A C^0 interior penalty method for mth-Laplace equation

1. Introduction

We consider the $m$ th-Laplace equation


	$(- 1)^{m} Δ^{m} u = f in Ω,$		(1.1a)
	$u = \frac{\partial u}{\partial ν} = \dots = \frac{\partial^{m - 1} u}{\partial ν^{m - 1}} = 0 on \partial Ω,$		(1.1b)

where $m$ is an arbitrary positive integer, $Ω$ is a bounded Lipschitz polyhedral domain in $R^{d}$ ( $d = 1, 2, 3, \dots$ ), and $ν$

is the outward unit normal vector field along

\partial Ω

. The source term

f \in H^{- 1} (Ω)

Several works have been done to solve numerically (1.1). Standard $H^{m}$ conforming finite elements space requires $C^{m - 1}$

continuity and leads to complicated construction of finite element space and lots of degrees of freedom when

m

is large. Bramble and Zlámal [2] studied the

H^{m}

conforming finite elements space on the two dimensional triangular meshes. Meanwhile, a

H^{m}

conforming finite element space is developed by Hu and Zhang on rectangular grids for arbitrary

d

in [6]

. Though up to this moment they have above mentioned restrictions, conforming

H^{m}

finite element space on simplicial meshes for

d \geq 3

are desirable in both theoretical analysis and practice. In order to simplify the construction of

H^{m}

finite element space, alternative

H^{m}

nonconforming finite element space is introduced in several works. In [9], a

H^{m}

nonconforming finite element space (named Morley-Wang-Xu elements) is introduced for

m \leq d

. Besides, Hu and Zhang also considered the

H^{m}

nonconforming finite element space in [7] on triangular meshes for

d = 2

. The finite element space in [9] is generalized for

m = d + 1

by Wu and Xu in [10]. Recently in [11], it is further generalized for arbitrary

m

and

d

but with stabilization along mesh interface in order to balance the weak continuity and the penalty terms. In order to obtain stability and optimal convergence in some discrete

H^{m}

-norm, [9, 10, 11] propose to compute numerical approximation to

D^{m} u

, such that their implementation may become quite complicated as

m

is large. The finite element spaces in [2, 6] can be used to solve numerically (1.1) with any source term

f \in H^{- m} (Ω)

. However, the implementation of these conforming and nonconforming finite element spaces can be quite challenging for large

m

. Virtual element methods have been investigated for (1.1). In [1], a conforming

H^{m}

virtual element method is introduced for convex polygonal domain in

R^{2}

. The finite element space in [1] contains piecewise

r

-th order polynomials, where

r \geq 2 m - 1

. The virtual element method in [1] needs strong assumption on regularity of

f

(

f \in H^{r - m + 1} (Ω)

) to achieve optimal convergence (see [1, Theorem

4.2

]). In [4], a nonconforming

H^{m}

virtual element method is developed for bounded Lipschitz polyhedral domain in

R^{d}

, where

d

can be any positive integer. The design of finite element space in [4], which contains piecewise

r

-th (

r \geq m

) order polynomials, is based on a generalized Green’s identity for

H^{m}

inner product. It is assumed that

m \leq d

in [4]. Besides above numerical methods based on primary formulations of (1.1

), a mixed formulation based on Helmholtz decomposition for tensor valued function is introduced in

[8] for two dimensional domain.

We propose a $C^{0}$ interior penalty method (2.2) for (1.1) for arbitrary positive integers $m$ and $d$ . The finite element space of (2.2) is the standard $H^{1}$ -conforming piecewise $r$ -th order polynomials, where $r \geq m$ . The design of (2.2) avoids computing $D^{m}$ of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. In fact, (2.2) only gets involved with calculation of high order multiplicity of Laplace of numerical solution ( $Δ^{i} u_{h}$ for $1 \leq i \leq m$ ) and the gradient of high order multiplicity of Laplace of numerical solution ( $\nabla Δ^{i} u_{h}$ for $0 \leq i \leq m - 1$ ) on both elements and mesh interfaces. Therefore our method (2.2) can be easily implemented, even when $m$ is large and $d = 3$ . After proving (Theorem 3.4) that discrete $H^{m}$ -norm (see Definition 3.1) is bounded by the natural energy semi-norm associated with (2.2), we manage to show our method (2.2) has stability and optimal convergence on bounded Lipschitz polyhedral domain in $R^{d}$ with respect to the discrete $H^{m}$ -norm, for any positive integers $m$ and $d$ . Roughly speaking, we have

