The lowest-order stabilizer free Weak Galerkin Finite Element Method

1 Introduction

In this paper, we are concerned with the development of an SFWG finite element method using the following Poisson equation

(1)		$- Δ u$	$=$	$f in Ω,$
(2)		$u$	$=$	$0 on \partial Ω,$

as the model problem, where $Ω$ is a polygonal domain in $R^{2}$ . The variational formulation of the Poisson problem (1)-(2) is to seek $u \in H_{0}^{1} (Ω)$ such that

(3)

(\nabla u, \nabla v)

=

(f, v), \forall v \in H_{0}^{1} (Ω) .

The standard weak Glaerkin (WG) method for the problem (1)-(2) seeks weak Galerkin finite element solution $u_{h} = {u_{0}, u_{b}}$ such that

(4)

(\nabla_{w} u_{h}, \nabla_{w} v) + s (u_{h}, v)

=

(f, v),

for all $v = {v_{0}, v_{b}}$ with $v_{b} = 0 on \partial Ω$ , where $\nabla_{w}$ is the weak gradient operator and $s (u_{h}, v)$ in (4) is a stabilizer term that ensures a sufficient weak continuity for the numerical approximation. The WG method has been developed and applied to different types of problems, including convection-diffusion equations [7, 6], Helmholtz equations [9, 12, 5], Stokes flow [11, 10], and biharmonic problems [8]. Recently, Al-Taweel and Wang in [2], proposed the lowest-order weak Galerkin finite element method for solving reaction-diffusion equations with singular perturbations in $2 D$ . One of major sources of the complexities of the WG methods and other discontinuous finite element methods is the stabilization term.

A stabilizer free weak Galerkin finite element method is proposed by Ye and Zhang in [3] as a new method for the solution of the Poisson equation on polytopal meshes in 2D or 3D, where $(P_{k} (T), P_{k} (e), [P_{j} (T)]^{2})$ elements are used. It is shown that there is a $j_{0} > 0$ so that the SFWG method converges with optimal order of convergence for any $j \geq j_{0}$ . However, when $j$ is too large, the magnitude of the weak gradient can be extremely large, causing numerical instability. In [1], the optimal $j_{0}$ is given to improve the efficiency and avoid unnecessary numerical difficulties. In this setting, if $(P_{k} (T), P_{k} (e), [P_{j} (T)]^{2})$ elements are used for a triangular mesh, $j_{0} = k + 1$ , where $k \geq 1$ . In this paper, we propose a scheme using $(P_{0} (T), P_{1} (e), [P_{1} (T)]^{2})$ elements for triangular meshes with the optimal order of convergence, which is more efficient than using the regular WG method $(P_{0} (T), P_{1} (e), [P_{1} (T)]^{2})$ elements.

The goal of this paper is to develop the theoretical foundation for using the lowest-order SFWG scheme to solve the Poisson equation (1)-(2) on a triangle mesh in $2 D$ . The rest of this paper is organized as follows. In Section 2, the notations and finite element space are introduced. Section 3 is devoted to investigating the error equations and several other required inequalities. The main error estimate is studied in Section 4. In Section 5, we will derive the optimal order $L^{2}$ error estimates for the SFWG finite element method for solving the equations (1)-(2). Several numerical tests are presented in Section 6. Conclusions and some future research plans are summarized in Section 7.

2 Notations

In this section, we shall introduce some notations, and definitions.

Suppose $T_{h}$ is a quasi uniform triangular partition of $Ω$ . For every element $T \in T_{h}$ , denote $h_{T}$ as its diameter and $h = {max}_{T \in T_{h}} h_{T}$ . Let $E_{h}$ be the set of all the edges in $T_{h} .$ The weak Galerkin finite element space is defined as follows:

(5)

V_{h} = {(v_{0}, v_{b}) : v_{0} \in P_{0} (T), \forall T \in T_{h}, and v_{b} \in P_{1} (e), \forall e \in E_{h}} .

