#### Abstract

A Godunov-mixed finite element method on changing meshes is presented to simulate the nonlinear Sobolev equations. The convection term of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and the diffusion term by an expanded mixed finite element method. The method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. Almost optimal error estimates in -norm under very general changes in the mesh can be obtained. Finally, a numerical experiment is given to illustrate the efficiency of the method.

#### 1. Introduction

We consider the following nonlinear Sobolev equations: where is a bounded subset of () with smooth boundary , and are known functions, and the coefficients satisfy the following condition: where , , , , , , and are positive constants. We assume that satisfy the smooth condition in the following analysis.

For time-changing localized phenomena, such as sharp fronts and layers, the finite element method on changing meshes [1–3] is advantageous over fixed finite element method. The reason is that the former treats the problem with the finite element method on space domain by using different meshes and different basic functions at different time levels so that it has the capability of self-adaptive local grid modification (refinement or unrefinement) to efficiently capture propagating fronts or moving layers. The work [4] had combined this method with mixed finite element method to study parabolic problems. In [5], an upwind-mixed method on changing meshes was considered for two-phase miscible flow in porous media.

Sobolev equations have important applications in many mathematical and physical problems, such as the percolation theory when the fluid flows through the cracks [6], the transfer problem of the moisture in the soil [7], and the heat conduction problem in different materials [8]. So there exists great and actual significance to research Sobolev equations. Many works had researched on numerical treatments for Sobolev equations. More attentions were paid for treating a damping term , which is a distinct character of Sobolev equations different from parabolic equation. For example, time stepping Galerkin method was presented for nonlinear Sobolev equations in [9, 10]. In [11, 12], nonlinear Sobolev equations with convection term were researched by using finite difference streamline-diffusion method and discontinuous Galerkin method, respectively. Two new least-squares mixed finite element procedures were formulated for solving convection-dominated Sobolev equations in [13].

Methods which combined Godunov-type schemes for advection with mixed finite elements for diffusion were introduced in [14] and had been applied to flow problems in reservoir engineering, contaminant transport, and computational fluid dynamics. Applications of these types of methods to single and two-phase flow in oil reservoirs were discussed in [15, 16]; application to the Navier-Stokes equations was given in [17]. These methods had proven useful for advective flow problems because they combined element-by-element conservation of mass with numerical stability and minimal numerical diffusion. Dawson [18] researched advective flow problems in one space dimension by high-order Godunov-mixed method. In 1993, he expanded this method to multidimensions [19] and presented three variations. In these methods, advection was approximated by a Godunov-type procedure, and diffusion was approximated by a low-order-mixed finite element method.

The object of this paper is to present a Godunov-mixed finite element method on changing meshes for the nonlinear Sobolev equations. The convection term of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and the diffusion term by an expanded mixed finite element method. This method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. We describe this method in Section 2. In Section 3, we introduce three projections and a lemma. We derive almost optimal error estimates in -norm under very general changes in the mesh in Section 4. In Section 5, we present results of numerical experiment, which confirm our theoretical results.

Throughout the analysis, the symbol will denote a generic constant, which is independent of mesh parameters and and not necessarily the same at different occurrences.

#### 2. The Godunov-Mixed Method on Changing Meshes

At first we give some notation and basic assumptions. The usual Sobolev spaces and norms are adopted on . The inner product on is denoted by . Define the following spaces and norms:

Let denote different time steps, , , . Assume that the time steps do not change too rapidly; that is, we assume there exist positive constants and which are independent of and such that For a given function , let .

Assume . At each time level , we construct a quasiuniform rectangular partition of : Let , , , and . And is the midpoint of , , . Let be defined analogously.

Let where and is the set of all polynomials of degree less than or equal to defined on . Similar definitions are given to and .

Then the lowest order Raviart-Thomas spaces and are given by That is, the space is the space of functions which are constant on each element , and is the space of vector valued functions whose components are continuous and linear on each element . The degrees of freedom of a function correspond to the values of at the midpoints of the sides of , where is the unit outward normal to . It is easy to see that and .

By introducing variables , , , and , we modify the first equation in (1.1) as

Here we are using the so-called “expanded” mixed finite element method, proposed by Arbogast et al. [20], which gives a gradient approximation as well as an approximation to the diffusion term .

The weak form of (2.6) is so that

The Godunov-mixed method on changing meshes is presented as follows: at each time level , find , such that When different finite element spaces are used at time levels and , the second and fifth equations of (2.9) give the -projection of the previous approximate solution into the current finite element space . This projection is used in the first and third equations of (2.9) as initial value to calculate . If the finite element spaces are the same at time levels and , we know that , .

