The Time Discontinuous H 1-Galerkin Mixed Finite Element Method for Linear Sobolev Equations

We combine the H-Galerkin mixed finite element method with the time discontinuous Galerkin method to approximate linear Sobolev equations. The advantages of these two methods are fully utilized. The approximate schemes are established to get the approximate solutions by a piecewise polynomial of degree at most q − 1 with the time variable. The existence and uniqueness of the solutions are proved, and the optimalH-norm error estimates are derived. We get high accuracy for both the space and time variables.

There has been a wide range of practical applications of Sobolev equations in engineering, such as the porous theories concerned with percolation into rocks with cracks [1], the transport problems of humidity in soil [2], and the heat conduction problems between different media [3,4].So many numerical simulation methods have been proposed to solve these equations; for example, see [5][6][7][8][9][10][11].
The time discontinuous Galerkin method was proposed by Delfour et al. in 1981 [12] to analyze the ordinary differential equations.The purpose of this method is to find the approximate solutions as a piecewise polynomial of degree at most  − 1 with time variable by discretizing the time variable with Galerkin method.So the similar processing method can be used to analyze the space and time variables in the definition and analysis.This method considered the space and time variables together in order to utilize the advantages of the finite element method on both the space and time variables.Hence, the accuracy of numerical solutions is improved.Jamet [13] and Lesaint and Raviart [14] used the time discontinuous Galerkin method to study the partial differential equations.Eriksson et al. promoted and enriched this method further; for example, see [15][16][17][18][19].
It is well known that standard mixed finite element spaces have to satisfy the LBB-consistency condition (also called Inf-Sup condition), which restricts the choice of finite element spaces.For example, in [20], the Raviart-Thomas spaces of index  ≥ 1 are used for the second-order elliptic problems.
In order to circumvent the LBB-consistency condition,  1 -Galerkin mixed finite element method was introduced by Pani [21] to parabolic equations.By introducing a flux variable  of the primary variable , parabolic equations are changed into a first-order system.Then,  1 -Galerkin finite element method is used to approximate the system.The finite element spaces  ℎ × W ℎ for variable  and flux  are allowed to be of different polynomial degrees and not subject to the LBB-consistency condition.Hence, the obtained estimations can distinguish the better approximation properties of  ℎ and W ℎ .Moreover, the quasi-uniformity condition was not imposed on the finite element mesh for  2 and  1 -norm error estimates.More applications of  1 -Galerkin mixed finite element method had been done to, for example, the parabolic integro-differential equations [22], the secondorder hyperbolic equations [23], and Sobolev equations [8].Recently, some researchers applied this method to other types of problems, such as RLW equation [24], heat transport equations [25], and pseudo-parabolic equation [26].
In this paper, we combine  1 -Galerkin mixed finite element method and the time discontinuous Galerkin method to approximate linear Sobolev equations.We establish the time discontinuous  1 -Galerkin mixed finite element schemes and expect to utilize the advantages of the two above-mentioned methods to obtain a high-accuracy numerical method.
The rest of this paper is organized as follows.In Section 2, we present the time discontinuous  1 -Galerkin mixed finite element schemes.In Section 3, we prove the existence and uniqueness of the solutions.In Section 4, the optimal  1norm error estimates are derived.In Section 5, we draw some conclusions of this paper.
Clearly (6a) is obtained by multiplying (3a) by ∇V and integrating the resulting equation with respect to .Multiplying (3b) by ∇ ⋅ w and integrating the first term by parts give where the condition   | Ω = 0 is used.From (3a), we know ∇  =  − ∇.Substituting this expression into the above equation yields (6b).Assume that  ℎ and W ℎ are the finite dimensional subspaces of  1 0 and H(div; Ω), respectively, with the following approximation properties: for ,  positive integers inf inf Let 0 =  0 <  1 < ⋅ ⋅ ⋅ <   =  be an unnecessarily uniform subdivision of [0, ] and With a given positive integer  ≥ 1, we will look for the approximation solutions {, Σ} of (6a) and (6b), which reduce to a polynomial of degree at most  − 1 with time variable  on each subinterval   with coefficients in  ℎ and W ℎ , respectively.That is to say, they belong to the following finite element spaces: Note that the functions of these two spaces are allowed to be discontinuous at the time nodal points but are taken to be continuous to the left there.For  ∈   ℎ, , we denote Notice that Here, we do not consider the continuity of V at  =   in the above equation, which means the time discontinuous Galerkin finite element space can be adopted.Hence, the time discontinuous  1 -Galerkin mixed finite element scheme of (6a) and (6b) is to find {, Σ} ∈  ℎ, ×W ℎ, such that Here, we take  0 =  0,ℎ and  0,ℎ is a certain initial approximation to  0 () in  ℎ . Since (14) can be rewritten as Due to the discontinuities of  ℎ, and W ℎ, with respect to , the local forms of ( 17) and ( 15) can be given as follows: The advantage of the local forms ( 18)-( 19) is that they can be solved locally in each time interval   ( = 0, . . .,  − 1) gradually.

Existence and Uniqueness of the Solutions
In this section, we prove the existence and uniqueness of the solutions of ( 18)- (19).First, we adopt the following lemma [27].
By Lemma 2, we can have the following theorem.
Proof.Since ( 18)-( 19) are linear equations about  and Σ, the existence and uniqueness of the solutions are equivalent to the homogenous linear equations only that have zero solutions on each time interval   [28].These proofs are different from well-known Brouwer's fixed point theorem [29], which is used to consider the nonlinear boundary problems.
Then, we have some properties of  ℎ  ∈  ℎ as the following lemmas.
Finally, by the triangle inequality and Lemmas 5, 7, 8, 9, and 12, we can establish the following theorem.
Theorem 13.Let  be the solution of the original problem (1a), (1b), and (1c);  and Σ are the finite element solutions of the approximate schemes ( 14)- (15).Assume that  is smooth enough to satisfy the required regularities in the analyses, and ‖ 0,ℎ −  0 ‖ + ℎ‖ 0,ℎ −  0 ‖ 1 ≤ ℎ +1 ‖ 0 ‖ +1 .Then, we have the following error estimates: From this theorem, we see that with respect to space variable  the optimal  1 -norm error estimates of both  and  are derived.Also, -order accuracy for time variable  is obtained.That is to say, we get high accuracy for both the space and time variables.

Conclusions
In this paper, we have solved the linear Sobolev equations (1a), (1b), and (1c) by the time discontinuous  1 -Galerkin mixed finite element method.We have presented the approximate schemes, proved the existence and uniqueness of the solutions, and derived the optimal  1 -norm error estimates for both the primary variable  and intermediate variable .In particular, high-order accuracy for time variable also is obtained.In our forthcoming work, some numerical experiments will be considered to verify our analysis for the schemes.