A locally calculable P^3-pressure using a P^4-velocity for incompressible Stokes equations

1 Introduction

High order finite element methods for incompressible Stokes equations have been developed well in 2 dimensional domain and analyzed along to the inf-sup 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 Scott-Vogelius 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 post-process [7].

In this paper, employing the precedented new error analysis, we will suggest a new finite element method to find a $P^{4}$ -velocity and a $P^{3}$ -pressure solving incompressible Stokes equations at low cost.

The method will solve first the decoupled equation for a divergence-free $P^{4}$ -velocity. Then, using the calculated velocity, we will define a $P^{3}$ -pressure by exploiting locally calculable components in the Falk-Neilan and Scott-Vogelius finite element methods [4, 6, 8]. The resulting $P^{3}$ -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 $P^{4}$ -velocity. Besides, the method overcomes the problem arising from the singular vertices or corners.

The suggested $P^{3}$ -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 $P^{4}$ -velocity will be offered. Based on the orthogonality of sting and non-sting functions, we will introduce an orthogonal decomposition of the space of $P^{3}$ -pressures in Section 3. The non-sting and constant components will be defined in Section 4. Then, Section 5 will be devoted to defining 3 sting components for regular vertices, singular non-corner vertices, singular corners, respectively. It will be done on clustering sting functions vertex-wisely. Finally, after summing up all the components to define a $P^{3}$ -pressure in Section 6, numerical tests will be presented in the last section.

Throughout the paper, for a set $S \subset R^{2}$ , standard notations for Sobolev spaces are employed and $L_{0}^{2} (S)$ is the space of all $f \in L^{2} (S)$ whose integrals over $S$ vanish. We will use $∥ \cdot ∥_{m, S}$ , $| \cdot |_{m, S}$ and $(\cdot, \cdot)_{S}$ for the norm, seminorm for $H^{m} (S)$ and $L^{2} (S)$ inner product, respectively. If $S = Ω$ , it may be omitted in the subscript. Denoting by $P^{k}$ , the space of all polynomials of degree less than or equals $k$ , $f {∣ ∣}_{S} \in P^{k}$ will mean that $f \in L^{2} (Ω)$ coincides with a polynomial in $P^{k}$ on $S$ .

2 Velocity from the decoupled equation

Let $Ω$ be a simply connected polygonal domain in $R^{2}$ . In this paper, we will approximate a pair of velocity and pressure $(u, p) \in [H_{0}^{1} (Ω)]^{2} \times L_{0}^{2} (Ω)$ which satisfies an incompressible Stokes equation:

(\nabla u, \nabla v) + (p, d i v v) + (q, d i v u) = (f, v) for all (v, q) \in [H_{0}^{1} (Ω)]^{2} \times L_{0}^{2} (Ω),

(1)

for a body force $f \in [L^{2} (Ω)]^{2}$ .

Given a family of shape-regular triangulations ${T_{h}}_{h > 0}$ of $Ω$ , define $P_{h}^{k} (Ω)$ as the following space of piecewise polynomials:

P_{h}^{k} (Ω) = {v_{h} \in L^{2} (Ω) : v_{h} {∣ ∣}_{K} \in P^{k} for all triangles K \in T_{h}}, k \geq 0.

Let $Σ_{h, 0}$ be a space of $C^{1}$ -Argyris $P^{5}$ triangle elements [2, 3] such that

Σ_{h, 0} = P_{h}^{5} (Ω) \cap H_{0}^{2} (Ω),

(2)

where

H_{0}^{2} (Ω) = {ϕ \in H^{2} (Ω) : ϕ, ϕ_{x}, ϕ_{y} \in H_{0}^{1} (Ω)} .

The degrees of freedom of $ϕ \in Σ_{h, 0}$ are the Hessians, gradients and values of $ϕ$ at vertices and its normal derivatives at midpoints of edges.

