American Journal of Computational Mathematics
Vol.08 No.01(2018), Article ID:83365,12 pages
10.4236/ajcm.2018.81009
Weak Galerkin Finite Element Method for the Unsteady Stokes Equation
Chen Ning, Haiming Gu
College of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, China
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: February 25, 2018; Accepted: March 25, 2018; Published: March 28, 2018
ABSTRACT
The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.
Keywords:
Weak Galerkin Finite Element Methods, Unsteady Stokes Equations, Stokes Projection
1. Introduction
The finite element method for the unsteady Stokes equations developed over the last several decades is based on the weak formulation by constructing a pair of finite element spaces satisfying the inf-sup condition of Babuska [1] and Brezzi [2] . Readers are referred to [3] [4] [5] [6] [7] for specific examples and details in the different finite element methods for the Stokes equations. The idea of weak Galerkin method was first introduced by the Professor Junping Wang in June 2011. Weak Galerkin refers to a general finite element technique for partial differential equations in which differential operators are approximated by weak forms as distributions for generalized functions. Thus, two of the key features in weak Galerkin methods are 1) the approximating functions are discontinuous, and 2) the usual derivatives are taken as distributions or approximations of distributions. The method was successfully applied to the second order elliptic equations [8] [9] , the Stokes equations [10] , Parabolic equations [11] , and Maxwell equations [12] . A posteriori error is effectively estimated, and proved the convergence of the WG finite element method in this paper.
2. Preliminaries
In this paper, we study the initial-boundary value problems of the Stokes.
(1)
where is fluid velocity, p is pressure, is volumetric power density.
The solution of the Stokes equations forms an important aspect of both theoretical and computational fluid dynamics. A limited number of solutions of these non-linear partial differential equations mostly involving spatially one-dimensional problems are given in the literature. Solutions of practical interest have been obtained for cases where, with suitable approximations, the equations are reduced to linear partial differential equations.
Let W be a bounded domain in R2. We introduce function spaces
, , , then the unsteady Stokes problem would take the following form: seek satisfying
(2)
We use and to be denote the norm and Semi-norm in the Sobolev space for any , respectively. The inner product in is denoted by . For example, for each , the Semi-norm is given by
and is said to be the norm of .
For w is to , the definition is given by
for , we have
The space and the norm defined in the defined as
3. Weak Galerkin Finite Element Approximation Scheme
Let K be any polygonal or polyhedral domain with boundary . A weak vector-valued function on the region K refers to a vector-valued function such that and . The first component can be understood as the value of v in K, and the second component represents v on the boundary of K. Note that may not necessarily be related to the trace of on should a trace be well-defined. Denote by the space of weak functions on K;
(3)
Definition 1. For any , the weak gradient of v is defined as a linear functional in the dual space of , whose action on each is given by
(4)
where n is the outward normal direction to , is the action of on , and is the action of on .
The Sobolev space can be embedded into the space by an inclusion map defined as follows
With the help of the inclusion map , the Sobolev space can be viewed as a subspace of by identifying each with .
Let be the set of polynomials on K with degree no more than r.
Definition 2. The discrete weak gradient operator, denoted by , is defined as the unique polynomial satisfying the following equation,
(5)
for all .
In what follows, we give the definition of weak divergence, first of all, we require weak function such that an Denote by the space of weak vector-valued functions on K;
(6)
Definition 3. For any , the weak divergence of v is defined as a linear functional in the dual space of whose action on each is given by
(7)
where n is the outward normal direction to , is the action of
on , and is the action of on .
The Sobolev space can be embedded into the space by an inclusion map defined as follows
Definition 4. A discrete weak divergence operator, denoted by , is defined as the unique polynomial that satisfies the following equation.
(8)
for all .
4. Weak Galerkin Finite Element Scheme
Let be a partition of the domain W with mesh size h that consists of arbitrary polygons/polyhedra. In this paper, we assume that the partition is WG shape regular-defined by a set of conditions as detailed in references. Denote by the set of all edges/flat faces in , and let be the set of all interior edges/faces. For any integer , we define a weak Galerkin finite element space for the velocity variable as follows,
We would like to emphasize that there is only a single value defined on each edge . For the pressure variable, we have the following finite element space
Denote by the subspace of consisting of discrete weak functions with vanishing boundary value;
The discrete weak gradient and the discrete weak divergence on the finite element space can be computed by using (5) and (8) on each element T, respectively. More precisely, they are given by
For simplicity of notation, from now on we shall drop the subscript in the notation and for the discrete weak gradient and the discrete weak divergence. The usual inner product can be written locally on each element as follows
Denote by the L2 projection operator from onto . For each edge/face , denote by the L2 projection from onto . We shall combine with by writing .