	$∥ u_{h} ∥_{m, h} \leq$	$C ∥ f ∥_{H^{- 1} (Ω)},$
	$∥ u_{h} - u ∥_{m, h} \leq$	$C h^{min (r + 1 - m, s - m)} ∥ u ∥_{H^{s} (Ω)},$

where $s \geq 2 m - 1$ . We refer to Theorem 3.6 and Theorem 3.7 for detailed descriptions on stability and optimal convergence. The design and analysis of our method (2.2

) can be easily generalized for nonlinear partial differential equations with

(- 1)^{m} Δ^{m} u

as their leading term. We would like to point out that our method (2.2) is not a generalization of the interior penalty method for sixth-order elliptic equation (

m = 3

) in [5, (

3.4

3.5

)]. Actually, the method in [5] needs to calculate numerical approximation to

D^{3} u

In next section, we present the $C^{0}$ interior penalty method. In section 3, we prove stability and optimal convergence with respect to discrete $H^{m}$ -norm (see Definition 3.1). In section 4, we provide numerical experiments.

2. $C^{0}$ interior penalty method

In this section, firstly we give notations to define the $C^{0}$ interior penalty method for (1.1). Then in section 2.1, we derive the $C^{0}$ interior penalty method for any $m \geq 1$ . Finally in section 2.2, we provide concrete examples of the method for $m = 1, 2, 3, 4$ .

Let $T_{h}$ be a quasi-uniform conforming simplicial mesh of $Ω$ . Here we define $h = {max}_{K \in T_{h}} h_{K}$ where $h_{K}$ is the diameter of the element $K \in T_{h}$ . We denote by $F_{h}$ , $F_{h}^{int}$ and $F_{h}^{\partial}$ the collections of all $(d - 1)$ -dimensional faces, interior faces and boundary faces of $T_{h}$ , respectively. Obviously, $F_{h} = F_{h}^{int} \cup F_{h}^{\partial}$ . For any positive integer $r$ , we define $V_{h}^{r} = H_{0}^{1} (Ω) \cap P_{r} (T_{h})$ , where $P_{r} (T_{h}) = {v_{h} \in L^{2} (Ω) : v_{h} |_{K} \in P_{r} (K), \forall K \in T_{h}}$ .

We introduce some trace operators. For any interior face $F \in F_{h}^{int}$ , let $K^{-}, K^{+} \in T_{h}$ be two elements sharing $F$ . We denote by $ν^{-}$ and $ν^{+}$ the outward unit normal vectors along $\partial K^{-}$ and $\partial K^{+}$ , respectively. For scalar function $v : Ω \to R$ and vector field $ϕ : Ω \to R^{d}$ , which may be discontinuous across $F_{h}^{int}$ , we define the following quantities. For $v^{-} := v |_{K^{-}}$ , $v^{+} := v |_{K^{+}}$ , $ϕ^{-} := ϕ |_{K^{-}}$ and $ϕ^{+} := ϕ |_{K^{+}}$ , we define

	${{v}} = \frac{1}{2} (v^{-} \|_{F} + v^{+} \|_{F}), {{ϕ}} = \frac{1}{2} (ϕ^{-} \|_{F} + ϕ^{+} \|_{F}),$
	$⟦ v ⟧ = v^{-} ν^{-} \|_{F} + v^{+} ν^{+} \|_{F}, ⟦ ϕ ⟧ = ϕ^{-} \cdot ν^{-} \|_{F} + ϕ^{+} \cdot ν^{+} \|_{F};$

if $F \in \partial K^{+} \cap \partial Ω$ , we define

{{v}} = v^{+} |_{F}, {{ϕ}} = ϕ^{+} |_{F}, ⟦ v ⟧ = v^{+} ν |_{F}, ⟦ ϕ ⟧ = ϕ^{+} \cdot ν |_{F} .

