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.
Given an open bounded domain
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 (
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 [
We concentrate on MMOC-Galerkin. It is an implicit scheme, so reasonably large time steps may be used, and it does not numerically diffuse the fronts to a particularly excessive degree. Unfortunately, it has certain inherent difficulties, especially with regard to local mass balance.
Mixed finite element method has been found to be very useful for solving flow equations not only for its optimal approximation of the unknown function and the vector flux, but also for its mass conservation due to the use of piecewise test function. Its theories and applications have been extensively discussed, for example, see [
Therefore, to purse the high performance of an algorithm, one should take a nice combination of MMOC and the mixed finite element method. In this way, characteristics-mixed finite element method was considered. However, most integrals in this method involve the transformation images of the test function, so it is difficult to compute in practice.
In this paper, we propose a new characteristics-mixed finite element scheme. It is similar to MMOC-Galerkin in that we approximate the hyperbolic part of the equation along the characteristics. We use, however, a mixed finite element spatial discretization of the equation. Since piecewise constant functions can be used as test functions, the scheme conserves mass locally and globally in the discrete level. One of our goals in the paper is to yield
An outline of the paper is as follows. In Section
We begin this section by introducing some notations. We denote by
We also use the following spaces that incorporate time dependence. If
We list the assumptions about the coefficients as follows:
We also assume that the solution
Define the spaces:
The restriction
Let
Let
This scheme is an implicit scheme and it is difficult to compute in practice. For simplicity, we can use
In this section, we give the proof of the existence and uniqueness of the solution of the discrete problem (
There exists a unique solution
Let
Find
In order to analysis the convergence of the method, it is convenient to introduce the mixed elliptic projection associated with our equations.
Let
Let
To obtain the optimal error estimates, we introduce the following three basic lemmas. These three lemmas are crucial to our main arguments.
For any function
For any
Let
Let
We now obtain (
Under the above assumptions about
Let
Lets first show the estimate of (
For the first term on the right-hand side of the above equation, following the treatment manner in [
By using (
In order to show the estimate (
Similarly, we get
We see that an application of Gronwall's lemma would complete the second part of our argument if we did not have the term
Finally by Gronwall's lemma, it follows from (
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.
This research is partially supported by the China Postdoctoral Science Foundation funded project (2011M501149), the Humanity and Social Science Foundation of the Ministry of Education of China (12YJCZH303), the Special Fund Project for Post Doctoral Innovation of Shandong Province (201103061), and Independent Innovation Foundation of Shandong University, IIFSDU (IFW12109).