A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

1 Introduction

The weak Galerkin finite element method is an effective and flexible numerical technique for solving partial differential equations. The main idea of weak Galerkin finite element methods is the use of weak functions and their corresponding discrete weak derivatives in algorithm design. The WG method was first introduced in [13, 14] and then has been applied to solve various partial differential equations [2, 4, 3, 5, 7, 8, 9, 10, 6, 11, 12, 15].

A stabilizing/penalty term is often essential in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous finite element methods simplifies finite element formulations and reduces programming complexity. Stabilizer free WG finite element methods have been studied in [16, 1, 20]. The idea is increasing the connectivity of a weak function across element boundary by raising the degree of polynomials for computing weak derivatives. In [16], we have proved that a stabilizer can be removed from the WG finite element formulation for the WG element $(P_{k} (T), P_{k} (e), [P_{j} (T)]^{d})$ if $j \geq k + n - 1$ , where $n$ is the number of edges/faces of an element. The condition $j \geq k + n - 1$ has been relaxed in [1]. Stabilizer free DG methods have also been developed in [17, 18].

For simplicity, we demonstrate the idea by using the second order elliptic problem that seeks an unknown function $u$ satisfying

(1.1)		$- \nabla \cdot (a \nabla u)$	$=$	$f in Ω,$
(1.2)		$u$	$=$	$g on \partial Ω,$

where $Ω$ is a polytopal domain in $R^{d}$ , $\nabla u$ denotes the gradient of the function $u$ , and $a$ is a symmetric $d \times d$ matrix-valued function in $Ω$ . We shall assume that there exist two positive numbers $λ_{1}, λ_{2} > 0$ such that

(1.3)

λ_{1} ξ^{t} ξ \leq ξ^{t} a ξ \leq λ_{2} ξ^{t} ξ, \forall ξ \in R^{d} .

Here $ξ$

is understood as a column vector and

ξ^{t}

is the transpose of

ξ

The goal of this paper is to propose and analyze a stabilizer free weak Galerkin method for (1.1)-(1.2) by using less number of unknowns than that of [16] without compromising the order of convergence. The WG scheme will use the configuration of $(P_{k} (T), P_{k - 1} (e)$ , $P_{j} (T)^{d})$ . For the WG element $(P_{k} (T), P_{k - 1} (e)$ , $P_{j} (T)^{d})$ , the stabilizer free WG method with the standard definition of weak gradient only produces suboptimal convergence rates in both energy norm and the $L^{2}$ norm, shown in Table 1.1 ([19]).

$P_{k} (T)$	$P_{k - 1} (e)$	$[P_{k + 1} (T)]^{d}$	$r_{1}$	$r_{2}$
$P_{1} (T)$	$P_{0} (e)$	$[P_{2} (T)]^{2}$	$0$	$0$
$P_{2} (T)$	$P_{1} (e)$	$[P_{3} (T)]^{2}$	$1$	$2$
$P_{3} (T)$	$P_{2} (e)$	$[P_{4} (T)]^{2}$	$2$	$3$

Table 1.1: Weak gradient calculated by (1.4),

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

The standard definition for a weak gradient $\nabla_{w} v$ of a weak function $v = {v_{0}, v_{b}}$ is a piecewise vector valued polynomial such that on each $T \in T_{h}$ , $\nabla_{w} v \in [P_{k + 1} (T)]^{2}$ satisfies ([13, 14, 8])

(1.4)

(\nabla_{w} v, q)_{T} = - (v_{0}, \nabla \cdot q)_{T} + ⟨ v_{b}, q \cdot n ⟩_{\partial T} \forall q \in [P_{j} (T)]^{2} .

A new way of defining weak gradient is introduced in (2.3) for the WG element $(P_{k} (T), P_{k - 1} (e), [P_{j} (T)]^{2})$ such that the our new corresponding stabilizer free WG approximation converges to the true solution with optimal order convergence rates, shown in Table 1.2.