Define a divergence-free space $Z_{h, 0}$ as

Z_{h, 0} = {(ϕ_{h, y}, - ϕ_{h, x}) : ϕ_{h} \in Σ_{h, 0}} \subset [P_{h}^{4} (Ω) \cap H_{0}^{1} (Ω)]^{2} .

We note that

dim Σ_{h, 0} = dim Z_{h, 0} = 6 V^{in} + E^{in} + V^{bdy} - V^{cnr},

where $V^{in}, E^{in}, V^{bdy}$ , and $V^{cnr}$ are the numbers of interior vertices, interior edges, boundary vertices and corners, respectively [4].

Then, we can solve $u_{h} \in Z_{h, 0}$ satisfying the following decoupled equation:

(\nabla u_{h}, \nabla v_{h}) = (f, v_{h}) for all v_{h} \in Z_{h, 0} .

(3)

Theorem 2.1.

Let $(u, p) \in [H_{0}^{1} (Ω)]^{2} \times L_{0}^{2} (Ω)$ and $u_{h} \in Z_{h, 0}$ satisfy (1), (3

), respectively. Then we estimate

| u - u_{h} |_{1} \leq C h^{4} | u |_{5},

(4)

if $u \in [H^{5} (Ω)]^{2}$ , where $C$ is a constant independent of $h$ .

Proof.

Since $(u, p) \in [H_{0}^{1} (Ω)]^{2} \times L_{0}^{2} (Ω)$ satisfies (1), we have $d i v u = 0$ . Thus, there exists a stream function $ϕ \in H_{0}^{2} (Ω)$ such that [5]

u = (ϕ_{y}, - ϕ_{x}) .

(5)

Let $Π_{h} ϕ \in Σ_{h, 0}$ be a projection of $ϕ$ such that the Hessians, gradients and values of $ϕ - Π_{h} ϕ$ at vertices and its normal derivatives at midpoints of edges vanish. Then, since $u \in [H^{5} (Ω)]^{2}$ , by Bramble-Hilbert lemma, we have

| ϕ - Π_{h} ϕ |_{2} \leq C h^{4} | ϕ |_{6} .

(6)

If we denote $Π_{h} u = ((Π_{h} ϕ)_{y}, - (Π_{h} ϕ)_{x}) \in Z_{h, 0}$ , then from (5) and (6), we estimate

| u - Π_{h} u |_{1} \leq C h^{4} | u |_{5} .

(7)

We note that $(p, d i v v_{h}) = 0$ for all $v_{h} \in Z_{h, 0}$ . Thus, from (1) and (3), we deduce

(\nabla u - \nabla u_{h}, \nabla v_{h}) = 0 for % all v_{h} \in Z_{h, 0} .

It is written in the form:

(\nabla Π_{h} u - \nabla u_{h}, \nabla v_{h}) = (\nabla Π_{h} u - \nabla u, \nabla v_{h}) for all v_{h} \in Z_{h, 0} .

(8)

Then we can establish (4) from (7) and (8) with $v_{h} = Π_{h} u - u_{h} \in Z_{h, 0}$ . ∎

3 Orthogonal decomposition of $P^{3}$

For a triangle $K \in T_{h}$ and an integer $k \geq 0$ , define

P^{k} (K) = {q \in L^{2} (Ω) : q {∣ ∣}_{K} \in P^{k}, q = 0 on Ω ∖ K} .

In the remaining of the paper, we will use the following notations:

$C$ : a generic constant which does not depend on $h$ of $T_{h}$ ,
$K (V)$ : the union of all triangles in $T_{h}$ sharing a vertex $V$ as in Figure 4,
$ξ^{⊥}$ : the counterclockwise $90^{\circ}$
rotation of a vector
$ξ$ ,
$| S |$ : the area or length of a set $S$ ,
$\mathpzcm(f)$ : the average of a function $f$ over $Ω$ .

We assume the following on $T_{h}$ to exclude pathological meshes.

Assumption 3.1.

Every triangles in $T_{h}$ has at most one corner point of $\partial Ω$ .