In this instance, the component $v_{0}$ symbolizes the interior value of $v$ , and the component $v_{b}$ symbolizes the edge value of $v$ on each $T$ and $e$ , respectively. Let $V_{h}^{0}$ be the subspace of $V_{h}$ defined as:

(6)

V_{h}^{0}

=

{v : v \in V_{h}, v_{b} = 0 on \partial Ω} .

For each element $T \in T_{h}$ , let $Q_{0}$ be the $L^{2}$ -projection onto $P_{0} (T)$ and let $Q_{h}$ be the $L^{2}$ -projection onto $[P_{1} (T)]^{2}$ . On each edge $e$ , denote by $Q_{b}$ the $L^{2}$ -projection operator onto $P_{1} (e)$ . Combining $Q_{0}$ and $Q_{b}$ , denote by $Q_{h} = {Q_{0}, Q_{b}}$ the $L^{2}$ -projection operator onto $V_{h} .$

For any $v = {v_{0}, v_{b}} \in V_{h}$ , the weak gradient $\nabla_{w} v \in [P_{1} (T)]^{2}$ is defined on $T$ as the unique polynomial satisfying

(7)

(\nabla_{w} v, \to q)_{T}

=

- (v_{0}, \nabla \cdot \to q)_{T} + ⟨ v_{b}, \to q \cdot \to n ⟩_{\partial T}, \forall \to q \in [P_{1} (T)]^{2},

where $\to n$

is the unit outward normal vector of

\partial T

For simplicity, we adopt the following notations,

(v, w)_{T_{h}} = \sum T \in T_{h} (v, w)_{T} = \sum T \in T_{h} \int_{T} v w d x,

{⟨ v, w ⟩}_{\partial T_{h}} = \sum T \in T_{h} {⟨ v, w ⟩}_{\partial T} = \sum T \in T_{h} \int_{\partial T} v w d x .

Stabilizer free Weak Galerkin Algorithm 1

A numerical solution for (1)-(2) can be obtain by finding $u_{h} = {v_{0}, v_{b}} \in V_{h}^{0}$ , such that the following equation holds

(8)

(\nabla_{w} u_{h}, \nabla_{w} v)_{T_{h}} = (f, v_{0}), \forall v = {v_{0}, v_{b}} \in V_{h}^{0} .

We define an energy norm $| | | \cdot | | |$ on $V_{h}$ as:

(9)

| | | v {| | |}^{2} = \sum T \in T_{h} (\nabla_{w} v, \nabla_{w} v)_{T} .

An $H^{1}$ semi norm on $V_{h}$ is defined as:

∥ v ∥_{1, h}^{2} = \sum T \in T_{h} (∥ \nabla v_{0} ∥_{T}^{2} + h_{T}^{- 1} ∥ v_{0} - v_{b} ∥_{\partial T}^{2}) .

Remark 1

$\nabla v_{0}$ in the above definition is simply a placeholder, since $\nabla v_{0} |_{T} = 0 for all T \in T_{h}$ .

Let

(10)

^Vh={(v0,vb)|v0∈Pk(T),vb∈Pk(e),∀T∈Th,e∈∂T}.

The following lemma shows that $∥ \cdot ∥_{1, h}$ and $| | | \cdot | | |$ defined in (9) are equivalent on ${^V}_{h}$ . (see [1]) Suppose that $\forall T \in T_{h}, T$ is convex with at most $μ$ edges and satisfies the following regularity conditions: for all edges $e_{t}$ and $e_{s}$ of $T$

(11)

| e_{s} | < α_{0} | e_{t} |;

for any two adjacent edges $e_{t} and e_{s}$ the angle $θ$ between them satisfies

(12)

θ_{0} < θ < π - θ_{0},

where $1 \leq α_{0} and θ_{0} > 0$ are independent of $T$ and $h$ . Let $j_{0} = k + μ - 2 or j_{0} = k + μ - 3$ when each edge of $T$ is parallel to another edge of $T$ . Denote by $j$ the degree of weak gradient. When $j \geq j_{0}$ , then there exist two constants $C_{1}, C_{2} > 0$ , such that for each $v={v0,vb}∈^Vh$ , the following hold true

C_{1} ∥ v ∥_{1, h, T} \leq | | | v {| | |}_{T} \leq C_{2} ∥ v ∥_{1, h, T}, \forall T \in T_{h},

where $C_{1} and C_{2}$ depend only on $α_{0} and θ_{0}$ .