$P_{k} (T)$	$P_{k - 1} (e)$	$[P_{k + 1} (T)]^{d}$	$r_{1}$	$r_{2}$
$P_{1} (T)$	$P_{0} (e)$	$[P_{2} (T)]^{2}$	$1$	$2$
$P_{2} (T)$	$P_{1} (e)$	$[P_{3} (T)]^{2}$	$2$	$3$
$P_{3} (T)$	$P_{2} (e)$	$[P_{4} (T)]^{2}$	$3$	$4$

Table 1.2: Weak gradient calculated by (2.3),

| | | \cdot | | | = O (h^{r_{1}})

and

∥ \cdot ∥ = O (h^{r_{2}})

We also prove the optimal convergence rates theoretically for the stabilizer free WG approximation in an energy norm and in the $L^{2}$ norm. The numerical examples are tested on different finite element partitions.

2 Weak Galerkin Finite Element Schemes

Let $T_{h}$ be a partition of the domain $Ω$ consisting of polygons in two dimension or polyhedra in three dimension satisfying a set of conditions specified in [14]. Denote by $E_{h}$ the set of all edges/faces in $T_{h}$ , and let $E_{h}^{0} = E_{h} ∖ \partial Ω$ be the set of all interior edges/faces. For simplicity, we will use term edge for edge/face without confusion.

For a given integer $k \geq 1$ , let $V_{h}$ be the weak Galerkin finite element space associated with $T_{h}$ defined as follows

(2.1)

V_{h} = {v = {v_{0}, v_{b}} : v_{0} |_{T} \in P_{k} (T), v_{b} |_{e} \in P_{k - 1} (e), e \subset \partial T, T \in T_{h}}

and its subspace $V_{h}^{0}$ is defined as

(2.2)

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

We would like to emphasize that any function $v \in V_{h}$ has a single value $v_{b}$ on each edge $e \in E_{h}$ .

For $v = {v_{0}, v_{b}} \in V_{h} \cup H^{1} (Ω)$ , a weak gradient $\nabla_{w} v$ is a piecewise vector valued polynomial such that on each $T \in T_{h}$ , $\nabla_{w} v |_{T} \in [P_{j} (T)]^{d}$ satisfies

(2.3)

(\nabla_{w} v, q)_{T} = (\nabla v_{0}, q)_{T} + ⟨ Q_{b} (v_{b} - v_{0}), q \cdot n ⟩_{\partial T} \forall q \in [P_{j} (T)]^{d},

where $j > k$ depends on the shape of the elements and will be determined later. In the above equation, we let $v_{0} = v$ and $v_{b} = v$ if $v \in H^{1} (Ω)$ .

For simplicity, we adopt the following notations,

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

Let $Q_{0}$ and $Q_{b}$ be the two element-wise defined $L^{2}$ projections onto $P_{k} (T)$ and $P_{k - 1} (e)$ on each $T \in T_{h}$ , respectively. Define $Q_{h} u = {Q_{0} u, Q_{b} u} \in V_{h}$ . Let $Q_{h}$ be the element-wise defined $L^{2}$ projection onto $[P_{j} (T)]^{d}$ on each element $T \in T_{h}$ .

Weak Galerkin Algorithm 1

A numerical approximation for (1.1)-(1.2) can be obtained by seeking $u_{h} = {u_{0}, u_{b}} \in V_{h}$ satisfying $u_{b} = Q_{b} g$ on $\partial Ω$ and the following equation:

(2.4)

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

The following lemma reveals a nice property of the weak gradient. Let $ϕ \in H^{1} (Ω)$ , then on any $T \in T_{h}$ , we have

(2.5)

\nabla_{w} ϕ

=

Q_{h} \nabla ϕ .

The definition of weak gradient (2.3) implies that for any $q \in [P_{j} (T)]^{d}$

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

which implies (2.5).

For any function $φ \in H^{1} (T)$ , the following trace inequality holds true (see [14] for details):

(2.6)

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

3 Well Posedness

For any $v \in V_{h} \cup H^{1} (Ω)$ , define two semi-norms

(3.1)			$=$	$(\nabla_{w} v, \nabla_{w} v)_{T_{h}},$
(3.2)			$=$	$(a \nabla_{w} v, \nabla_{w} v)_{T_{h}} .$

It follows from (1.3) that there exist two positive constants $α$ and $β$ such that

(3.3)

We introduce a discrete $H^{1}$ semi-norm as follows:

(3.4)

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

It is easy to see that $∥ v ∥_{1, h}$ defines a norm in $V_{h}^{0}$ .

Next we will show that $| | | \cdot | | |$ also defines a norm for $V_{h}^{0}$ by proving the equivalence of $| | | \cdot | | |$ and $∥ \cdot ∥_{1, h}$ in $V_{h}$ . First we need the following lemma.