3.1 sting function

Let $V$ be a vertex of a triangle $K$ . Then there exists a unique function $s_{V K} \in P^{3} (K)$ satisfying the following quadrature rule:

\int_{K} s_{V K} q d x d y = \frac{| K |}{100} q (V) for all q \in P^{3},

(9)

since the both sides of (9) are linear functionals on $P^{3}$ . If $ˆ K$ is a reference triangle with vertices $(0, 0), (1, 0), (0, 1)$ and $ˆ V = (0, 1)$ , we have

s_{ˆ V ˆ K} (x, y) = \frac{28}{5} y^{3} - \frac{63}{10} y^{2} + \frac{9}{5} y - \frac{1}{10},

(10)

as depicted in Figure 1-(a). Given a vertex $V$ of $K$ , we note that

s_{V K} = s_{ˆ V ˆ K} \circ F^{- 1} for an affine transformation F : ˆ K ⟶ K .

(11)

Thus, the values of $s_{V K}$ are inherited from those of $s_{ˆ V ˆ K}$ as

(12)

If $q = α s_{V K}$ for a scalar $α$ , we call it a sting function of $V$ on $K$ , named after the shape of its graph as in Figure 1-(a).

For a triangle $K$ , define a subspace of $P^{3} (K)$ as

S (K) =< s_{V_{1} K}, s_{V_{2} K}, s_{V_{3} K} >,

(13)

where $V_{1}, V_{2}, V_{3}$ are 3 vertices of $K$ . From (12), it is easy to prove that

dim S (K) = 3.

(14)

(a) a sting function $s_{V K}$ of $V$ on $K$

3.2 non-sting function

For a triangle $K$ , let

B (K) = {v \in [P^{4} (K)]^{2} : v = 0 on % \partial K},

(15)

and define a subspace of $P^{3} (K)$ as

N (K) = {d i v v : v \in B (K)} .

(16)

If $q \in N (K)$ , we will call it a non-sting function on $K$ . By definition in (15), (16), every non-sting function $q \in N (K)$ has the following properties:

q (V) = 0 for every vertex V of K, \int_{K} q d x d y = 0.

(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),

N (K) ⊥ S (K) .

(18)

The fact $dim B (K) = 6$ induces the following, with an aid of Lemma 3.1 below,

dim N (K) = 6.

(19)

Lemma 3.1.

If $v \in B (K)$ and $d i v v = 0$ , then $v = 0$ .

Proof.

Since $d i v v = 0$ on $K$ and $v = 0$ on $\partial K$ , there exists $ϕ \in P^{5}$ such that

(ϕ_{y}, - ϕ_{x}) = v on K, ϕ = 0 on \partial K .

Let $ℓ_{1}, ℓ_{2}, ℓ_{3}$ be 3 infinite lines containing 3 line segments of $\partial K$ , respectively. Then $ϕ, \nabla ϕ$ vanish on $ℓ_{1}, ℓ_{2}, ℓ_{3}$ . It implies that $ϕ$ vanishes on any line which passes 3 points in $ℓ_{1} \cup ℓ_{2} \cup ℓ_{3}$ . Thus we have $ϕ = 0$ and $v = 0$ on $K$ . ∎

The above lemma tells that $∥ d i v v ∥_{0, K}$ is a norm of $v \in B (K)$ . Furthermore, we can show that

| v |_{1, K} \leq C ∥ d i v v ∥_{0, K} for all v \in B (K) .

(20)

Actually, $N (K)$ in (16) is the space of all function $q \in P^{3} (K)$ satisfying (17) [6].

3.3 orthogonal decomposition

Let $1^{K} \in P^{3} (K)$ be a constant function of value 1 on $K$ . Then, we can decompose $P^{3} (K)$ as in the following lemma. We will notate

A ⊥ ⨁ B for A ⨁ B, if A ⊥ B .

Lemma 3.2.

P^{3} (K) = N (K) ⊥ ⨁ (< 1^{K} > ⨁ S (K)) .

