear evolution equations. And the new characteristic expanded mixed finite element method for Sobolev equation will be studied. The new expanded characteristic-mixed weak formulation is to find { , , } : [0, ] → 1 0 × ( 2 (Ω)) 2 × ( 2 (Ω)) 2 such that where (x, ) = (1 + |c(x)| 2 ) 1/2 , In another article, we will give the error estimates for the new characteristic expanded mixed finite element method.