Full-Discrete Weak Galerkin Finite Element Method for Solving Diffusion- Convection Problem

This paper applied and analyzes full-discrete weak Galerkin (WG) finite element method for non steady two dimensional convection-diffusion problems on conforming polygon. We approximate the time derivative by backward finite difference method and the elliptic form by WG finite element method. The main idea of WG finite element methods is the use of weak functions and their corresponding discrete weak derivatives in standard weak form of the model problem. The theoretical evidence proved that the error estimate in 2 L -norm, the properties of the bilinear form ) , ( v u a , (v-elliptic and continuity), stability, and the energy conservation law.


Introduction:
Weak Galerkin (WG) finite element methods refer to general finite element methods for partial differential equations (PDEs). The novel idea of WG finite element methods is on the use of weak functions, weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play important roles in numerical methods for PDEs [6]. The key to WG finite element methods is the use of a discrete weak gradients operator which is defined and computed by solving inexpensive problems locally on each element. The fundamental difference between the weak Galerkin finite element method and other existing methods, such as Discontinuous Galerkin Finite Element Method ( [1,3,17]) and the Finite Volume Method ( [4,7,18]), is used of weak function and weak gradient in the design of numerical scheme based on existing weak forms for the underlying PDEs.
WG finite element methods were first introduced in [6] for solving steady second order elliptic problem and later on in [8] for shape regular polytopal meshes. The method has been successfully applied to elliptic interface problems [10], Helmholtz equations [13], and biharmonic equations [11,12], in this paper, we applied the WG finite element methods for non steady diffusion convection problem and proved that the error estimate in 2 L -norm, the elliptic property, and the energy conservation law. This paper is organized as follows: Section 2 is devoted to a description some preliminaries and notations, In section 3 we show the Convection-Diffusion Problem. The weak function the weak gradient and the weak Galerkin method. In section 4 we discuses the full-discrete weak Galerkin method. In section 5 we proved the energy conservation law of WG finite element method. The error analysis for Semi-discrete weak Galerkin finite element method proved in section 6. for any 0  s [5,14]. For example, for any integer 0  s , the semi-norm

Preliminaries and Notations:
The Sobolev norm

The Convection-Diffusion Problem:
We consider the non-state diffusion-convection problem, Subject to I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 3 N u m b e r 4 J o u r n a l o f A d v a n c e i n M a t h e m a t i c s 7335 | P a g e S e p t e m b e r 2 0 1 6 w w w . c i r w o r l d . c o m , and  is known as the gradient operator. And  is diffusion coefficient, For simplicity, let the function f in (1) be locally integrable in  . We shall consider solutions of (1) with a nonhomogeneous Dirichlet boundary condition is a function defined on the boundary of  .
is the Sobolev space consisting of functions which, together with their gradients, are square integrable on the boundary of  , The standard weak form for seeks

THE WEAK GRADIENT:
The main idea of weak Galerkin methods is the use of discrete weak derivatives in the place of strong derivatives in the variational form for the underlying partial differential equations. The weak gradient operator will then be employed to discretization the problem (3) through the use of a discrete weak gradient operator as building bricks [6]. The discrete weak gradient is given by approximating the weak gradient operator with piecewise polynomial functions. To introduced the weak gradient operator; Firstly we define the weak function. Let K be any polygonal domain with interior 0 K and boundary K  . A weak function on the region K refers to a vector-valued function the space of weak functions on K [9]; i.e.
Secondly, we define the weak gradient, for any , the weak gradient of v is defined as a linear functional v w  in the dual space of ) ; ( K div H whose action on each ) ; where n is the outward normal direction to K  . The discrete weak gradient operator was defined by approximating v w  in a polynomial subspace of the dual of . More precisely, for any non-negative integer 0  r , denote by ) (K P r the set of polynomials on K with degree no more than r . Let be a subspace of the space of vector-valued polynomials of degree r . The discrete weak gradient operator, denoted by satisfying the following equation:

WEAK GALERKIN FINITE ELEMENT SPACE:
Let h T be a triangular partition of the domain  with T  as its boundary. For each h T T  , Denote by ) ( 0 T P j the set of polynomials on 0 T with degree no more than j , and The corresponding finite element space would be defined by patching According to (6) , the discrete weak gradient of v , on each element T is given by the following equation: For each element is given by the following equation: L -projection operators, then the right-hand side of (12) is given by; This implies the desired identity (11).
One can also define the projection h  be the usual projection operator in the Dirichlet finite element method such that The following result is based on the above property of h  .

Lemma 3.2.2 [6]: For any
For any ) , we introduce the following bilinear form where I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 3 N u m b e r 4 J o u r n a l o f A d v a n c e i n M a t h e m a t i c s 7337 | P a g e S e p t e m b e r 2 0 1 6

FULL-DISCRETE WEAK GALERKIN FINITE ELEMENT METHOD [15]:
We now turn our attention to some discrete time weak Galerkin procedure. We introduce a time step k and the time Where n is a nonnegative integer , and denote by ) , to be determined .The backward Euler weak Galerkin method is define by a backward difference equation , if  Proof: By using Cauchy -Schwarz inequality in equation (16), we have Now, by using Cauchy -Schwarz inequality by  -inequality in the second term.
By Cauchy-Schwarz inequality and using the fact that of the first term, we have

ENERGY CONSERVATION OF WEAK GALERKIN:
Theorem (5.1): The numerical solution of equation (16) satisfies the energy conservation law.
The second term of equation,

Error analysis for Full-discrete weak Galerkin Finite Element Method:
In this section we show that the theoretical analysis of error between the exact solution of equation (3) Now, addition and subtraction ) ( u   and by using the triangle inequality we have and u be the solution of (15) and n h U be the weak Galerkin approximation of equation (16). Then there exists a constant C such that. To estimate , from equation (15) and lemma (3.2.2), we have Now, addition and subtraction ) ), on the left hand side of the above equation and then using (13), we have Combining equation (16) with the above equation, we have We have

Conclusion:
The present paper presents the full-discrete weak Galerkin finite element method for the non-steady diffusion-convection problem. by using this method we proved the energy conservation law to verified that the numerical flux to be continuous a cross the edge of each element K through a selection of the test function .We have proved the properties of the bilinear form ) , ( v u a , (v-elliptic and continuity ),stability, also we prove the convergence of the scheme.