An Upwind Finite Volume Element Method for Nonlinear Convection Diffusion Problem

where is a bounded region with piecewise smooth boundary . R    is a small positive constant and is a smooth vector function on   u f   F x x  1 2 u        3 f u f u    x x  R  ,   0 0   F x . The finite volume element method (FVEM) is a discrete technique for partial differential equations, especially for those arising from physical conservation laws, including mass, momentum and energy. This method has been introduced and analyzed by R. Li and his collaborators since 1980s, see [1] for details. The FVEM uses a volume integral formulation of the original problem and a finite partitioning set of covolumes to discretize the equations.The approximate solution is chosen out of a finite element spaces [1-3] The FVEM is widely used in computational fluid mechanics and heat transfer problems [2-5]. It possesses the important and crucial property of inheriting the physical conservation laws of the original problem locally. Thus it can be expected to capture shocks, or to study other physical phenomena more effectively. On the other hand, the convection-dominated diffusion problem has strong hyperbolic characteristics, and therefore the numerical method is very difficult in mathematics and mechanics. when the central difference method, though it has second-order accuracy, is used to solve the convection-dominated diffusion problem, it produces numerical diffusion and oscillation near the discontinuous domain, making numerical simulation failure. The case usually occurs when the finite element methods (FEM) and FVEM are used for solve the convectiondominated diffusion problem. For the two-phase plane incompressible displacement problem which is assumed to be -periodic, J. Douglas, Jr., and T.F.Russell have published some articles on the characteristic finite difference method and FEM to solve the convection-dominated diffusion problems and to overcome oscillation and faults likely to occur in the traditional method [6]. Tabata and his collaborators have been studying upwind schemes based triangulation for convection-diffusion problem since 1977 [7-11]. Yuan, starting from the practical exploration and development of oil-gas resources, put forward the upwind finite difference fractional steps methods for the two-phase threedimensional compressible displacement problem [12]. 


Introduction
Consider the following nonlinear convection-diffusion problem:   0     where is a bounded region with piecewise smooth boundary .

R  
  is a small positive constant and is a smooth vector function on . The finite volume element method (FVEM) is a discrete technique for partial differential equations, especially for those arising from physical conservation laws, including mass, momentum and energy.This method has been introduced and analyzed by R. Li and his collaborators since 1980s, see [1] for details.The FVEM uses a volume integral formulation of the original problem and a finite partitioning set of covolumes to discretize the equations.The approximate solution is chosen out of a finite element spaces [1][2][3] The FVEM is widely used in computational fluid mechanics and heat transfer problems [2][3][4][5].It possesses the important and crucial property of inheriting the physical conservation laws of the original problem locally.Thus it can be expected to capture shocks, or to study other physical phenomena more effectively.
On the other hand, the convection-dominated diffusion problem has strong hyperbolic characteristics, and there-fore the numerical method is very difficult in mathematics and mechanics.when the central difference method, though it has second-order accuracy, is used to solve the convection-dominated diffusion problem, it produces numerical diffusion and oscillation near the discontinuous domain, making numerical simulation failure.The case usually occurs when the finite element methods (FEM) and FVEM are used for solve the convectiondominated diffusion problem.
For the two-phase plane incompressible displacement problem which is assumed to be -periodic, J. Douglas, Jr., and T.F.Russell have published some articles on the characteristic finite difference method and FEM to solve the convection-dominated diffusion problems and to overcome oscillation and faults likely to occur in the traditional method [6].Tabata and his collaborators have been studying upwind schemes based triangulation for convection-diffusion problem since 1977 [7][8][9][10][11].Yuan, starting from the practical exploration and development of oil-gas resources, put forward the upwind finite difference fractional steps methods for the two-phase threedimensional compressible displacement problem [12].


Most of the papers known concern on the FVEM for one-and two-dimensional linear partial differential equations [1][2][3][4]13,14].In recent years, M. Feistauer [15,16], by introducing lumping operator, constructed finite volume-finite element method for nonlinear convectiondiffusion problems.On the other hand, because the FEM costs great expense to solve the three-dimensional problems, finite difference methods (FDM) are usually used to approximate the problems [12].These works inspire us to look into the subject how to use the upwind FVEM to solve three-dimensional nonlinear convection-dominated diffusion problems.In this article, we continue to our work [17] and put forward an upwind FVEM for three-dimensional nonlinear convection-dominated diffusion problems based on tetrahedron partition and its dual partition of .Some techniques, such as calculus of variations, commutating operator and the a priori error estimate, are adopted.The a priori error estimate in -norm and

L
H -norm is derived to determine the error between the approximate solution and the true solution.
The remainder of this paper is organized as follows.In Section 2, we put forward the upwind FVEM for problem (1).In this section, we introduce notations, construct tetrahedron mesh partition h of and its dual partition.Some auxiliary lemmas and the a priori error estimate in -norm and H -norm of the scheme are shown In Section 3 and Section 4, respectively.In Section 5, some concluding remarks are presented.
Throughout this paper we use C (without or with subscript) to denote a generic constant independent of discrete parameters.We also adopt the standard notations of Sobolev spaces and norms and semi-norms as in [18,19].
 