2.1. Derivation of $C^{0}$ interior penalty method

We assume the exact solution $u \in H^{2 m - 1} (Ω)$ . For any $v_{h} \in V_{h}$ , via $m$ -times integrating by parts,

		$((- 1)^{m} Δ^{m} u, v_{h})_{Ω}$
	$=$	$⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (Δ^{~ m} u, Δ^{~ m} v_{h})_{T_{h}} - Σ_{i = 0}^{~ m - 1} ⟨ Δ^{~ m + i} u, \partial_{ν} Δ^{~ m - i - 1} v_{h} ⟩_{\partial T_{h}} + Σ_{i = 0}^{~ m - 2} ⟨ \partial_{ν} Δ^{~ m + i} u, Δ^{~ m - i - 1} v_{h} ⟩_{\partial T_{h}}, if m = 2 ~ m (m is an even number); (\nabla Δ^{~ m} u, \nabla Δ^{~ m} v_{h})_{T_{h}} + Σ_{i = 0}^{~ m - 1} ⟨ Δ^{~ m + i + 1} u, \partial_{ν} Δ^{~ m - i - 1} v_{h} ⟩_{\partial T_{h}} - Σ_{i = 0}^{~ m - 1} ⟨ \partial_{ν} Δ^{~ m + i} u, Δ^{~ m - i} v_{h} ⟩_{\partial T_{h}}, if m = 2 ~ m + 1 (m is an odd number) . \end{matrix}$

Since $u \in H^{2 m - 1} (Ω)$ , for any $v_{h} \in V_{h}$ ,

		$((- 1)^{m} Δ^{m} u, v_{h})_{Ω}$		(2.1)
	$=$

(2.1) inspires us to define the coupling term $C_{h}$ in Definition 2.1.

Definition 2.1.

For any $w_{h}, v_{h} \in V_{h}$ , we define the coupling term $C_{h} (w_{h}, v_{h})$ along mesh interface $F_{h}$ by

		$C_{h} (w_{h}, v_{h})$
	$=$	$⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} - Σ_{i = 0}^{~ m - 1} ⟨ {{Δ^{~ m + i} w_{h}}}, ⟦ \nabla Δ^{~ m - i - 1} v_{h} ⟧ ⟩_{F_{h}} + Σ_{i = 0}^{~ m - 2} ⟨ {{\nabla Δ^{~ m + i} w_{h}}}, ⟦ Δ^{~ m - i - 1} v_{h} ⟧ ⟩_{F_{h}}, if m = 2 ~ m; Σ_{i = 0}^{~ m - 1} ⟨ {{Δ^{~ m + i + 1} w_{h}}}, ⟦ \nabla Δ^{~ m - i - 1} v_{h} ⟧ ⟩_{F_{h}} - Σ_{i = 0}^{~ m - 1} ⟨ {{\nabla Δ^{~ m + i} w_{h}}}, ⟦ Δ^{~ m - i} v_{h} ⟧ ⟩_{F_{h}}, if m = 2 ~ m + 1. \end{matrix}$

In order to define $C^{0}$ interior penalty method, we need the stabilization term $S_{h}$ in Definition 2.2.

Definition 2.2.

For any $w_{h}, v_{h} \in V_{h}$ , we define the stabilization term $S_{h} (w_{h}, v_{h})$ along mesh interface $F_{h}$ by

		$S_{h} (w_{h}, v_{h})$
	$=$	$⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} Σ_{i = 0}^{~ m - 1} h^{- (4 i + 1)} ⟨ ⟦ \nabla Δ^{~ m - i - 1} w_{h} ⟧, ⟦ \nabla Δ^{~ m - i - 1} v_{h} ⟧ ⟩_{F_{h}} + Σ_{i = 0}^{~ m - 2} h^{- (4 i + 3)} ⟨ ⟦ Δ^{~ m - i - 1} w_{h} ⟧, ⟦ Δ^{~ m - i - 1} v_{h} ⟧ ⟩_{F_{h}}, if m = 2 ~ m (m is an even number); Σ_{i = 0}^{~ m - 1} h^{- (4 i + 3)} ⟨ ⟦ \nabla Δ^{~ m - i - 1} w_{h} ⟧, ⟦ \nabla Δ^{~ m - i - 1} v_{h} ⟧ ⟩_{F_{h}} + Σ_{i = 0}^{~ m - 1} h^{- (4 i + 1)} ⟨ ⟦ Δ^{~ m - i} w_{h} ⟧, ⟦ Δ^{~ m - i} v_{h} ⟧ ⟩_{F_{h}}, if m = 2 ~ m + 1 (m is an odd number) . \end{matrix}$