(21)

Proof.

From (17), we have $< 1^{K} >⊥ N (K)$ . Thus, by (14), (18), (19), it is enough to prove (21) that

1^{K} \notin S (K) .

For the vertices $V_{1}, V_{2}, V_{3}$ of $K$ , let

s_{1}^{K} = \frac{5}{4} (s_{V_{1} K} + s_{V_{2} K} + s_{V_{3} K}) \in S (K) .

(22)

Then, by (12), $1^{K} - s_{1}^{K}$ vanishes at all vertices of $K$ . By (9), it means that

(23)

Assume $1^{K} \in S (K)$ . Then, $1^{K} - s_{1}^{K} \in S (K)$ and $1^{K} - s_{1}^{K} = 0$ from (23). It contradicts to

\int_{K} s_{1}^{K} d x d y = \frac{5}{4} \int_{K} s_{V_{1} K} + s_{V_{2} K} + s_{V_{3} K} d x d y = \frac{3}{80} | K | \neq | K | = \int_{K} 1^{K} d x d y .

∎

Let’s define the following subspaces of $P_{h}^{3} (Ω) = ⨁ K \in T_{h} P^{3} (K)$ :

N_{h} = ⨁ K \in T_{h} N (K), C_{h} = ⨁ K \in T_{h} < 1^{K} >, S_{h} = ⨁ K \in T_{h} S (K) .

(24)

Then by Lemma 3.2, we have

P_{h}^{3} (Ω) = N_{h} ⊥ ⨁ (C_{h} ⨁ S_{h}) .

(25)

3.4 decomposition of $Π_{h} p$

For $(u, p)$ satisfying (1), let $Π_{h} p \in P_{h}^{3} (Ω)$

be a Hermite interpolation of

p

such that

\nabla Π_{h} p (V) = \nabla p (V), Π_{h} p (V) = p (V), Π_{h} p (G) = p (G),

(26)

at all vertices $V$ and gravity centers $G$ of triangles in $T_{h}$ . Then, if $p \in H^{4} (Ω)$ , we have

∥ p - Π_{h} p ∥_{0} \leq C h^{4} | p |_{4} .

(27)

By (25), we can decompose $Π_{h} p$ into

Π_{h} p = Π_{h} p^{N} + Π_{h} p^{C} + Π_{h} p^{S},

(28)

for

Π_{h} p^{N} \in N_{h}, Π_{h} p^{C} \in C_{h}, Π_{h} p^{S} \in S_{h},

called the non-sting, constant and sting components of $Π_{h} p$ , respectively. We will approximate them component-wisely in next two sections, exploiting the following equation for $Π_{h} p$ ,

(Π_{h} p, d i v v) = (f, v) - (\nabla u, \nabla v) - (p - Π_{h} p, d i v v) for all v \in {[H_{0}^{1} (Ω)]}^{2} .

(29)

4 Non-sting and constant components

Define the following spaces:

\begin{matrix} V_{h, 0} & = {v_{h} \in [P_{h}^{4} (Ω) \cap H_{0}^{1} (Ω)]^{2} : \nabla v_{h} is continuous at all vertices in T_{h}}, Q_{h, 0} & = {q_{h} \in P_{h}^{3} (Ω) \cap L_{0}^{2} (Ω) : q_{h} is continuous at all vertices in T_{h}}, Q_{h, 00} & = {q_{h} \in Q_{h, 0} : q_{h} % vanishes at all corners of \partial Ω} . \end{matrix}

(30)

Then, for $f$ in (1), there exists a unique $(u_{h}, r_{h}) \in V_{h, 0} \times Q_{h, 00}$ satisfying

(31)

since the following Stokes complex is exact with $Σ_{h, 0}$ in (2) [4]:

0 ⟶ Σ_{h, 0} c u r l - - \to V_{h, 0} d i v - - \to Q_{h, 00} ⟶ 0.