([18]) Let $T$ be a convex, shape-regular $n$ -polygon/polyhedron of size $h_{T}$ . Let $e \in \partial T$ be an edge/face-polygon of $T$ , of size $C h_{T}$ . Let $v_{h} \in V_{h}$ and $v_{h} = {v_{0}, v_{b}}$ on $T$ . Then there exists a polynomial $q \in [P_{j} (T)]^{d}$ , $j = n + k - 1$ , such that

(3.5)	$- (\nabla v_{0}, q)_{T}$	$= 0,$
(3.6)	$⟨ Q_{b} (v_{0} - v_{b}), q \cdot n ⟩_{e}$	$= ∥ Q_{b} (v_{0} - v_{b}) ∥_{e}^{2} \forall e \subset \partial T,$
(3.7)	$∥ q ∥_{T}^{2}$	$\leq C h_{T} ∥ Q_{b} (v_{0} - v_{b}) ∥_{e}^{2} .$

There exist two positive constants $C_{1}$ and $C_{2}$ such that for any $v = {v_{0}, v_{b}} \in V_{h}$ , we have

(3.8)

C_{1} ∥ v ∥_{1, h} \leq | | | v | | | \leq C_{2} ∥ v ∥_{1, h} .

For any $v = {v_{0}, v_{b}} \in V_{h}$ , it follows from the definition of weak gradient (2.3) and integration by parts that on each $T \in T_{h}$

(3.9)

(\nabla_{w} v, q)_{T} = (\nabla v_{0}, q)_{T} + ⟨ Q_{b} (v_{b} - Q v_{0}), q \cdot n ⟩_{\partial T} \forall q \in [P_{j} (T)]^{d} .

By letting $q = \nabla_{w} v |_{T}$ in (3.9) we arrive at

(\nabla_{w} v, \nabla_{w} v)_{T} = (\nabla v_{0}, \nabla_{w} v)_{T} + ⟨ Q_{b} (v_{b} - v_{0}), \nabla_{w} v \cdot n ⟩_{\partial T} .

Letting $q = \nabla v_{0} |_{T}$ in (3.9) implies

(3.10)

(\nabla_{w} v, \nabla v_{0})_{T} = (\nabla v_{0}, \nabla v_{0})_{T} + ⟨ Q_{b} (v_{b} - v_{0}), \nabla v_{0} \cdot n ⟩_{\partial T} .

From the trace inequality (2.6) and the inverse inequality we have

	$∥ \nabla_{w} v ∥_{T}^{2}$	$\leq$	$∥ \nabla v_{0} ∥_{T} ∥ \nabla_{w} v ∥_{T} + ∥ Q_{b} (v_{0} - v_{b}) ∥_{\partial T} ∥ \nabla_{w} v ∥_{\partial T}$
		$\leq$	$∥ \nabla v_{0} ∥_{T} ∥ \nabla_{w} v ∥_{T} + C h_{T}^{- 1 / 2} ∥ Q_{b} (v_{0} - v_{b}) ∥_{\partial T} ∥ \nabla_{w} v ∥_{T},$

which implies

∥ \nabla_{w} v ∥_{T} \leq C (∥ \nabla v_{0} ∥_{T} + h_{T}^{- 1 / 2} ∥ Q_{b} (v_{0} - v_{b}) ∥_{\partial T}),

and consequently

| | | v | | | \leq C_{2} ∥ v ∥_{1, h} .

Next we will prove $C_{1} ∥ v ∥_{1, h} \leq | | | v | | |$ . First we need to prove

(3.11)

h_{e}^{- 1 / 2} ∥ Q_{b} (v_{0} - v_{b}) ∥_{e} \leq C ∥ \nabla_{w} v ∥_{T} .

For $e \in E_{h}$ and $T \in T_{h}$ with $e \subset \partial T$ , it has been proved in Lemma 3 that there exists $q_{0} \in [P_{j} (T)]^{d}$ such that

(3.12)

(\nabla v_{0}, q_{0})_{T} = 0, ⟨ Q_{b} (v_{b} - v_{0}), q_{0} \cdot n ⟩_{\partial T} = ∥ Q_{b} (v_{0} - v_{b}) ∥_{\partial T}^{2},

and

(3.13)

∥ q_{0} ∥_{T} \leq C h_{T}^{1 / 2} ∥ Q_{b} (v_{b} - v_{0}) ∥_{\partial T} .