Let in (2.9) be constructed by the Godunov method. We adopt the Godunov flux [19] for a given flux function : where are the left and right states, respectively. It is easily seen that is Lipschitz in its arguments if is Lipschitz in and is consistent, that is, .

Let , , . Now we give the calculation of by by the following three steps.

*First Step *

We construct a piecewise linear function on each element :
where and are slopes. The slope calculation can be performed as
where and are forward and backward difference operators in the direction, respectively:
and satisfy
where and are positive integers independent of . The slope is defined analogously.

* Second Step *

By the Taylor expansion, for with smooth second derivatives, we have
By (2.6), we see that
where

Based on (2.17) and a similar expansion about , we define :
where

*Third Step *

With the above definitions, is calculated as follows.

If , define

If , define
For , the definition is similar.

The equations (2.19)–(2.22) hold for elements at least one element away from the boundary. At the left and right boundaries, we can set
and in the slope calculation procedure, we define and using (2.12) with
On the bottom and top boundaries, we set
in (2.19).

#### 3. Projections and Lemma

We introduce three projections and a lemma to obtain error estimates. Let be the projection of in mixed finite element space such that and define as the -projection of such that

These projections satisfy ([21, 22]): and it is easy to see that .

Let satisfies that the value of at the midpoint of the boundaries is equal to the average value of on boundaries, that is, Let . From (3.4), we can define Then, can be defined similarly.

Lemma 3.1. *Let and assume that is sufficiently smooth. Then
*

*Proof. *Firstly, construct from given in (3.1) and (3.2) by
at interior edges, where
And define similarly.

Then, assuming is twice differentiable and Lipschitz continuous, and using the Lipschitz continuity of the Godunov flux and (2.14), it can be shown that
A similar bound holds for .

At the boundaries, we follow (2.23) and (2.25) and define
Define analogous to .

If , we consider

By the Lipschitz continuity of and (3.9), we have
The similar arguments are applied to to get
By the consistency of , we see
For , it is easy to have

Using (3.3), (3.12)–(3.15), and equivalence of norms, we have

The similar approach can be used when . Defining and similarly and following analogous arguments yields
Thus, Lemma 3.1 holds.

#### 4. Error Estimates

At time level , for all , , the exact solutions satisfy where , .

Let

Using projections (3.1) and (3.2), we subtract (2.9) from (4.1) to get where the last term of the first equation in (4.3) is related to the changing meshes. If the meshes do not change, this term will be zero.

Taking , in the second equation and in the third, then adding them together, we have For (4.4), we see

(I)

(II)

(III) The last second term in (4.6) is related to the changing meshes. If the meshes do not change, this term will be zero.

Substitute (4.5)–(4.7) into (4.4) to get Furthermore, for the first and third terms on the left-hand side of (4.8), we have By (4.9), we modify (4.8) as

By the Lipschitz continuity of , , , , and (3.3), the following terms on the right-hand side of (4.10) can be obtained:

We turn to consider . Letting in the second equation of (4.10), then we have By Lemma 3.1, we have

Substituting (4.11) and (4.13) into (4.10), we have

Let be the time step at which is maximum, that is, Multiplying (4.14) by and summing on from to , we obtain

Assuming that the mesh is modified at most times, and , where is independent of and , we get [3]: where is a sufficiently small positive constant.

Substituting (4.17) into (4.16), we find Using the discrete Gronwall's lemma, we have Similarly as (4.14), we can derive

By the triangle inequality and (3.3), we can obtain the following result.

Theorem 4.1. *Assuming that the coefficients satisfy condition (1.2), the mesh is modified at most times, , and , , , then we have
**
where and are positive constants independent of and .*

#### 5. Numerical Example

In this section, we give numerical experiment of the following nonlinear equations to illustrate the efficiency of the Godunov-mixed finite element method on changing meshes: where , , , , and is chosen properly so that the exact solution is .

In the numerical example, and change as the following four cases.

*Case 1. *, set ; , set ; , set and calculate 40 steps at every time interval.

*Case 2. *, set ; , set ; , set and calculate 80 steps at every time interval.

*Case 3. *, set ; , set ; , set and calculate 160 steps at every time interval.

*Case 4. *, set ; , set ; , set and calculate 320 steps at every time interval.

The numerical solutions , are calculated and the error estimates of are obtained; see Tables 1 and 2.