We note that $u_{h}$ in (31) is the previously calculated $u_{h} \in Z_{h, 0}$ in (3) and $r_{h} \in Q_{h, 00}$ is the unique solution of the following equation,

(r_{h}, d i v v_{h}) = (f, v_{h}) - (\nabla u_{h}, \nabla v_{h}) for all v_{h} \in V_{h, 0} .

(32)

Similarly to (28), decompose $r_{h}$ into

r_{h} = r_{h}^{N} + r_{h}^{C} + r_{h}^{S} % for r_{h}^{N} \in N_{h}, r_{h}^{C} \in C_{h}, r_{h}^{S} \in S_{h} .

(33)

4.1 non-sting component $p_{h}^{N}$

For a triangle $K \in T_{h}$ , we note $B (K) \subset V_{h, 0}$ . Thus, by (24), (33) and orthogonality in Lemma 3.2, we can rewrite (32) into

(r_{h}^{N}, d i v v_{h}) = (f, v_{h}) - (\nabla u_{h}, \nabla v_{h}) for all% v_{h} \in B (K) .

(34)

Then by definition of $N (K)$ in (16), $r_{h}^{N} {∣ ∣}_{K} \in N (K)$ is determined by (34). Thus we can calculate $r_{h}^{N}$ locally on each triangle $K$ in $T_{h}$ .

Lemma 4.1.

Define $p_{h}^{N} = r_{h}^{N}$ in (33), then for $Π_{h} p^{N}$ in (28), we estimate

(35)

Proof.

From (29) and Lemma 3.2, $Π_{h} p^{N}$ satisfies that

(36)

If we set $e_{h}^{K} = Π_{h} p^{N} {∣ ∣}_{K} - r_{h}^{N} {∣ ∣}_{K} \in N (K)$ , from (34) and (36), we have

(37)

Then (35) comes from (20), (37) and the definition of $N (K)$ in (16). ∎

4.2 piecewise constant component $p_{h}^{C}$

Define a subspace $V_{h, 00}$ of $V_{h, 0}$ as

V_{h, 00} = {v_{h} \in V_{h, 0} : \nabla v_{h} vanishes at all vertices in T_{h}} .

(38)

Then, by quadrature rule in (9) and definition of $V_{h, 00}$ in (38), we can reduce (32) into

(r_{h}^{C}, d i v v_{h}) = (f, v_{h}) - (\nabla u_{h}, \nabla v_{h}) - (p_{h}^{N}, d i v v_{h}) for all v_{h} \in V_{h, 00},

(39)

where $p_{h}^{N} = r_{h}^{N}$ defined in Lemma 4.1.

For each triangle $K \in T_{h}$ , let $C_{K}$ be a constant such that

C_{K} = r_{h}^{C} {∣ ∣}_{K} .

(40)

If $K_{1}, K_{2}$ are two adjacent triangles sharing an edge, there exists a test function $v_{h} \in V_{h, 00}$ such that

the support of v_{h} is in K_{1} \cup K_{2}, \int_{K_{j}} d i v v_{h} d x d y = (- 1)^{j + 1}, j = 1, 2.

(41)

Then from (39)-(41), we have

C_{K_{1}} - C_{K_{2}} = (f, v_{h}) - (\nabla u_{h}, \nabla v_{h}) - (p_{h}^{N}, d i v v_{h}) .

(42)

Fix a triangle $K_{0}$ . Then for each triangle $K \in T_{h}$ , we can choose a telescoping sequence of triangles:

K_{0}, K_{1}, K_{2}, \dots, K_{n} = K,

(43)

so that $K_{j}, K_{j + 1}$ share an edge for $j = 0, 1, 2, \dots, n - 1$ . Then, $C_{K} - C_{K_{0}}$ can be obtained from

C_{K_{0}} - C_{K_{1}}, C_{K_{1}} - C_{K_{2}}, \dots, C_{K_{n - 1}} - C_{K_{n}},

which are calculated locally as in (42).

