Quasi-Linear Convection-Dominated Transport Problem Based on Characteristics-Mixed Finite Element Method

We propose the characteristics-mixed method for approximating the solution to a convection-dominated transport problem. The new method is a combination of characteristic approximation to handle the convection part in time and a mixed finite element spatial approximation to deal with the diffusion part. Boundary conditions are incorporated in a natural fashion. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Analysis shows that the scheme has much smaller time-truncation errors than those of standard methods.

This equation governs such phenomena as the flow of heat within a moving fluid, the transport of dissolved nutrients or contaminants within the groundwater, and the transport of a surfactant or tracer within an incompressible oil in a petroleum reservoir.
For convenience, we assume (1) is Ω-periodic, this is physically reasonable since no-flow boundaries are generally treated by reflection, and because in general interior flow patterns are much more important than boundary effects.Thus, the no-flow boundary condition above can be dropped.Because of molecular diffusion, () is uniformly positive.Although this implies that the equation is uniformly parabolic, in many applications the Peclet number is quite high.Thus, convection dominates diffusion and the equation is nearly hyperbolic in nature.The concentration often develops sharp fronts that are nearly shocking.It is well known that strictly parabolic discretization schemes applied to the problem do not work well when it is convection dominated.It is especially difficult to approximate well the sharp fronts and conserve the material or mass in the system.
Effective discretization schemes should recognize to some extent the hyperbolic nature of the equation.Many such schemes have been developed, such as the explicit method of characteristics, upstream-weighted finite difference schemes [1], higher-order Godunov schemes [2,3], the streamline diffusion method [4], the least-squares-mixed finite element method [5,6], the modified method of

The Characteristics-Mixed Finite Element Formulation
We begin this section by introducing some notations.We denote by  , () the standard Sobolev space of differential functions in   ().Let ‖ ⋅ ‖ ,, be its norm and let ‖ ⋅ ‖ , be the norm of   () =  ,2 () or   () 2 , where we omit  if  = Ω.When  = 0, we let  2 denote the corresponding space defined on Ω and ‖⋅‖ let denote its norm.We also use the following spaces that incorporate time dependence.If where if  = ∞, the integral is replaced by the essential supremum.
We list the assumptions about the coefficients as follows: here and throughout this paper,  denotes different constants in different places.
Remark 1.The restriction  ⋅ ⃗  = 0 on the space  can be removed if the Dirichlet boundary condition is imposed, that is, the results presented in the following section also hold in the case of the first or the second type boundary value problems.
Let  ℎ be a quasi-regular polygonaization of Ω and let  ℎ ×  ℎ ⊂  ×  be the associated Raviart-Thomas-Nedelec space [15] of index  ≥ 0. In the procedure to be used, we will consider a time step Δ > 0 and approximate the solution at times   = Δ.The characteristic derivative will be approximated basically in the following manner.

The Existence and Uniqueness of the Solution of the Discrete Problem
In this section, we give the proof of the existence and uniqueness of the solution of the discrete problem (10).
Proof.Let {  }  =1 ⊂  ℎ and {  }  =1 ⊂  ℎ be two sets of bases, respectively, and let then ( 10) may be written in the following matrix form. Find such that where ) . ( Since {  }  =1 and {  }  =1 are bases, respectively, and (  ℎ ) ≥  0 ,  and  are two symmetric positive definite matrices.Solve   from (13)(b), then substitute it into (13)(a), we have Then, where Since () and () are globally Lipschitz continuous about , F is globally Lipschitz continuous about   too.With  0 given by ( 10)(c) it is clear that this nonlinear system of algebraic equations may be solved for small Δ, so that (10) defines a unique discrete solution for Δ ≤ .
In order to analysis the convergence of the method, it is convenient to introduce the mixed elliptic projection associated with our equations.

Several Lemmas
To obtain the optimal error estimates, we introduce the following three basic lemmas.These three lemmas are crucial to our main arguments.
Lemma 4. For any function  ∈  2 (Ω), there exists a function where  is a constant.
then, there exists a constant  > 0 such that Then, Thus, consider About the transformation  = () =  − ()Δ, because of the smoothness and periodicity of ,  is a differentiable homeomorphism of Ω onto itself for Δ sufficiently small (see [9]).By using the Jacobian of this transformation and a change of variables, we have

Error Estimates
Under the above assumptions about Ω, , , and , we can derive the optimal order estimates of ( ℎ − ) and ( ℎ − ).
Theorem 7. Let ( ℎ ,  ℎ ), (, ) denote the solution of (10) and (8), respectively.Suppose that Δ = (ℎ), we have for sufficiently small Δ > 0 the following: (39) Proof.Lets first show the estimate of (38).Taking V ℎ =   ℎ and  ℎ =   ℎ in (23), we obtain For the first term on the right-hand side of the above equation, following the treatment manner in [8], we will bound ‖(  /)−(  − −1 )/Δ‖ through the representation below involving an integral in the parameter  along the tangent to the characteristics from (,  −1 ) to (,   ).Denote the coordinates of the point on the segment by ((), ()).The standard backward difference quotient error equation is given by analogously, along the tangent to the characteristics following: Taking the and this completes the treatment of the right-hand side of (40).By using (33), the left-hand side of ( 40) is bounded by 49) with (40) to give the following recursion relation: ( This completes the first part of our conclusion. In order to show the estimate (39), by (23)(b), we get Choosing V ℎ = (  ℎ −  −1 ℎ )/Δ and  ℎ =   ℎ , respectively, in (23)(a) and (54) and adding them to obtain The left-hand side satisfies the inequality: Substitute it into (55), we have Bound  1 - 5 on the right-hand side of (56) one by one, we have We see that an application of Gronwall's lemma would complete the second part of our argument if we did not have the term max 0≤≤−1 ‖  ℎ ‖ 0,∞ on the right hand side of (64).We will use an induction argument to prove the hypothesis as follows: (69) Note that our results in this paper do not cover the cases of nonlinear convection-dominated systems, which are of importance in some applications, particularly in reservoir simulations.