Remark 2

If all $T$ ’s are either triangles or parallelograms, then $j_{0} = k + 1$ .

There exists $C > 0$ so that

For any $T \in T_{h}$ and $\to q \in [P_{1} (T)]^{2}$ , it follows from integration by part, trace inequality, and inverse inequality that

	$(\nabla_{w} v, \to q)_{T}$	$=$	$(\nabla v, \to q)_{T} + {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{\partial T}$
		$\leq$	$∥ \nabla v ∥_{T} ∥ \to q ∥_{T} + C h_{T}^{\frac{- 1}{2}} ∥ v_{0} - v_{b} ∥_{\partial T} ∥ \to q ∥_{T} .$

Letting $\to q = \nabla_{w} v$ yields the result.

The following lemma is one of the major results of this paper. For any $v \in V_{h}^{0}, if \nabla_{w} v |_{T} \in [P_{1} (T)]^{2}, \forall T \in T_{h}$ , then,

(13)

∥ v ∥_{1, h} = \sum T \in T_{h} \frac{1}{h_{T}^{\frac{1}{2}}} ∥ v_{0} - v_{b} ∥_{\partial T} \leq C | | | \nabla_{w} v | | | .

Let $T_{1} \in T_{h}, e_{1} \cup e_{2} \cup e_{3} = \partial T, e_{2} \in \partial T \cap \partial Ω$ . Since $d e g (v_{0} - v_{b})_{e_{1}} = d e g v_{0} = 0$ , it follows from Lemma 1 that

(14)

∥ v_{0} - v_{b} ∥_{e_{2}} \leq C h_{T_{1}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{1}} .

After a linear transformation we may assume

e_{1} = {(x, 0) | 0 \leq x \leq 1}, and e_{2} = {(0, y) | 0 \leq y \leq 1}

. Let

\to t

be a unit tangent vector to

e_{3}

such that

\to t \cdot {\to n}_{1} = \frac{1}{\sqrt{2}}

. Denote

v_{b} |_{e_{i}} = v_{b}^{(i)}

. Let

v_{0} - v_{b}^{(1)} (x) = a + b x

and let

\to q = (a + b x + b y) \to t

. Then

	$\| {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{e_{2}} \|$	$\leq$	$∥ v_{0} - v_{b} ∥_{e_{2}} ∥ \to q ∥_{e_{2}}$
		$=$	$∥ v_{0} - v_{b} ∥_{e_{2}} ∥ v_{0} - v_{b} ∥_{e_{1}} .$

Thus

$\frac{1}{\sqrt{2}} ∥ v_{0} - v_{b} ∥_{e_{1}}^{2}$	$=$	${⟨ v_{0} - v_{b}, (v_{0} - v_{b} + b y) \to t \cdot_{1} ⟩}_{e_{1}}$
	$\leq$	$(\nabla_{w} v, \to q)_{T_{1}} + \| {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{e_{2}} \|$
	$\leq$	$∥ \nabla_{w} v ∥_{T_{1}} ∥ \to q ∥_{T_{1}} + ∥ v_{0} - v_{b} ∥_{e_{2}} ∥ v_{0} - v_{b} ∥_{e_{1}} .$

Note that

	$∥ \to q ∥_{T_{1}}^{2}$	$=$	$\iint_{T} (a + b x + b y)^{2} d A$
		$=$	$\frac{1}{2} (a^{2} + \frac{4}{3} a b + \frac{b^{2}}{2}) = [\begin{matrix} a & b \end{matrix}] ⎡ ⎣ \begin{matrix} \frac{1}{2} & \frac{1}{3} \frac{1}{3} & \frac{1}{4} \end{matrix} ⎤ ⎦ [\begin{matrix} a b \end{matrix}] = α^{T} A α .$

Note also that

Since $B - A$ is positive definite,

(15)

∥ \to q ∥_{T_{1}}^{2} \leq ∥ v_{0} - v_{b} ∥_{e_{1}}^{2} .