We would like to point out that $S_{h} (w_{h}, v_{h}) = 0$ if $m = 1$ .

The $C^{0}$ interior penalty method is to find $u_{h} \in V_{h}$ , such that for any $v_{h}$ ,

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (Δ^{~ m} u_{h}, Δ^{~ m} v_{h})_{T_{h}} + C_{h} (u_{h}, v_{h}) + C_{h} (v_{h}, u_{h}) + τ S_{h} (u_{h}, v_{h}) = (f, v_{h})_{Ω}, if m = 2 ~ m; (\nabla Δ^{~ m} u_{h}, \nabla Δ^{~ m} v_{h})_{T_{h}} + C_{h} (u_{h}, v_{h}) + C_{h} (v_{h}, u_{h}) + τ S_{h} (u_{h}, v_{h}) = (f, v_{h})_{Ω}, if m = 2 ~ m + 1. \end{matrix}

(2.2)

Here the parameter $τ \geq 1$ shall be large enough but independent of $h$ .

2.2. Examples of $C^{0}$ interior penalty method

$m = 1$

The $C^{0}$ interior penalty method for $- Δ u = f$ is to find $u_{h} \in V_{h}$ satisfying

$(\nabla u_{h}, \nabla v_{h})_{Ω} = (f, v_{h})_{Ω}, \forall v_{h} \in V_{h} .$ (2.3)
$m = 2$

The $C^{0}$ interior penalty method for $Δ^{2} u = f$ is to find $u_{h} \in V_{h}$ satisfying

(2.4)

$+ τ h^{- 1} ⟨ ⟦ \nabla u_{h} ⟧, ⟦ \nabla v_{h} ⟧ ⟩_{F_{h}} = (f, v_{h})_{Ω}, \forall v_{h} \in V_{h} .$

$m = 3$

The $C^{0}$ interior penalty method for $- Δ^{3} u = f$ is to find $u_{h} \in V_{h}$ satisfying

		$(\nabla Δ u_{h}, \nabla Δ v_{h})_{T_{h}} + (⟨ {{Δ^{2} u_{h}}}, ⟦ \nabla v_{h} ⟧ ⟩_{F_{h}} - ⟨ {{\nabla Δ u_{h}}}, ⟦ Δ v_{h} ⟧ ⟩_{F_{h}})$		(2.5)

		$+ τ (h^{- 3} ⟨ ⟦ \nabla u_{h} ⟧, ⟦ \nabla v_{h} ⟧ ⟩_{F_{h}} + h^{- 1} ⟨ ⟦ Δ u_{h} ⟧, ⟦ Δ v_{h} ⟧ ⟩_{F_{h}}) = (f, v_{h})_{Ω}, \forall v_{h} \in V_{h} .$

It is easy to see that (2.5) is quite different from the interior penalty method in [5].

$m = 4$

The $C^{0}$ interior penalty method for $Δ^{4} u = f$ is to find $u_{h} \in V_{h}$ satisfying

$(Δ^{2} u_{h}, Δ^{2} v_{h})_{T_{h}}$ (2.6)

$+ τ (h^{- 5} ⟨ ⟦ \nabla u_{h} ⟧, ⟦ \nabla v_{h} ⟧ ⟩_{F_{h}} + h^{- 3} ⟨ ⟦ Δ u_{h} ⟧, ⟦ Δ v_{h} ⟧ ⟩_{F_{h}} + h^{- 1} ⟨ ⟦ \nabla Δ u_{h} ⟧, ⟦ \nabla Δ v_{h} ⟧ ⟩_{F_{h}})$

$= (f, v_{h})_{Ω}, \forall v_{h} \in V_{h} .$

3. Analysis

In this section, firstly we prove Theorem 3.4, which states the discrete $H^{m}$ -norm (see Definition 3.1) bounded by the natural energy semi-norm associated with the $C^{0}$ interior penalty method (2.2). Then we prove Theorem 3.6, which shows the energy estimate of (2.2). Finally, we prove Theorem 3.7, which gives optimal convergence of numerical approximation to $u$ in the discrete $H^{m}$ -norm.