Substituting $q_{0}$ into (3.9), we get

(3.14)

(\nabla_{w} v, q_{0})_{T} = ∥ Q_{b} (v_{b} - v_{0}) ∥_{\partial T}^{2} .

It follows from Cauchy-Schwarz inequality and (3.13) that

∥ Q_{b} (v_{b} - v_{0}) ∥_{\partial T}^{2} \leq C ∥ \nabla_{w} v ∥_{T} ∥ q_{0} ∥_{T} \leq C h_{T}^{1 / 2} ∥ \nabla_{w} v ∥_{T} ∥ Q_{b} (v_{0} - v_{b}) ∥_{\partial T},

which implies

(3.15)

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

It follows from (3.10), the trace inequality, the inverse inequality and (3.15),

	$∥ \nabla v_{0} ∥_{T}^{2}$	$\leq ∥ \nabla_{w} v ∥_{T} ∥ \nabla v_{0} ∥_{T} + C h_{T}^{- 1 / 2} ∥ Q_{b} (v_{0} - v_{b}) ∥_{\partial T} ∥ \nabla v_{0} ∥_{T}$
		$\leq C ∥ \nabla_{w} v ∥_{T} ∥ \nabla v_{0} ∥_{T},$

which implies

∥ \nabla v_{0} ∥_{T} \leq C ∥ \nabla_{w} v ∥_{T} .

Combining the above estimate and (3.15), we prove the lower bound of (3.8) and complete the proof of the lemma.

The weak Galerkin finite element scheme (2.4) has a unique solution.

If $u_{h}^{(1)}$ and $u_{h}^{(2)}$ are two solutions of (2.4), then $ε_{h} = u_{h}^{(1)} - u_{h}^{(2)} \in V_{h}^{0}$ would satisfy the following equation

(a \nabla_{w} ε_{h}, \nabla_{w} v)_{T_{h}} = 0, \forall v \in V_{h}^{0} .

Then by letting $v = ε_{h}$ in the above equation and (3.3), we arrive at

It follows from (3.8) that $∥ ε_{h} ∥_{1, h} = 0$ . Since $∥ \cdot ∥_{1, h}$ is a norm in $V_{h}^{0}$ , one has $ε_{h} = 0$ . This completes the proof of the lemma.

4 Error Analysis

The goal of this section is to establish error estimates for the weak Galerkin finite element solution $u_{h}$ arising from (2.4

). For simplicity of analysis, we assume that the coefficient tensor

a

in (1.1) is a piecewise constant matrix with respect to the finite element partition

T_{h}

. The result can be extended to variable tensors without any difficulty, provided that the tensor

a

is piecewise sufficiently smooth.

4.1 Error Equation

Let $e_{h} = u - u_{h}$ and $ϵ_{h} = Q_{h} u - u_{h} \in V_{h}$ . In this section, we derive an error equation that $e_{h}$ satisfies.

For any $v \in V_{h}^{0}$ , the following error equation holds true

(4.1)

(a \nabla_{w} e_{h}, \nabla_{w} v)_{T_{h}} = e_{1} (u, v) + e_{2} (u, v),

where

	$e_{1} (u, v)$	$=$	$⟨ a (\nabla u - Q_{h} \nabla u) \cdot n, Q_{b} v_{0} - v_{b} ⟩_{\partial T_{h}},$
	$e_{2} (u, v)$	$=$	$⟨ a \nabla u \cdot n, v_{0} - Q_{b} v_{0} ⟩_{\partial T_{h}} .$

For $v = {v_{0}, v_{b}} \in V_{h}^{0}$ , testing (1.1) by $v_{0}$ and using the fact that $⟨ a \nabla u \cdot n, v_{b} ⟩_{\partial T_{h}} = 0$ , we arrive at

(4.2)

(a \nabla u, \nabla v_{0})_{T_{h}} - ⟨ a \nabla u \cdot n, v_{0} - v_{b} ⟩_{\partial T_{h}} = (f, v_{0}) .

Obviously, we have

(4.3)

⟨ a \nabla u \cdot n, v_{0} - v_{b} ⟩_{\partial T_{h}}

=

⟨ a \nabla u \cdot n, Q_{b} v_{0} - v_{b} ⟩_{\partial T_{h}} + ⟨ a \nabla u \cdot n, v_{0} - Q_{b} v_{0} ⟩_{\partial T_{h}} .