The knowledge of $C_{K} - C_{K_{0}}$ for all $K \in T_{h}$ gives us $CK0−\mathpzcm(rCh)$ from

0=∫ΩrCh−\mathpzcm(rCh) dxdy=∑K∈ThCK|K|−\mathpzcm(rCh)|Ω|=∑K∈Th(CK−CK0)|K|+(CK0−\mathpzcm(rCh))|Ω|.

Now, we can obtain $rCh−\mathpzcm(rCh)∈L20(Ω)$ , since for each $K \in T_{h}$ ,

rCh∣∣K−\mathpzcm(rCh)=CK−CK0+CK0−\mathpzcm(rCh).

Remark 4.2.

The difference $C_{K} - C_{K_{0}}$ does not depend on the choice of telescoping sequence in (43) from the existence of $r_{h}^{C}$ satisfying (39).

Lemma 4.3.

Define $pCh=rCh−\mathpzcm(rCh)$ for $r_{h}^{C}$ in (33), then for $Π_{h} p^{C}$ in (28), we estimate

∥ΠhpC−\mathpzcm(ΠhpC)−pCh∥0≤C(|u−uh|1+∥p−Πhp∥0).

(44)

Proof.

By (29) and quadrature rule in (9), $Π_{h} p^{C}$ satisfies for all $v_{h} \in V_{h, 00}$ ,

(45)

If we set $eh=ΠhpC−\mathpzcm(ΠhpC)−pCh∈P0h(Ω)∩L20(Ω)$ , from (39) and (45), it satisfies

(46)

for all $v_{h} \in V_{h, 00}$ .

We can establish Lemma 4.4 below that tells the existence of a nontrivial $v_{h} \in V_{h, 00}$ such that

β ∥ e_{h} ∥_{0} | v_{h} |_{1} \leq (e_{h}, d i v v_{h}) for β > 0 regardless of h .

(47)

Then (44) comes from (46), (47) and Lemma 4.1. ∎

Lemma 4.4.

$V_{h, 00} \times P_{h}^{0} (Ω) \cap L_{0}^{2} (Ω)$ satisfies the inf-sup condition.

Proof.

Given $c_{h} \in P_{h}^{0} (Ω) \cap L_{0}^{2} (Ω)$ , there exists a nontrivial $w \in [P_{h}^{2} (Ω) \cap H_{0}^{1} (Ω)]^{2}$ such that [1]

β ∥ c_{h} ∥_{0} | w |_{1} \leq (c_{h}, d i v w),

(48)

for a constant $β > 0$ regardless of $h$ .

For each triangle $K$ in $T_{h}$ , define $z_{K} \in [P^{4}]^{2}$ so that

\begin{matrix} \nabla z_{K} & = & \nabla w & at all % vertices of K, \int_{E} z_{K} d ℓ & = & 0 & for each edge E of K, z_{K} & = & 0 & at all 3 vertices and midpoints of 3 medians of % K . \end{matrix}

(49)

Then, for a reference triangle $^K$ and an affine map $F :^K ⟶ K$ , we have

| z_{K} |_{1, K} \leq C | z_{K} \circ F |_{1,^K} \leq C | w \circ F |_{1,^K} \leq C | w |_{1, K} .

(50)

If we define $z \in [P_{h}^{4} (Ω)]^{2}$ by $z {∣ ∣}_{K} = z_{K}$ for all $K \in T_{h}$ , then $z$ belongs to $[H_{0}^{1}]^{2}$ , since derivatives of $w$ along to edges are continuous. We note that $w \neq z$ . If so, $w = 0$ from the second and third conditions in (49).

Thus, from (49) and (50), $z$ satisfies

w - z \in V_{h, 00} ∖ {0}, | z |_{1} \leq C | w |_{1}, (1, d i v z)_{K} = 0 for all K .