Definition 3.1.

For any integers $m \geq 2$ , we define the discrete $H^{m}$ -norm $∥ v ∥_{m, h}$ by

		$∥ v ∥_{m, h}^{2} = Σ_{i = 0}^{m} ∥ D^{i} v ∥_{L^{2} (T_{h})}^{2} + Σ_{j = 1}^{m - 1} h^{- (2 m - 2 j - 1)} ∥ ⟦ D^{j} v ⟧_{J} ∥_{L^{2} (F_{h})}^{2},$
	$:=$	$Σ_{i = 0}^{m} Σ_{K \in T_{h}} ∥ D^{i} v ∥_{L^{2} (K)}^{2} + Σ_{j = 1}^{m - 1} Σ_{F \in F_{h}} h^{- (2 m - 2 j - 1)} ∥ ⟦ D^{j} v ⟧_{J} ∥_{L^{2} (F)}^{2}, \forall v \in H_{0}^{1} (Ω) \cap H^{m} (T_{h}) .$

For any $F \in F_{h}^{int}$ , there are two elements $K^{-}, K^{+} \in T_{h}$ sharing the common face $F$ . We denote by $v^{-} := v |_{K^{-}}$ and $v^{+} := v |_{K^{+}}$ . We define

		$∥ ⟦ D^{j} v ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$=$	$\sum 1 \leq k_{1}, \dots, k_{j} \leq d ∥ ⟦ \partial_{x_{k_{1}}} \partial_{x_{k_{2}}} \dots \partial_{x_{k_{j}}} v ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$:=$	$\sum 1 \leq k_{1}, \dots, k_{j} \leq d ∥ (\partial_{x_{k_{1}}} \partial_{x_{k_{2}}} \dots \partial_{x_{k_{j}}} v^{-}) \|_{F} - (\partial_{x_{k_{1}}} \partial_{x_{k_{2}}} \dots \partial_{x_{k_{j}}} v^{+}) \|_{F} ∥_{L^{2} (F)}^{2} .$

For any $F \in F_{h}^{\partial}$ , we define

		$∥ ⟦ D^{j} v ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$=$	$\sum 1 \leq k_{1}, \dots, k_{j} \leq d ∥ ⟦ \partial_{x_{k_{1}}} \partial_{x_{k_{2}}} \dots \partial_{x_{k_{j}}} v ⟧_{J} ∥_{L^{2} (F)}^{2} := \sum 1 \leq k_{1}, \dots, k_{j} \leq d ∥ (\partial_{x_{k_{1}}} \partial_{x_{k_{2}}} \dots \partial_{x_{k_{j}}} v) \|_{F} ∥_{L^{2} (F)}^{2} .$

3.1. Theorem 3.4 : discrete $H^{m}$ -norm bounded by natural energy semi-norm

The proof of Theorem 3.4 is based on Lemma 3.2 and Lemma 3.3.

Lemma 3.2.

For any integers $r \geq m \geq 2$ , there is a constant $C > 0$ such that

Σ_{j = 1}^{m - 1} h^{- (2 m - 2 j - 1)} ∥ ⟦ D^{j} v_{h} ⟧_{J} ∥_{L^{2} (F_{h})}^{2} \leq C S_{h} (v_{h}, v_{h}), \forall v_{h} \in V_{h} .

(3.1)

Proof.

We choose $F \in F_{h}$ arbitrarily. There is an orthonormal coordinate system ${y_{k}}_{k = 1}^{d}$ such that the $y_{d}$ -axis is parallel to normal vector along $F$ . Therefore $y_{1}$ -axis, $\dots$ , $y_{d - 1}$ -axis are all parallel to $F$ .

