Weak Galerkin Finite Element Method for the Unsteady Stokes Equation ()
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.