We are now in a position to describe a weak Galerkin finite element scheme for the Stokes Equations (1). To this end, we first introduce three bilinear forms as follows
WG Algorithm. Seek satisfying
(9)
In the following, the proof process of Lemma 1-6 refers to reference [10] [11] [12] .
Lemma 1. For any , the following equation hold true,
Lemma 2. For any we have
In addition to the projection defined in the previous section, let and be two local L2 projections onto and , respectively.
Lemma 3. The projection operators , , and satisfy the following commutative properties
Lemma 4. There exists a positive constant b independent of h such that
for all .
Lemma 5. Poincare inequality of Weak gradient operator: If , then exists a constant c satisfying
First of all, we study the existence and uniqueness of the solution for (9). The space defined as follows
Then we need to seek satisfying
(10)
Let be the solution of (10) and which is unique, the linear bounded functional on defined as follows.
(11)
Then problem (9) is equivalent to seek satisfying
(12)
Using LBB condition and Lax-Milgram Lemma, we know that the solution of (12) is unique.
Combing (11) and (12), it is concluded that if initial approximation , the solution of (9) is unique.
In what follows, we introduce Stokes projection, which is the important approximation of projection.
Lemma 6. First of all, we introduce Stokes projection of , which is need satisfying
(13)
If let , easy to know that satisfying
(14)
Then is the finite element approximation of , so we have
(15)
5. Error Equations
In what follows, we list Lemma 7 to prove the error estimation of approximate solution for Semi-discrete scheme.
We know that and be solution of (1) and Galerkin finite element solution of (9), respectively. The L2 projection of u in the finite element space is given by . Similarly, the pressure p is projected into as . Denote by and the corresponding error given by
(16)
Lemma 7. Let be sufficiently smooth and satisfy the following equation
(17)
in the domain W. Let and be the L2 projection of into the finite element space . Then, the following equation holds true
(18)
for all . Where and are two linear functionals on defined by
Proof. Together Lemma 3, Equation (5) and integration by parts. we obtain
(19)
Next, Combing Lemma 3 and Equation (8), the fact that ,
then using integration by parts, we obtain
We can imply that
(20)
Next, we test (17) by using in to obtain, we can obtain
(21)
It follows from the usual integration by parts that
Where we have used the fact that . using Equations (19) and (20), we have
(22)
Substituting (20), (22) and into (21) yields
which completes the proof of the lemma.
In what follows, we give the derivation of the error equation of (9).
Lemma 8. Let and be the error of the weak Galerkin finite element solution arising from (9), as defined by (16). Then, we have
(23)
for all and , where is a linear functional defined on .
Proof. Since satisfies the Equation (17) with , then from Lemma 6 we have
Adding to both side of the above equation give
(24)
The difference of (24) and (9) yields the following equation,
for all , where . This completes the derivation of (23).
As to (24), we test Equation (1) by and use (9) to obtain
(25)
The difference of (25) and (9) yields the following equation
for all .
Which completes the proof of the lemma.
In the following, the proof process of Lemma 9 refers to reference [10] .
Lemma 9. If and , with the
precondition of regular-shape , we have the following estimation.
6. Error Estimates
The following theorem is the main result of this paper.
Theorem 1. Let and be the solution of (1) and (9), respectively. the following error estimates is true.
Proof. Let
By the error of Equation (23), we have
(26)
Substituting (13) into (26), we obtain
(27)
Let , combing the Equation (25) and (14), we have
That is
By Lemma 2 and Cauchy inequality, we have
(28)
By Gronwall Lemma, we have
(29)
By Cauchy inequality, we have
(30)
Then take the integration about t of both side of Equation (28)
Since , then
(31)
Combing the Equations (15), (29), (30) and triangle inequality, we have
(32)
(33)
Next, we proof the error estimate of pressure approximation , by using error Equation (23), we have
By using Lemma 2, Lemma 5 and Lemma 9, we obtain
By Lemma 4, we have
(34)
Next we seek error estimate , then take the derivation about t of both sides of Equation (27)
Let , take the derivation about t of both side of Equations (14) and (25), we obtain
That is
By Lemma 2 and Cauchy inequality, we have
That is
(35)
Since , that is
(36)
Combing the Equations (15) and triangle inequality, we have
(37)
Substituting (33) and (36) into (34), we have
This completes the proof. Thus, the error estimates of Theorem 1 hold. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.
Cite this paper
Ning, C. and Gu, H.M. (2018) Weak Galerkin Finite Element Method for the Unsteady Navier-Stokes Equation. American Journal of Computational Mathematics, 8, 108-119. https://doi.org/10.4236/ajcm.2018.81009
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