Thus

\frac{1}{\sqrt{2}} ∥ v_{0} - v_{b} ∥_{e_{1}}^{2} \leq ∥ \nabla_{w} v ∥_{T_{1}} ∥ v_{0} - v_{b} ∥_{e_{1}} + ∥ v_{0} - v_{b} ∥_{e_{2}} ∥ v_{0} - v_{b} ∥_{e_{1}} .

By a scaling argument and

∥ v_{0} - v_{b} ∥_{e_{1}} \leq C (h_{T_{1}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{1}} + ∥ v_{0} - v_{b} ∥_{e_{2}}) \leq C h_{T_{1}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{1}} .

Similarly, we can show that

∥ v_{0} - v_{b} ∥_{e_{3}} \leq C h_{T_{1}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{1}} .

Thus we have

∥ v_{0} - v_{b} ∥_{\partial T_{1}} \leq C h_{T_{1}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{1}} .

Similarly, for any $T \in T_{h}$ with at least one edge on $\partial Ω$ ,

∥ v_{0} - v_{b} ∥_{\partial T} \leq C h_{T}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T} .

Let $B_{h}^{1} = {T \in T_{h}; T has at leat one edge on \partial Ω}$ . Now, let’s look at $T_{2} \in T_{h} ∖ B_{h}^{1}$ and $T_{1} \in B_{h}^{1}$ so that they share an edge $e_{1}$ . Without loss, we may assume that the vertices of $T_{2}$ are $(0, 0), (1, 0), and (0, 1)$ ,

e_{1} = {(x, 0) | 0 \leq x \leq 1},

and the other edge of $T_{1}$ is $(a_{1}, b_{1})$ , where $b_{1} < 0$ . Denote $v_{0}^{(i)} = v_{0} |_{T_{i}}, i = 1, 2$ . Since

∥ v_{0}^{(2)} - v_{b} ∥_{e_{1}} \leq ∥ v_{0}^{(2)} - v_{0}^{(1)} ∥_{e_{1}} + ∥ v_{0}^{(1)} - v_{b} ∥_{e_{1}}

and

∥ v_{0}^{(1)} - v_{b} ∥_{e_{1}} \leq C h_{T_{1}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{1}},

we need to show that

∥ v_{0}^{(2)} - v_{0}^{(1)} ∥_{e_{1}} \leq C (∥ \nabla_{w} v ∥_{T_{1}} + ∥ \nabla_{w} v ∥_{T_{2}}) .

Let $_{2} and_{4}$ be unit tangents to $e_{2} and e_{4}$ , respectively; $L_{3} and L_{5}$ be linear functions such that $L_{3} |_{e_{3}} = 0 and L_{5} |_{e_{5}} = 0$ . Let

_{1} = \to q |_{T_{1}} = L_{3} (x, y) (v_{0}^{(2)} - v_{0}^{(1)})_{2}

and

_{2} = \to q |_{T_{2}} = L_{5} (x, y) (v_{0}^{(2)} - v_{0}^{(1)})_{4} .

Then

_{i} \cdot_{2 i} = 0,_{i} |_{e_{2 i + 1}} = 0, i = 1, 2.

Scale $L_{1} and L_{3}$ if necessary so that

- L_{3} (0, 0)_{2} \cdot {_{1}}^{(1)} = 1 = L_{5} (0, 0)_{4} \cdot {_{1}}^{(2)},

where $\to n_{1}^{(i)}$ is the unit outwards normal vector of $e_{1} \in \partial T_{i}, i = 1, 2$ .
Since $L_{5} (1, 0)_{2} \cdot {_{1}}^{(2)} = 0$ ,

L_{5} (x, y)_{4} \cdot {_{1}}^{(2)} = {~ L}_{5} (x, y) = 1 - x - y .

