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An approximate scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using immersed interface finite element method for the pressure equation which is based on the broken

Miscible displacement of one incompressible fluid in a porous medium

In the formulation, we have utilized the fact that in applications the vertical dimension of a subsurface geological formation is often much smaller than the horizontal dimension. We thus simplify the problem via a vertical average to rewrite the problem as a two-dimensional problem in the horizontal dimension, so the physical domain

For (

Let

There exist much literature concerning numerical method and numerical analysis of miscible placement problem in porous media, for example, [

For better approximation, the fitted finite element methods whose mesh depends on the smooth interface are developed [

In this paper, we apply

The rest of the paper is organized as follows: in Section

In this paper, without loss of generality, we consider the case in which

The proper jump conditions for elliptic (

Let

For the analysis, we introduce the space

We integrate (

By the Sobolev embedding theorem, for any

Assume that

In this section, firstly, we define a broken

In order to construct a broken

Let

If

If

The interface curve

Then we can separate the triangles of partition

As usual, we will define linear finite element spaces on each element of the partition

For this space, we define the interpolation operator

For interface element

Let

By similar calculation to that given in Theorem 2.2 of [

Therefore, the coefficients of (

Therefore we can define the following finite element space on an interface element

Also we use

Although for functions in

Combining the definitions of

Now, assume that the approximation of concentration

The question at hand is to discretize the concentration equation.

Multiplying the concentration equation (

Let

For convergence analysis conducted in the following section, we need an interpolation operator

Next, let

The function

For convergence analysis conducted in Section

Let

Let

In this section, we will present convergence analysis for the pressure and concentration.

Let

Since the immersed finite element formulation (

For the second part, it is a consistency error estimate. By the definition of

Combining (

We now are in the position to prove the error estimate for concentration.

Assume that the true solution

We decompose the error

We choose the test function as

The terms on the left side can be easily bounded by

Next, we should estimate terms on the right side one by one.

By (

Similarly, we can bound the second and third terms as follows:

For the fifth term, we can derive the estimate by using the boundness of

Similarly, we can derive

In order to prove the forth term, we need the following induction hypothesis:

As the statement in [

Therefore, collecting all the bounds derived above and using Gronwall inequality leads to

To complete the argument, we have to check hypothesis (

Combining (

Assume that the true solution

In this paper, we just get a priori error estimates for the coefficient

This work is supported in part by the National Natural Science Foundation of China (no. 10971254), National Natural Science Foundation of Shandong Province (no. Y2007A14), and Young Scientists Fund of Shandong Province (no. 2008BS01008).