1 Introduction
High order finite element methods for incompressible Stokes equations have been developed well in 2 dimensional domain and analyzed along to the infsup condition [4, 6, 8]
. They, however, endure their large degrees of freedom and have to avoid singular vertices or corners.
Recently, we have found a so called sting function causing the discrete Stokes equation to be singular on the presence of exactly singular vertices in the ScottVogelius finite element method. Even on nearly singular vertices, the pressure solution is easy to be spoiled. To fix the problem, a new error analysis was devised in a successive way and restored the ruined pressure by simple postprocess [7].
In this paper, employing the precedented new error analysis, we will suggest a new finite element method to find a velocity and a pressure solving incompressible Stokes equations at low cost.
The method will solve first the decoupled equation for a divergencefree velocity. Then, using the calculated velocity, we will define a pressure by exploiting locally calculable components in the FalkNeilan and ScottVogelius finite element methods [4, 6, 8]. The resulting pressure is analyzed to have the optimal order of convergence.
Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the velocity. Besides, the method overcomes the problem arising from the singular vertices or corners.
The suggested pressure consists of several components, each of whom needs others to be defined in a successive way. In the overall paper, the characteristics of sting functions depicted in Figure 1(a) will also play a key role as in the precedented work in [7].
The paper is organized as follows. In the next section, the detail on finding a velocity will be offered. Based on the orthogonality of sting and nonsting functions, we will introduce an orthogonal decomposition of the space of pressures in Section 3. The nonsting and constant components will be defined in Section 4. Then, Section 5 will be devoted to defining 3 sting components for regular vertices, singular noncorner vertices, singular corners, respectively. It will be done on clustering sting functions vertexwisely. Finally, after summing up all the components to define a pressure in Section 6, numerical tests will be presented in the last section.
Throughout the paper, for a set , standard notations for Sobolev spaces are employed and is the space of all whose integrals over vanish. We will use , and for the norm, seminorm for and inner product, respectively. If , it may be omitted in the subscript. Denoting by , the space of all polynomials of degree less than or equals , will mean that coincides with a polynomial in on .
2 Velocity from the decoupled equation
Let be a simply connected polygonal domain in . In this paper, we will approximate a pair of velocity and pressure which satisfies an incompressible Stokes equation:
(1) 
for a body force .
Given a family of shaperegular triangulations of , define as the following space of piecewise polynomials:
Let be a space of Argyris triangle elements [2, 3] such that
(2) 
where
The degrees of freedom of are the Hessians, gradients and values of at vertices and its normal derivatives at midpoints of edges.
Define a divergencefree space as
We note that
where , and are the numbers of interior vertices, interior edges, boundary vertices and corners, respectively [4].
Then, we can solve satisfying the following decoupled equation:
(3) 
Theorem 2.1.
3 Orthogonal decomposition of
For a triangle and an integer , define
In the remaining of the paper, we will use the following notations:

: a generic constant which does not depend on of ,

: the union of all triangles in sharing a vertex as in Figure 4,

: the area or length of a set ,

: the average of a function over .
We assume the following on to exclude pathological meshes.
Assumption 3.1.
Every triangles in has at most one corner point of .
3.1 sting function
Let be a vertex of a triangle . Then there exists a unique function satisfying the following quadrature rule:
(9) 
since the both sides of (9) are linear functionals on . If is a reference triangle with vertices and , we have
(10) 
as depicted in Figure 1(a). Given a vertex of , we note that
(11) 
Thus, the values of are inherited from those of as
(12) 
If for a scalar , we call it a sting function of on , named after the shape of its graph as in Figure 1(a).
For a triangle , define a subspace of as
(13) 
where are 3 vertices of . From (12), it is easy to prove that
(14) 
3.2 nonsting function
For a triangle , let
(15) 
and define a subspace of as
(16) 
If , we will call it a nonsting function on . By definition in (15), (16), every nonsting function has the following properties:
(17) 
An example of its graph is depicted in Figure 1(b).
Then, the following orthogonality is clear from (13), (17) and the quadrature rule in (9),
(18) 
The fact induces the following, with an aid of Lemma 3.1 below,
(19) 
Lemma 3.1.
If and , then .
Proof.
Since on and on , there exists such that
Let be 3 infinite lines containing 3 line segments of , respectively. Then vanish on . It implies that vanishes on any line which passes 3 points in . Thus we have and on . ∎
3.3 orthogonal decomposition
Let be a constant function of value 1 on . Then, we can decompose as in the following lemma. We will notate
Lemma 3.2.
(21) 
Proof.
3.4 decomposition of
For satisfying (1), let
be a Hermite interpolation of
such that(26) 
at all vertices and gravity centers of triangles in . Then, if , we have
(27) 
By (25), we can decompose into
(28) 
for
called the nonsting, constant and sting components of , respectively. We will approximate them componentwisely in next two sections, exploiting the following equation for ,
(29) 
4 Nonsting and constant components
Define the following spaces:
(30) 
Then, for in (1), there exists a unique satisfying
(31) 
since the following Stokes complex is exact with in (2) [4]:
We note that in (31) is the previously calculated in (3) and is the unique solution of the following equation,
(32) 
Similarly to (28), decompose into
(33) 
4.1 nonsting component
4.2 piecewise constant component
Define a subspace of as
(38) 
Then, by quadrature rule in (9) and definition of in (38), we can reduce (32) into
(39) 
where defined in Lemma 4.1.
For each triangle , let be a constant such that
(40) 
If are two adjacent triangles sharing an edge, there exists a test function such that
(41) 
(42) 
Fix a triangle . Then for each triangle , we can choose a telescoping sequence of triangles:
(43) 
so that share an edge for . Then, can be obtained from
which are calculated locally as in (42).
The knowledge of for all gives us from
Now, we can obtain , since for each ,
Remark 4.2.
Proof.
Lemma 4.4.
satisfies the infsup condition.
Proof.
For each triangle in , define so that
(49) 
Then, for a reference triangle and an affine map , we have
(50) 
If we define by for all , then belongs to , since derivatives of along to edges are continuous. We note that . If so, from the second and third conditions in (49).
For each triangle , define as in (22) so that vanishes at all vertices of . If we modify in (33) by
we have
(52) 
We note is continuous at every vertex, since so is and vanish at all vertices in . By (32) and (52), the vertexcontinuous sting function satisfies the following system:
(53) 
If has no singular corner, we could revise (53) slightly with the FalkNeilan finite element space [4], to obtain a sting component in approximating of in (28).
To avoid a global system such as (53), we will seek locally calculable sting components vertexwisely in the next section by unleashing the vertexcontinuity.
5 Sting component
For each vertex , let be the space of all sting functions of , that is,
where are all triangles in sharing . An example of a function in is represented in Figure 2.
In this section, we will define to approximate for regular vertex , singular noncorner vertex , singular corner , respectively in order: .
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