Similarly

L_{3} (x, y)_{2} \cdot {_{1}}^{(2)} = {^L}_{3} (x, y) = 1 - x + α y, for some α .

It follows from the shape regularity assumptions that the slope of $e_{3}, \frac{1}{α}$ , satisfies $| \frac{1}{α} | \geq α_{0} > 0$ for some $α_{0}$ . Since $_{3} |_{e_{1}} =_{5} |_{e_{1}}$ ,

	$(\nabla_{w} v, \to q)_{T_{1} \cup T_{2}}$	$=$	${⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{\partial T_{1}} + {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{\partial T_{2}}$
		$=$	${⟨ v_{0}^{(2)} - v_{0}^{(1)}, (v_{0}^{(2)} - v_{0}^{(1)})_{3} ⟩}_{e_{1}} .$

Note that $0 \leq_{5} (x, y) \leq 1 on T_{1} and 0 \leq_{3} (x, y) \leq 1 on T_{2}$ . Then

$(\nabla_{w} v, \to q)_{T_{1} \cup T_{2}}$	$=$	${⟨ v_{0}^{(2)} - v_{0}^{(1)}, (v_{0}^{(2)} - v_{0}^{(1)})_{3} (x, y) ⟩}_{e_{1}}$
	$=$	$\int_{0}^{1} (v_{0}^{(2)} - v_{0}^{(1)})^{2} (1 - x) d x$
	$=$	$\frac{1}{2} ∥ v_{0}^{(2)} - v_{0}^{(1)} ∥_{e_{1}}^{2} .$

Thus

∥ \to q ∥_{T_{2}}^{2} = \iint_{T_{2}} (1 - x - y)^{2} (v_{0}^{(2)} - v_{0}^{(1)})^{2} d A = \frac{1}{12} (v_{0}^{(2)} - v_{0}^{(1)})^{2} \leq ∥ v_{0}^{(2)} - v_{0}^{(1)} ∥_{e_{1}}^{2} .

Similarly,

∥ \to q ∥_{T_{1}}^{2} = \frac{1}{4} (1 + α + \frac{α^{2}}{3}) (v_{0}^{(2)} - v_{0}^{(1)})^{2} \leq \frac{1}{4} (1 + \frac{1}{α_{0}} + \frac{1}{3 α_{0}^{2}}) ∥ v_{0}^{(2)} - v_{0}^{(1)} ∥_{e_{1}}^{2} .

Thus

	$\| (\nabla_{w} v, \to q)_{T_{1} \cup T_{2}} \|$	$\leq$	$∥ \nabla_{w} v ∥_{T_{1}} ∥ \to q ∥_{T_{1}} + ∥ \nabla_{w} v ∥_{T_{2}} ∥ \to q ∥_{T_{2}}$
		$\leq$	$C ∥ \nabla_{w} v ∥_{T_{1} \cup T_{2}} \cdot ∥ v_{0}^{(2)} - v_{0}^{(1)} ∥_{e_{1}} .$

Thus after a scaling we have

∥ v_{0}^{(2)} - v_{0}^{(1)} ∥_{e_{1}} \leq C h_{T_{2}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{1} \cup T_{2}} .

Now let’s look at $e_{4} \in \partial T_{2}$ . Let

\to q = (v_{0} - v_{b}^{(4)} + b x)_{5}, where v_{0} - v_{b}^{(4)} = a + b y .

Then

(\nabla_{w} v, \to q)_{T_{2}} = {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{e_{1}} + {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{e_{4}},

which implies

| (\nabla_{w} v, \to q)_{T_{2}} | \geq | {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{e_{4}} | - | {⟨ v_{0} - v_{b}, \to q \cdot \to n ⟩}_{e_{1}} | .