Combining (4.2) and (4.3) gives

(4.4)

(a \nabla u, \nabla v_{0})_{T_{h}} - ⟨ a \nabla u \cdot n, Q_{b} v_{0} - v_{b} ⟩_{\partial T_{h}} = (f, v_{0}) + e_{2} (u, v) .

It follows from (2.3) and (2.5) that

(4.5)	$(a \nabla u, \nabla v_{0})_{T_{h}}$	$=$	$(a Q_{h} \nabla u, \nabla v_{0})_{T_{h}}$
		$=$	$(a Q_{h} \nabla u, \nabla_{w} v)_{T_{h}} + ⟨ Q_{b} v_{0} - v_{b}, a Q_{h} \nabla u \cdot n ⟩_{\partial T_{h}}$
		$=$	$(a \nabla_{w} u, \nabla_{w} v)_{T_{h}} + ⟨ Q_{b} v_{0} - v_{b}, a Q_{h} \nabla u \cdot n ⟩_{\partial T_{h}} .$

Using (4.4) and (4.5), we have

(4.6)

(a \nabla_{w} u, \nabla_{w} v)_{T_{h}}

=

(f, v_{0}) + e_{1} (u, v) + e_{2} (u, v) .

Subtracting (2.4) from (4.6) yields,

(a \nabla_{w} e_{h}, \nabla_{w} v)_{T_{h}} = e_{1} (u, v) + e_{2} (u, v) \forall v \in V_{h}^{0} .

This completes the proof of the lemma.

4.2 Error Estimates in Energy Norm

Optimal convergence rate of the WG approximation in energy norm will be obtained in this section. First we will bound the two terms $e_{1} (u, v)$ and $e_{2} (u, v)$ in the error equation (4.1).

For any $w \in H^{k + 1} (Ω)$ and $v = {v_{0}, v_{b}} \in V_{h}^{0}$ , we have

(4.7)		$\| e_{1} (w, v) \|$	$\leq$
(4.8)		$\| e_{2} (w, v) \|$	$\leq$

Using the Cauchy-Schwarz inequality, the trace inequality (2.6), (1.3) and (3.8), we have

$\| e_{1} (w, v) \|$	$=$	$∣ ∣ ∣ ∣ \sum T \in T_{h} ⟨ a (\nabla w - Q_{h} \nabla w) \cdot n, Q_{b} v_{0} - v_{b} ⟩_{\partial T} ∣ ∣ ∣ ∣$
	$\leq$	$C \sum T \in T_{h} ∥ \nabla w - Q_{h} \nabla w ∥_{\partial T} ∥ Q_{b} v_{0} - v_{b} ∥_{\partial T}$
	$\leq$	$C {⎛ ⎝ \sum T \in T_{h} h_{T} ∥ (\nabla w - Q_{h} \nabla w) ∥_{\partial T}^{2} ⎞ ⎠}_{T}^{\frac{1}{2}} {⎛ ⎝ \sum T \in T_{h} h_{T}^{- 1} ∥ Q_{b} v_{0} - v_{b} ∥_{\partial T}^{2} ⎞ ⎠}_{b}^{\frac{}{1} 2}$
	$\leq$

Let $Q_{k - 1}$ be the element-wise defined $L^{2}$ projection onto $[P_{k - 1} (T)]^{d}$ on each $T \in T_{h}$ . Using the Cauchy-Schwarz inequality, the trace inequality (2.6), (1.3) and the inverse inequality, we have

$\| e_{2} (w, v) \|$	$=$	$∣ ∣ ∣ ∣ \sum T \in T_{h} ⟨ a \nabla w \cdot n, v_{0} - Q_{b} v_{0} ⟩_{\partial T} ∣ ∣ ∣ ∣$
	$=$	$∣ ∣ ∣ ∣ \sum T \in T_{h} ⟨ a (\nabla w - Q_{k - 1} \nabla w) \cdot n, v_{0} - Q_{b} v_{0} ⟩_{\partial T} ∣ ∣ ∣ ∣$
	$\leq$	$C \sum T \in T_{h} ∥ \nabla w - Q_{k - 1} \nabla w ∥_{\partial T} ∥ v_{0} - Q_{b} v_{0} ∥_{\partial T}$
	$\leq$	$C {⎛ ⎝ \sum T \in T_{h} h_{T} ∥ (\nabla w - Q_{k - 1} \nabla w) ∥_{\partial T}^{2} ⎞ ⎠}_{T}^{\frac{1}{2}} {⎛ ⎝ \sum T \in T_{h} h_{T}^{- 1} h_{T}^{2} ∥ \nabla v_{0} ∥_{\partial T}^{2} ⎞ ⎠}_{0}^{\frac{1}{2}}$
	$\leq$

We have proved the lemma.