We claim that for any $1 \leq l \leq m$ , there is a positive integer $C^{'}$ such that

∥ ⟦ D^{l} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} \leq C^{'} (Σ_{j = 0}^{l} h^{- (2 l - 2 j)} ∥ ⟦ \partial_{y_{d}}^{j} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}), \forall {~ v}_{h} \in P_{r} (T_{h}) .

(3.2)

When $l = 1$ , it is easy to see

∥ ⟦ D {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} = Σ_{k = 1}^{d} ∥ ⟦ \partial_{x_{k}} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} = Σ_{k = 1}^{d} ∥ ⟦ \partial_{y_{k}} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} .

By discrete inverse inequality and the fact that $y_{1}$ -axis, $\dots$ , $y_{d - 1}$ -axis are all parallel to $F$ , we have that

Σ_{k = 1}^{d - 1} ∥ ⟦ \partial_{y_{k}} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} \leq C h^{- 2} ∥ ⟦ {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} .

Therefore we have

∥ ⟦ D {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} \leq C (h^{- 2} ∥ ⟦ {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} + ∥ ⟦ \partial_{y_{d}} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}) .

Thus (3.2) holds when $l = 1$ . We assume that (3.2) holds for any $l < m$ . Then by discrete inverse inequality and the fact that $y_{1}$ -axis, $\dots$ , $y_{d - 1}$ -axis are all parallel to $F$ ,

		$∥ ⟦ D^{l + 1} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} = Σ_{k = 1}^{d} ∥ ⟦ \partial_{x_{k}} D^{l} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} = Σ_{k = 1}^{d} ∥ ⟦ \partial_{y_{k}} D^{l} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$\leq$	$C (h^{- 2} ∥ ⟦ D^{l} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} + ∥ D^{l} (\partial_{y_{d}} {~ v}_{h}) ∥_{L^{2} (F)}^{2}) .$

Since ${~ v}_{h} \in P_{r} (T_{h})$ , then $\partial_{y_{d}} {~ v}_{h} \in P_{r} (T_{h})$ . Since we assume (3.2) holds for $l$ , we have

	$∥ D^{l} (\partial_{y_{d}} {~ v}_{h}) ∥_{L^{2} (F)}^{2} \leq C (Σ_{j = 0}^{l} h^{- (2 l - 2 j)} ∥ ⟦ \partial_{y_{d}}^{j} (\partial_{y_{d}} {~ v}_{h}) ⟧_{J} ∥_{L^{2} (F)}^{2}),$
	$h^{- 2} ∥ ⟦ D^{l} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} \leq C (Σ_{j = 0}^{l} h^{- (2 (l + 1) - 2 j)} ∥ ⟦ \partial_{y_{d}}^{j} {~ v}_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}) .$

Therefore (3.2) holds for $l + 1$ . Thus we can conclude that the claim (3.2) is true.

Since $∥ ⟦ \partial_{y_{d}} v_{h} ⟧_{J} ∥_{L^{2} (F)} = ∥ ⟦ \nabla v_{h} ⟧ ∥_{L^{2} (F)}$ , (3.2) and the fact $v_{h} \in H_{0}^{1} (Ω)$ imply

∥ ⟦ D v_{h} ⟧_{J} ∥_{L^{2} (F)} = ∥ ⟦ \nabla v_{h} ⟧ ∥_{L^{2} (F)} .

(3.3)

Applying (3.2) with $l = 2$ , we have

	$∥ ⟦ D^{2} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}$	(3.4)
$\leq$	$C (h^{- 4} ∥ ⟦ v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} + h^{- 2} ∥ ⟦ \partial_{y_{d}} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} + ∥ ⟦ \partial_{y_{d}}^{2} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2})$
$=$	$C (h^{- 2} ∥ ⟦ \nabla v_{h} ⟧ ∥_{L^{2} (F)}^{2} + ∥ ⟦ \partial_{y_{d}}^{2} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}) .$

The last equality in (3.4) holds since $v_{h} \in H_{0}^{1} (Ω)$ and $∥ ⟦ \partial_{y_{d}} v_{h} ⟧_{J} ∥_{L^{2} (F)} = ∥ ⟦ \nabla v_{h} ⟧ ∥_{L^{2} (F)}$ . We notice that

Δ v_{h} = (\partial_{y_{1}}^{2} v_{h} + \dots + \partial_{y_{d - 1}}^{2} v_{h}) + \partial_{y_{d}}^{2} v_{h} .