Thus

|_{5} \cdot_{4} | ∥ v_{0} - v_{b} ∥_{e_{4}}^{2} \leq ∥ \nabla_{w} v ∥_{T_{2}} ∥ \to q ∥_{T_{2}} + ∥ v_{0} - v_{b} ∥_{e_{1}} ∥ \to q ∥_{e_{1}} .

Since

∥ \to q ∥_{T_{2}}^{2} = \frac{1}{2} (a^{2} + \frac{4}{3} a b + \frac{b^{2}}{2}) and ∥ v_{0} - v_{b} ∥_{e_{4}}^{2} = a^{2} + a b + \frac{b^{2}}{3},

∥ v_{0} - v_{b} ∥_{e_{4}}^{2} \geq ∥ \to q ∥_{T_{2}}^{2} .

Note that

∥ \to q ∥_{e_{1}}^{2} = ∥ \to q ∥_{e_{4}}^{2} = ∥ v_{0} - v_{b ∥_{e}}^{2} .

It follows from

|_{5} \cdot_{4} | ∥ v_{0} - v_{b} ∥_{e_{4}}^{2} \leq ∥ \nabla_{w} v ∥_{T_{2}} ∥ \to q ∥_{T_{2}} + C ∥ v_{0} - v_{b} ∥_{e_{1}} ∥ \to q ∥_{e_{1}},

that

∥ v_{0} - v_{b} ∥_{e_{4}} \leq C ∥ \nabla_{w} v ∥_{T_{2}} .

By a scaling argument,

∥ v_{0} - v_{b} ∥_{e_{4}} \leq C h_{T_{2}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{2}} .

Similarly,

∥ v_{0} - v_{b} ∥_{e_{5}} \leq C h_{T_{2}}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{2}} .

Thus

∥ v_{0} - v_{b} ∥_{\partial T_{2}} \leq C h_{T}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{T_{2} \cup T_{1}} \leq C h_{T_{2}}^{\frac{1}{2}} (∥ \nabla_{w} v ∥_{T_{1}} + ∥ \nabla_{w} v ∥_{T_{2}}) .

Now let $B_{h}^{2} = {T \in T_{h}; T \in B_{h}^{1} or T shares one % edge with some T^{'} \in B_{h}^{1}}$ . So far we have shown that $\forall T \in B_{h}^{2}$ ,

∥ v_{0} - v_{b} ∥_{\partial T} \leq C h_{T}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{\cup T_{i}} \leq C h_{T}^{\frac{1}{2}} \sum ∥ \nabla_{w} v ∥_{T_{i}},

where $T_{i} \in B_{h}^{2}$ and shares at least one edge with $T$ . This process can be continued, until we have visited every $T \in T_{h}$ and have

∥ v_{0} - v_{b} ∥_{\partial T} \leq C h_{T}^{\frac{1}{2}} ∥ \nabla_{w} v ∥_{\cup T_{i}} \leq C h_{T}^{\frac{1}{2}} \sum ∥ \nabla_{w} v ∥_{T_{i}},

where $T_{i}$ shares at least $1$ edge with $T$ . (13) follows from this immediately.

Combining Lemmas 2 and 2, we have the following theorem. There exists $C_{1} > 0 and C_{2} > 0$ such that

3 Error equation

In this section, we derive the error estimate Algorithm 1.

For any function $ψ \in H^{1} (T)$ , the following trace inequality holds true:

(16)

∥ ψ ∥_{e}^{2} \leq C (h_{T}^{- 1} ∥ ψ ∥_{T}^{2} + h_{T} ∥ \nabla ψ ∥_{T}^{2}) .

(Inverse Inequality) There exists a constants $C$ such that for any piecewise polynomial $ψ |_{T} \in P_{k} (T)$ ,

(17)

| | \nabla ψ | |_{T} \leq C h_{T}^{- 1} | | ψ | |_{T}, \forall T \in T_{h} .