(51)

Then, the following comes from (48) and (51), which completes the proof:

∎

For each triangle $K$ , define $s_{1}^{K}$ as in (22) so that $1^{K} - s_{1}^{K}$ vanishes at all vertices of $K$ . If we modify $r_{h}^{C}, r_{h}^{S}$ in (33) by

{r_{h}^{C}}^{'} = \sum K \in T_{h} r_{h}^{C} {∣ ∣}_{K} (1^{K} - s_{1}^{K}), {r_{h}^{S}}^{'} = r_{h}^{S} + \sum K \in T_{h} r_{h}^{C} {∣ ∣}_{K} s_{1}^{K},

we have

r_{h} = r_{h}^{N} + {r_{h}^{C}}^{'} + {r_{h}^{S}}^{'} .

(52)

We note ${r_{h}^{S}}^{'}$ is continuous at every vertex, since so is $r_{h} \in Q_{h, 00}$ and $r_{h}^{N}, {r_{h}^{C}}^{'}$ vanish at all vertices in $T_{h}$ . By (32) and (52), the vertex-continuous sting function ${r_{h}^{S}}^{'} \in S_{h}$ satisfies the following system:

({r_{h}^{S}}^{'}, d i v v_{h}) = (f, v_{h}) - (\nabla u_{h}, \nabla v_{h}) - (r_{h}^{N} + {r_{h}^{C}}^{'}, d i v v_{h}) for all v_{h} \in V_{h, 0} .

(53)

If $T_{h}$ has no singular corner, we could revise (53) slightly with the Falk-Neilan finite element space [4], to obtain a sting component in $S_{h}$ approximating $Π_{h} p^{S}$ of $Π_{h} p$ in (28).

To avoid a global system such as (53), we will seek locally calculable sting components vertex-wisely in the next section by unleashing the vertex-continuity.

5 Sting component

For each vertex $V$ , let $S (V)$ be the space of all sting functions of $V$ , that is,

S (V) =< s_{V K_{1}}, s_{V K_{2}}, \dots, s_{V K_{J}} >,

where $K_{1}, K_{2}, \dots, K_{J}$ are all triangles in $T_{h}$ sharing $V$ . An example of a function in $S (V)$ is represented in Figure 2.

Figure 2: a sting function of $V$ , $α_{1} s_{V K_{1}} + α_{2} s_{V K_{2}} + \dots + α_{5} s_{V K_{5}} \in S (V)$

Then we note that

S_{h} = ⨁ K \in T_{h} S (K) = ⨁ V : vertex S (V) .

Thus the sting component $Π_{h} p^{S}$ of $Π_{h} p$ in (28) can be decomposed by vertex into

Π_{h} p^{S} = \sum V : vertex Π_{h} p^{V} for Π_{h} p^{V} \in S (V) .

(54)

In this section, we will define $p_{h}^{V} \in S (V)$ to approximate $Π_{h} p^{V}$ for regular vertex $V = V_{r}$ , singular non-corner vertex $V = V_{s}$ , singular corner $V = V_{c}$ , respectively in order: $V_{r}, V_{s}, V_{c}$ .

5.1 two test functions for two adjacent triangles

Let $K_{1}, K_{2}$ be two adjacent triangles sharing an edge and a vertex $V$ as in Figure 3. Denote other 3 vertices and a unit tangent vector by $W_{0}, W_{1}, W_{2}, τ$ so that

¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ V W_{0} = K_{1} \cap K_{2}, τ = \frac{- --- \to V W_{0}}{| ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ V W_{0} |}, W_{j} \in K_{j} ∖ {V, W_{0}}, j = 1, 2.

Then, there exists a function $w \in P_{h}^{4} (Ω) \cap H_{0}^{1} (Ω)$ such that [7]

\frac{\partial w}{\partial τ} (V) = 1, \frac{\partial w}{} \partial τ (W_{0}) = 0, \int_{¯ ¯¯¯¯¯¯¯¯¯¯ ¯ V W_{0}} w d ℓ = 0, the support of w is K_{1} \cup K_{2} .