(3.5)

Since $y_{1}$ -axis, $\dots$ , $y_{d - 1}$ -axis are all parallel to $F$ , discrete inverse inequality implies

		$∥ ⟦ (\partial_{y_{1}}^{2} v_{h} + \dots + \partial_{y_{d - 1}}^{2} v_{h}) ⟧ ∥_{L^{2} (F)}^{2}$
	$=$	$∥ ⟦ (\partial_{y_{1}}^{2} v_{h} + \dots + \partial_{y_{d - 1}}^{2} v_{h}) ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$\leq$	$C h^{- 2} ∥ ⟦ D v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} = C h^{- 2} ∥ ⟦ \nabla v_{h} ⟧ ∥_{L^{2} (F)}^{2} .$

By (3.5) and the above inequality, we have

∥ ⟦ \partial_{y_{d}}^{2} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} \leq C (h^{- 2} ∥ ⟦ \nabla v_{h} ⟧ ∥_{L^{2} (F)}^{2} + ∥ ⟦ Δ v_{h} ⟧ ∥_{L^{2} (F)}^{2}) .

(3.6)

By (3.4, 3.6), we have

(3.7)

Since $F \in F_{h}$ is chosen arbitrarily, (3.3, 3.7) imply that (3.1) holds when $m = 2$ .

We assume that $1 \leq l < m$ is an odd number, $l = 2 ~ l + 1$ and

		$∥ ⟦ D^{l} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}$		(3.8)
	$\leq$	$C (Σ_{i = 0}^{~ l} h^{- 4 i} ∥ ⟦ \nabla Δ^{~ l - i} v_{h} ⟧ ∥_{L^{2} (F)}^{2} + Σ_{i = 0}^{~ l - 1} h^{- (4 i + 2)} ∥ ⟦ Δ^{~ l - i} v_{h} ⟧ ∥_{L^{2} (F)}^{2}) .$

Then by applying (3.8) for each $\partial_{y_{k}} v_{h}$ , we have

		$∥ ⟦ D^{l + 1} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2} = Σ_{k = 1}^{d} ∥ ⟦ D^{l} (\partial_{x_{k}} v_{h}) ⟧_{J} ∥_{L^{2} (F)}^{2} = Σ_{k = 1}^{d} ∥ ⟦ D^{l} (\partial_{y_{k}} v_{h}) ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$\leq$	$C Σ_{k = 1}^{d} (Σ_{i = 0}^{~ l} h^{- 4 i} ∥ ⟦ \nabla Δ^{~ l - i} (\partial_{y_{k}} v_{h}) ⟧ ∥_{L^{2} (F)}^{2} + Σ_{i = 0}^{~ l - 1} h^{- (4 i + 2)} ∥ ⟦ Δ^{~ l - i} (\partial_{y_{k}} v_{h}) ⟧ ∥_{L^{2} (F)}^{2}) .$

Here $2 ~ l + 1 = l$ . Since $y_{1}$ -axis, $\dots$ , $y_{d - 1}$ -axis are all parallel to $F$ , discrete inverse inequality implies

		$∥ ⟦ D^{l + 1} v_{h} ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$\leq$	$C h^{- 2} (Σ_{i = 0}^{~ l} h^{- 4 i} ∥ ⟦ \nabla Δ^{~ l - i} v_{h} ⟧ ∥_{L^{2} (F)}^{2} + Σ_{i = 0}^{~ l - 1} h^{- (4 i + 2)} ∥ ⟦ Δ^{~ l - i} v_{h} ⟧ ∥_{L^{2} (F)}^{2})$
		$+ C (Σ_{i = 0}^{~ l} h^{- 4 i} ∥ ⟦ \nabla Δ^{~ l - i} (\partial_{y_{d}} v_{h}) ⟧ ∥_{L^{2} (F)}^{2} + Σ_{i = 0}^{~ l - 1} h^{- (4 i + 2)} ∥ ⟦ Δ^{~ l - i} (\partial_{y_{d}} v_{h}) ⟧ ∥_{L^{2} (F)}^{2}) .$