Let $u \in H_{0}^{2 + i} (Ω), i = 0, 1,$ be the solution of the problem and $T_{h}$ be a finite element partition of $Ω$ satisfying the shape regularity assumptions. Then, the $L^{2}$ projections $Q_{0} and Q_{h}$ satisfies

(18)		$\sum T \in T_{h} (∥ u - Q_{0} u ∥_{T}^{2} + h_{T}^{2} ∥ \nabla (u - Q_{0} u) ∥_{T}^{2}) \leq C h^{2} ∥ u ∥_{1}^{2},$
(19)		$\sum T \in T_{h} (∥ \nabla u - Q_{h} \nabla u ∥_{T}^{2} + h_{T}^{2} ∥ \nabla u - Q_{h} \nabla u) ∥_{1, T}^{2}) \leq C h^{2 (1 + i)} ∥ u ∥_{2 + i}^{2}, i = 0, 1.$

Let $ϕ \in H^{1} (Ω)$ . Then for each element $T \in T_{h}$ , we have

(20)

Q_{h} (\nabla ϕ) = \nabla_{w} Q_{h} ϕ .

By definition (7) and integration by parts, for each $\to q \in [P_{1} (T)]^{2}$ we have

$(Q_{h} (\nabla ϕ), \to q)_{T}$	$=$	$- (ϕ, \nabla \cdot \to q)_{T} + {⟨ ϕ, \to q \cdot \to n ⟩}_{\partial T}$
	$=$	$- (Q_{0} ϕ, \nabla \cdot \to q)_{T} + {⟨ Q_{b} ϕ, \to q \cdot \to n ⟩}_{\partial T}$
	$=$	$(\nabla_{w} Q_{h} ϕ, \to q)_{T} .$

which implies (20). Let $ϕ \in H^{1} (Ω)$ . Then for all $v \in V_{h}^{0}$ , we have

(21)

(\nabla ϕ, \nabla v_{0})_{T} = (\nabla_{w} (Q_{h} ϕ), \nabla_{w} v)_{T}

+

{⟨ (Q_{h} (\nabla ϕ) \cdot \to n, v_{0} - v_{b} ⟩}_{\partial T} .

Let $Q_{h} (\nabla ϕ) = \to q$ and $Q_{h} ϕ = P$ . By Lemma 3 $\to q = \nabla_{w} P$ .

(22)		$(\to q, \nabla v_{0})_{T}$	$=$	$- (\nabla \cdot \to q, v_{0})_{T} + {⟨ \to q \cdot \to n, v_{0} ⟩}_{\partial T},$
(23)		$(\to q, \nabla_{w} v)_{T}$	$=$	$- (\nabla \cdot \to q, v_{0})_{T} + {⟨ \to q \cdot \to n, v_{b} ⟩}_{\partial T},$

implies

	$(\to q, \nabla v_{0})_{T}$	$=$	$(\to q, \nabla_{w} v)_{T} + {⟨ \to q \cdot \to n, v_{0} - v_{b} ⟩}_{\partial T}$
		$=$	$(\nabla_{w} P, \nabla_{w} v)_{T} + {⟨ \to q \cdot \to n, v_{0} - v_{b} ⟩}_{\partial T},$

which completes the proof. Let $e_{h} = Q_{h} u - u_{h} \in V_{h}$ . Then for any $v \in V_{h}^{0}$ , we have

(24)

(\nabla_{w} e_{h}, \nabla_{w} v)_{T_{h}} = ℓ (u, v),

where $ℓ (u, v)$ is defined as follows,

ℓ (u, v) = \sum T \in T_{h} ⟨ (\nabla u - Q_{h} \nabla u) \cdot \to n, v_{0} - v_{b} ⟩_{\partial T} .

Testing the equation (1) by $v = {v_{0}, v_{b}} \in V_{h}$ and using the fact that $\sum_{T \in T_{h}} {⟨ \nabla u \cdot \to n, v_{b} ⟩}_{\partial T} = 0$ , we arrive at

The lowest-order stabilizer free Weak Galerkin Finite Element Method

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The lowest-order stabilizer free Weak Galerkin Finite Element Method

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2 Notations

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