Expanded Mixed Finite Element Method for the Two-dimensional Sobolev Equation

Expanded mixed finite element method is introduced to approximate the two-dimensional Sobolev equation. This formulation expands the standard mixed formulation in the sense that three unknown variables are explicitly treated. Existence and uniqueness of the numerical solution are demonstrated. Optimal order error estimates for both the scalar and two vector functions are established.

The mixed finite element method, which is a finite element method [5] with constrained conditions, plays an important role in the research of the numerical solution for partial differential equations.Its general theory was proposed by Babuska [6] and Brezzi [7].Falk and Osborn [8] improved their theory and expanded the adaptability of the mixed finite element method.The mixed finite element method (see [9,10] for instance) is wildly used for the modeling of fluid flow and transport, as it provides accurate and locally mass conservative velocities.
The main motivation of the expanded mixed finite element method [11][12][13][14][15] is to introduce three (or more) auxiliary variables for practical problems, while the traditional finite element method and mixed finite element method can only approximate one and two variables, respectively.The expanded mixed finite element method also has some other advantages except introducing more variables.It can treat individual boundary conditions.Also, this method is suitable for differential equation with small coefficient (close to zero) which does not need to be inverted.Consequently, this method works for the problems with small diffusion or low permeability terms in fluid problems.Using this method, we can get optimal order error estimates for certain nonlinear problems, while standard mixed formulation sometimes gives only suboptimal error estimates.
The object of this paper is to present the expanded mixed finite element method for the Sobolev equation.We conduct theoretical analysis to study the existence and uniqueness and obtain optimal order error estimates.The rest of this paper is organized as follows.In Section 2, the mixed weak formulation and its mixed element approximation are considered.In Section 3, we prove the existence and uniqueness of approximation form.In Section 4, some lemmas are given.
In Section 5, optimal order semidiscrete error estimates are established.
Throughout the paper, we will use , with or without subscript, to denote a generic positive constant which does not depend on the discretization parameter ℎ.Vectors will be expressed in boldface.At the same time, we show a useful -Cauchy inequality

Mixed Weak Form and Mixed Element Approximation
For 1 ≤  ≤ +∞ and  any nonnegative integer, let denote the Sobolev spaces [16] endowed with the norm (the subscript Ω will always be omitted). 2.We denote by ( , ) the inner product in either  2 (Ω) or  2 (Ω) 2 ; that is, The notation ⟨, ⟩ denotes the  2 -inner product on the bound- To formulate the weak form, let  =  1 /; we introduce two vector variables Letting c = −∇, we rewrite (1) as the following system: −  = 0, (x, ) ∈ Ω × (0, ] , (x, ) = 0, (x, ) ∈ Ω × (0, ] , Let n be the unit exterior normal vector to the boundary of Ω.Then ( 8) is formulated in the following expanded mixed weak form: find (, , ) ∈  × Λ × V, such that where Let  ℎ be a quasiregular polygonalization of Ω (by triangles or rectangles), with ℎ being the maximum diameter of the elements of the polygonalization.Let  ℎ × Λ ℎ × V ℎ be a conforming mixed element space with index  and discretization parameter ℎ.  ℎ × Λ ℎ × V ℎ is an approximation to  × Λ × V.There are many conforming (or compatible) mixed element function spaces such as Raviart-Thomas elements [17], BDFM elements [18,19].Some RT type mixed elements are listed in Table 1.Here,   () is the polynomial up to  order in two-dimensional domain used in triangle, while  , () is the polynomial up to ,  in each dimension used in rectangle.
Replacing the three variables by their approximation, we get the expanded mixed finite element approximation problem: find where The error analysis next makes use of three projection operators.The first operator is the Raviart-Thomas projection (or Brezzi-Douglas-Marini projection) The following approximation properties are well known.
The other two operators are the standard  2 -projection [5]  ℎ and  ℎ onto  ℎ and Λ ℎ , respectively, They have the approximation properties The two projections Π ℎ and  ℎ preserve the commuting property

Existence and Uniqueness of Approximation Form
In this section, we consider the existence and uniqueness of the solution of (11).
Proof.In fact, this equation is linear; it suffices to show that the associated homogeneous system has only the trivial solution.In the first equation of ( 19), if we take  =  ℎ, and  = div  ℎ , respectively, then we have that By (20), it is easy to see that Choosing  = div  ℎ , k =  ℎ , and  =  ℎ in (19), we get In the third equation of (19), letting  =  ℎ and  =  ℎ , respectively, then Using the -Cauchy inequality to (22), we have

Some Lemmas
In the study of parabolic equations, we usually introduce a mixed elliptic projection associated with our equations.Define a map: find Similarly to Lemma 1, we can prove that system (29) has a unique solution.Now we give some error estimates of (ũ ℎ , λℎ , σℎ ).Define System (29) can be rewritten as follows: Now we consider the estimates of  1 and where  0 = 0 for  = 0, and  0 = 0 for  ≥ 1.
Proof.Let  ∈  2 (Ω) ⋂  1 0 (Ω) be the solution of the following problem: Then we know For 0 <  ≤ , from the second equation of (31), we have that It is easy to see that where  0 ℎ  is the piecewise constant interpolation of function .Using the previous estimates, we obtain      Proof.For 0 <  ≤ , the proof proceeds in three steps as follows.

Main Result
In this section, we consider error estimates for the continuous-in-time mixed finite element approximation.Define Proof.From ( 9) and ( 19), we have the following error equation: In (85), choosing  = Together with the results of Lemmas 2 and 3, the proof is completed.

Table 1 :
Some RT type mixed elements.