Let $u \in H^{k + 1} (Ω)$ , then

(4.9)

| | | u - Q_{h} u | | | \leq C h^{k} | u |_{k + 1} .

It follows from (2.3) and (2.6),

$\| (\nabla_{w} (u - Q_{h} u), q)_{T} \|$	$=$	$\| (\nabla (u - Q_{0} u), q)_{T} + ⟨ Q_{b} (Q_{0} u - Q_{b} u), q \cdot n ⟩_{\partial T} \|$
	$=$	$\| (\nabla (u - Q_{0} u), q)_{T} + ⟨ Q_{b} (Q_{0} u - u), q \cdot n ⟩_{\partial T} \|$
	$\leq$	$∥ \nabla (u - Q_{0} u) ∥_{T} ∥ q ∥_{T} + C h^{- 1 / 2} ∥ Q_{0} u - u ∥_{\partial T} ∥ q ∥_{T}$
	$\leq$	$C h^{k} \| u \|_{k + 1, T} ∥ q ∥_{T} .$

Letting $q = \nabla_{w} (u - Q_{h} u)$ in the above equation and taking summation over $T$ , we have

| | | u - Q_{h} u | | | \leq C h^{k} | u |_{k + 1} .

We have proved the lemma.

Let $u_{h} \in V_{h}$ be the weak Galerkin finite element solution of (2.4). Assume the exact solution $u \in H^{k + 1} (Ω)$ . Then, there exists a constant $C$ such that

(4.10)

| | | u - u_{h} | | | \leq C h^{k} | u |_{k + 1} .

It is straightforward to obtain

		$=$	$(a \nabla_{w} e_{h}, \nabla_{w} e_{h})_{T_{h}}$
		$=$	$(a (\nabla_{w} u - \nabla_{w} u_{h}), \nabla_{w} e_{h})_{T_{h}}$
		$=$	$(a (\nabla_{w} Q_{h} u - \nabla_{w} u_{h}), \nabla_{w} e_{h})_{T_{h}} + (a (\nabla_{w} u - \nabla_{w} Q_{h} u), \nabla_{w} e_{h})_{T_{h}}$
		$=$	$(a \nabla_{w} e_{h}, \nabla_{w} ϵ_{h})_{T_{h}} + (a \nabla_{w} (u - Q_{h} u), \nabla_{w} e_{h})_{T_{h}} .$

We will bound the two terms in (4.2). Letting $v = ϵ_{h} \in V_{h}$ in (4.1) and using (4.7), (4.8) and (4.9), we have

(4.12)	$\| (a \nabla_{w} e_{h}, \nabla_{w} ϵ_{h})_{T_{h}} \|$	$=$	$\| e_{1} (u, ϵ_{h}) + e_{2} (u, ϵ_{h}) \|$
		$\leq$	$C h^{k} \| u \|_{k + 1} \| \| \| ϵ_{h} \| \| \|$
		$\leq$	$C h^{k} \| u \|_{k + 1} \| \| \| Q_{h} u - u_{h} \| \| \|$
		$\leq$	$C h^{k} \| u \|_{k + 1} (\| \| \| Q_{h} u - u \| \| \| + \| \| \| u - u_{h} \| \| \|)$
		$\leq$

The estimate (4.9) implies

(4.13)		$\| (\nabla_{w} (u - Q_{h} u), \nabla_{w} e_{h})_{T_{h}} \|$	$\leq$	$C \| \| \| u - Q_{h} u \| \| \| \| \| \| e_{h} \| \| \|$
(4.13)			$\leq$

Combining the estimates (4.12) and (4.13) with (4.2), we arrive

| | | e_{h} | | | \leq C h^{k} | u |_{k + 1},

which completes the proof.