C
The vector function has 1-order continuous partial derivative w.r.t. and u .
 u  F x x  Suppose the true solution of problem (1) possess certain smooth and satisfy: Before presenting the numerical scheme we introduce some notations.For simplicity we assume  is the domain . Firstly, Let us consider a family of regular tetrahedron partition in the domain h T  , which is a closure of .Let be maximum diameter of cell of h T .For a fixed tetrahedron partition For a given tetrahedron partition h T with nodes we construct two kind of dual partitions.First, we will construct the circumcenter dual partition of .
. Connecting j Q of the two face-adjacent tetrahedron cells which belong to , then we can derive a polyhedron i P K  which surrounds the node i .P j Q are vertices of the polyhe- is the circumcenter dual partition of h T .Denote by the midpoint of and its adjacent node Denote by .
is the other dual partition to .
As follows, we assume that the partition family h is regular, i.e., there exist positive constants 1 2

T C C
 independent of , such that the following condition (A2) satisfies: Suppose that a trial function space whose basis functions are based on [15], and Multiplying both sides of (1) by , integrating on dual partition cell , using Green formula, and summing with respect to , we have where   where ij  is the unit outward normal vector of .

For we introduce bilinear form
where ij  is the area of .
ij So far, we can obtain the semi-discrete upwind finite volume element scheme: Find such that where   If approximate solution h is known, then can be found by the following full-discrete upwind finite volume element scheme.

Auxiliary Lemmas
Define the discrete norm and the discrete semi-norm [1] as follows.
obviously, the discrete norm and the discrete semi-norm are equivalent to the continuous norm and the full-norm on , respectively.

   
Remarks: 1) From Lemma 1, we know that is symmetrical and positive definite in .
  The proof of lemma 2 can be completed by computing integral on cell Q K , directly.Theorem 1. (Trace Theorem) [20].Suppose that  has a piecewise Lipschitz boundary, and that is a real number in range p 1 p    .Then there exists a constant , such that Proof.From Hölder inequality, we can get that Using Taylor expansion, trace theory in which we choose and Hölder inequality, we can complete the proof of lemma 3.
Proof.From the properties of the functions in , for each partition cell where   is the volume of tetrahedron i.e., Ve , whose coordinates are x y z   P P , are four vertices of tetrahedron cell 0 1 2 3 i i i i which belongs to h .0 1 2 3 l are the volume coordinates which are corresponding to tetrahedron cell .For , Analogously, we can define the remaining coefficients   For simplifying numerical integral, we divide the polyhedron integral domain For simplicity, we will omit the variable in function . From volume coordinate formula, noting 2 3 From the above equality, we can complete the proof of

Convergence Analysis
Now we consider the error estimates of the approximate solution.Let Choosing in ( 7), then we have Subtracting ( 14) from (25), we obtain that where .
  Choosing h h in Equality (26), denote by 1 2 and 1 2 3 4 T T the left and right hand side terms of Equality (26), respectively.We will analyze the six terms successively.
From (20) of Lemma 1, we can get the estimate to as follows. 23 From ( 27)-(29), we have For each terms of the right hand side of (26).Using interpolation theory, triangulation inequality and lemma 4, we know that    

T C e e h u h u h
Combining ( 31), ( 32), ( 33) with (34) and applying Sobolev space embedding theory, we know that the of (26) satisfies From ( 30) and (35), using inverse estimate we know Summing from 1 to with respect to in the above inequality, we can obtain that Noting the equivalence of 0  and 1  with 0  and 1 , respectively.Using the inverse estimate, we have that there exist three positive constants Further, (37) may be rewritten as Noting that , combining finite element space interpolation theory, we can obtain the resulting error estimates to the approximate solution as follows. where, Therefore we have the following theory.Theorem 2. Suppose that the solution to the problem (1) is sufficiently smooth.When and are small enough and satisfy the relationship .The initial value is chosen as interpolation of , then the Equation (44) holds.

Conclusions
In this paper, we continued our work [17] and presented a class of upwind FVEM based on tetrahedron partition for a three dimensional nonlinear convection diffusion equation, analyzed and derived error estimate in norm and 2 L 1 H -norm for the method.In the ongoing work, we will discuss how to derive optimal error estimate in -norm and how to code and present numerical results to demonstrate the performance.

1 k and 2 k
the vertices of the edge k and