(55)

Assuming $K_{1}, K_{2}$ are counterclockwisely numbered with respect to $V$ , by simple calculation, we have

\nabla w {∣ ∣}_{K_{1}} (V) = \frac{| ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ V W_{0} |}{2 | K_{1} |} {- --- \to V W_{1}}^{⊥}, \nabla w {∣ ∣}_{K_{2}} (V) = - \frac{| ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ V W_{0} |}{2 | K_{2} |} {- --- \to V W_{2}}^{⊥} .

(56)

For a vector $ξ = (ξ_{1}, ξ_{2})$ , denote $w_{h}^{ξ} = w ξ = (ξ_{1} w, ξ_{2} w)$ . Then from (55) and (56), $d i v w_{h}^{ξ}$ vanishes at all vertices in $T_{h}$ except

d i v w_{h}^{ξ} {∣ ∣}_{K_{1}} (V) = \frac{| ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ V W_{0} |}{2 | K_{1} |} {- --- \to V W_{1}}^{⊥} \cdot ξ, d i v w_{h}^{ξ} {∣ ∣}_{K_{2}} (V) = - \frac{| ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ V W_{0} |}{2 | K_{2} |} {- --- \to V W_{2}}^{⊥} \cdot ξ .

(57)

Let $q_{h} \in S (K_{1}) ⨁ S (K_{2})$ be a sting function on $K_{1} \cup K_{2}$ . It is represented with some constants $α_{j}, β_{j}, j = 1, 2, 3$ as

Then, by (57) and quadrature rule of sting functions in (9), we have that

It can be written in simpler form:

\begin{matrix} (q_{h}, d i v w_{h}^{τ}) & = & \frac{ℓ_{1} ℓ_{0} sin θ_{1}}{200} α_{1} + \frac{ℓ_{0} ℓ_{2} sin θ_{2}}{200} β_{1} = \frac{1}{100} (| K_{1} | α_{1} + | K_{2} | β_{1}), (q_{h}, d i v w_{h}^{τ^{⊥}}) & = & \frac{ℓ_{1} ℓ_{0} cos θ_{1}}{200} α_{1} - \frac{ℓ_{0} ℓ_{2} cos θ_{2}}{200} β_{1}, \end{matrix}

(58)

where $θ$

A locally calculable P^3-pressure using a P^4-velocity for incompressible Stokes equations

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A stabilized formulation for the solution of the incompressible unsteady Stokes equations in the frequency domain

1 Introduction

2 Velocity from the decoupled equation

Theorem 2.1.

Proof.

3 Orthogonal decomposition of $P^{3}$

Assumption 3.1.

3.1 sting function

3.2 non-sting function

Lemma 3.1.

Proof.

3.3 orthogonal decomposition

Lemma 3.2.

Proof.

3.4 decomposition of $Π_{h} p$

4 Non-sting and constant components

4.1 non-sting component $p_{h}^{N}$

Lemma 4.1.

Proof.

4.2 piecewise constant component $p_{h}^{C}$

Remark 4.2.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

5 Sting component

5.1 two test functions for two adjacent triangles

A locally calculable P^3-pressure using a P^4-velocity for incompressible Stokes equations

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This week in AI

1 Introduction

2 Velocity from the decoupled equation

Theorem 2.1.

Proof.

3 Orthogonal decomposition of P3

Assumption 3.1.

3.1 sting function

3.2 non-sting function

Lemma 3.1.

Proof.

3.3 orthogonal decomposition

Lemma 3.2.

Proof.

3.4 decomposition of Πhp

4 Non-sting and constant components

4.1 non-sting component pNh

Lemma 4.1.

Proof.

4.2 piecewise constant component pCh

Remark 4.2.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

5 Sting component

5.1 two test functions for two adjacent triangles

3 Orthogonal decomposition of $P^{3}$

3.4 decomposition of $Π_{h} p$

4.1 non-sting component $p_{h}^{N}$

4.2 piecewise constant component $p_{h}^{C}$