The estimates (4.9) and (4.10) imply

(4.14)

4.3 Error Estimates in $L^{2}$ Norm

The standard duality argument is used to obtain $L^{2}$ error estimate. Recall $e_{h} = {e_{0}, e_{b}} = u - u_{h}$ and $ϵ_{h} = {ϵ_{0}, ϵ_{b}} = Q_{h} u - u_{h}$ . The dual problem seeks $Φ \in H_{0}^{1} (Ω)$ satisfying

(4.15)

- \nabla \cdot a \nabla Φ

=

ϵ_{0} in Ω .

Assume that the following $H^{2}$ -regularity holds

(4.16)

∥ Φ ∥_{2} \leq C ∥ ϵ_{0} ∥ .

Let $u_{h} \in V_{h}$ be the weak Galerkin finite element solution of (2.4). Assume that the exact solution $u \in H^{k + 1} (Ω)$ and (4.16) holds true. Then, there exists a constant $C$ such that

(4.17)

∥ Q_{0} u - u_{0} ∥ \leq C h^{k + 1} | u |_{k + 1} .

By testing (4.15) with $ϵ_{0}$ we obtain

(4.18)	$∥ ϵ_{0} ∥^{2}$	$=$	$- (\nabla \cdot (a \nabla Φ), ϵ_{0})$
		$=$
		$=$	$(a \nabla Φ, \nabla ϵ_{0})_{T_{h}} - ⟨ a \nabla Φ \cdot n, Q_{b} ϵ_{0} - ϵ_{b} ⟩_{\partial T_{h}} - ⟨ a \nabla Φ \cdot n, ϵ_{0} - Q_{b} ϵ_{0} ⟩_{\partial T_{h}}$
		$=$	$(a \nabla Φ, \nabla ϵ_{0})_{T_{h}} - ⟨ a \nabla Φ \cdot n, Q_{b} ϵ_{0} - ϵ_{b} ⟩_{\partial T_{h}} - e_{2} (Φ, ϵ_{h}),$

where we have used the fact that $ϵ_{b} = 0$ on $\partial Ω$ . Setting $u = Φ$ and $v = ϵ_{h}$ in (4.5) yields

(4.19)

(a \nabla Φ, \nabla ϵ_{0})_{T_{h}}

=

(a \nabla_{w} Φ, \nabla_{w} ϵ_{h})_{T_{h}} + ⟨ Q_{b} ϵ_{0} - ϵ_{b}, a Q_{h} \nabla Φ \cdot n ⟩_{\partial T_{h}} .

Substituting (4.19) into (4.18) gives

(4.20)	$∥ ϵ_{0} ∥^{2}$	$=$	$(a \nabla_{w} ϵ_{h}, \nabla_{w} Φ)_{T_{h}} - e_{1} (Φ, ϵ_{h}) - e_{2} (Φ, ϵ_{h})$
		$=$	$(a \nabla_{w} e_{h}, \nabla_{w} Φ)_{T_{h}} + (a \nabla_{w} (Q_{h} u - u), \nabla_{w} Φ)_{T_{h}} - e_{1} (Φ, ϵ_{h}) - e_{2} (Φ, ϵ_{h})$
		$=$	$(a \nabla_{w} e_{h}, \nabla_{w} Q_{h} Φ)_{T_{h}} + (a \nabla_{w} e_{h}, \nabla_{w} (Φ - Q_{h} Φ))_{T_{h}}$
		$+$	$(a \nabla_{w} (Q_{h} u - u), \nabla_{w} Φ)_{T_{h}} - e_{1} (Φ, ϵ_{h}) - e_{2} (Φ, ϵ_{h})$
		$=$	$e_{1} (u, Q_{h} Φ) + e_{2} (u, Q_{h} Φ) - e_{1} (Φ, ϵ_{h}) - e_{2} (Φ, ϵ_{h})$
		$+$	$(a \nabla_{w} e_{h}, \nabla_{w} (Φ - Q_{h} Φ))_{T}$

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

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This week in AI

1 Introduction

2 Weak Galerkin Finite Element Schemes

Weak Galerkin Algorithm 1

3 Well Posedness

4 Error Analysis

4.1 Error Equation

4.2 Error Estimates in Energy Norm

4.3 Error Estimates in L2 Norm

4.3 Error Estimates in $L^{2}$ Norm