Again by the fact that $y_{1}$ -axis, $\dots$ , $y_{d - 1}$ -axis are all parallel to $F$ , we have that for any $0 \leq i \leq ~ l$ ,

		$∥ ⟦ \nabla Δ^{~ l - i} (\partial_{y_{d}} v_{h}) ⟧ ∥_{L^{2} (F)}^{2} = ∥ ⟦ \nabla (\partial_{y_{d}} Δ^{~ l - i} v_{h}) ⟧_{J} ∥_{L^{2} (F)}^{2}$
	$=$	$(∥ ⟦ \partial_{y_{1}} (\partial_{y_{d}}$

A C^0 interior penalty method for mth-Laplace equation

Authors

A mixed finite element scheme for biharmonic equation with variable coefficient and von Kármán equations

Convergence analysis of a finite difference method for stochastic Cahn–Hilliard equation

A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form

A Hybrid-High Order Method for Quasilinear Elliptic Problems of Nonmonotone Type

Dispersion Analysis of CIP-FEM for Helmholtz Equation

Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh

Convergence and Error Estimates for the Conservative Spectral Method for Fokker-Planck-Landau Equations

1. Introduction

2. $C^{0}$ interior penalty method

2.1. Derivation of $C^{0}$ interior penalty method

Definition 2.1.

Definition 2.2.

2.2. Examples of $C^{0}$ interior penalty method

3. Analysis

Definition 3.1.

3.1. Theorem 3.4 : discrete $H^{m}$ -norm bounded by natural energy semi-norm

Lemma 3.2.

Proof.

				(2.4)
		$+ τ h^{- 1} ⟨ ⟦ \nabla u_{h} ⟧, ⟦ \nabla v_{h} ⟧ ⟩_{F_{h}} = (f, v_{h})_{Ω}, \forall v_{h} \in V_{h} .$

		$(Δ^{2} u_{h}, Δ^{2} v_{h})_{T_{h}}$		(2.6)


		$+ τ (h^{- 5} ⟨ ⟦ \nabla u_{h} ⟧, ⟦ \nabla v_{h} ⟧ ⟩_{F_{h}} + h^{- 3} ⟨ ⟦ Δ u_{h} ⟧, ⟦ Δ v_{h} ⟧ ⟩_{F_{h}} + h^{- 1} ⟨ ⟦ \nabla Δ u_{h} ⟧, ⟦ \nabla Δ v_{h} ⟧ ⟩_{F_{h}})$
		$= (f, v_{h})_{Ω}, \forall v_{h} \in V_{h} .$

A C^0 interior penalty method for mth-Laplace equation

Authors

Related Research

A mixed finite element scheme for biharmonic equation with variable coefficient and von Kármán equations

Convergence analysis of a finite difference method for stochastic Cahn–Hilliard equation

A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form

A Hybrid-High Order Method for Quasilinear Elliptic Problems of Nonmonotone Type

Dispersion Analysis of CIP-FEM for Helmholtz Equation

Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh

Convergence and Error Estimates for the Conservative Spectral Method for Fokker-Planck-Landau Equations

This week in AI

1. Introduction

2. C0 interior penalty method

2.1. Derivation of C0 interior penalty method

Definition 2.1.

Definition 2.2.

2.2. Examples of C0 interior penalty method

3. Analysis

Definition 3.1.

3.1. Theorem 3.4 : discrete Hm-norm bounded by natural energy semi-norm

Lemma 3.2.

Proof.

2. $C^{0}$ interior penalty method

2.1. Derivation of $C^{0}$ interior penalty method

2.2. Examples of $C^{0}$ interior penalty method

3.1. Theorem 3.4 : discrete $H^{m}$ -norm bounded by natural energy semi-norm