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

In this paper, we propose a C^0 interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝ^d, where m and d can be any positive integers. The standard H^1-conforming piecewise r-th order polynomial space is used to approximate the exact solution u, where r can be any integer greater than or equal to m. Unlike the interior penalty method in [T. Gudi and M. Neilan, An interior penalty method for a sixth-order elliptic equation, IMA J. Numer. Anal., 31(4) (2011), pp. 1734–1753], we avoid computing D^m of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete H^m-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete H^m-norm. Numerical experiments validate our theoretical estimate.

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1. Introduction

We consider the th-Laplace equation

 (−1)mΔmu=f in Ω, (1.1a) u=∂u∂ν=⋯=∂m−1u∂νm−1=0 on ∂Ω, (1.1b)

where is an arbitrary positive integer, is a bounded Lipschitz polyhedral domain in (), and

is the outward unit normal vector field along

. The source term .

Several works have been done to solve numerically (1.1). Standard conforming finite elements space requires

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

is large. Bramble and Zlámal [2] studied the conforming finite elements space on the two dimensional triangular meshes. Meanwhile, a conforming finite element space is developed by Hu and Zhang on rectangular grids for arbitrary in [6]

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

finite element space on simplicial meshes for are desirable in both theoretical analysis and practice. In order to simplify the construction of finite element space, alternative nonconforming finite element space is introduced in several works. In [9], a nonconforming finite element space (named Morley-Wang-Xu elements) is introduced for . Besides, Hu and Zhang also considered the nonconforming finite element space in [7] on triangular meshes for . The finite element space in [9] is generalized for by Wu and Xu in [10]. Recently in [11], it is further generalized for arbitrary and 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 -norm, [9, 10, 11] propose to compute numerical approximation to , such that their implementation may become quite complicated as is large. The finite element spaces in [2, 6] can be used to solve numerically (1.1) with any source term . However, the implementation of these conforming and nonconforming finite element spaces can be quite challenging for large . Virtual element methods have been investigated for (1.1). In [1], a conforming virtual element method is introduced for convex polygonal domain in . The finite element space in [1] contains piecewise -th order polynomials, where . The virtual element method in [1] needs strong assumption on regularity of () to achieve optimal convergence (see [1, Theorem ]). In [4], a nonconforming virtual element method is developed for bounded Lipschitz polyhedral domain in , where can be any positive integer. The design of finite element space in [4], which contains piecewise -th () order polynomials, is based on a generalized Green’s identity for inner product. It is assumed that 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 interior penalty method (2.2) for (1.1) for arbitrary positive integers and . The finite element space of (2.2) is the standard -conforming piecewise -th order polynomials, where . The design of (2.2) avoids computing 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 ( for ) and the gradient of high order multiplicity of Laplace of numerical solution ( for ) on both elements and mesh interfaces. Therefore our method (2.2) can be easily implemented, even when is large and . After proving (Theorem 3.4) that discrete -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 with respect to the discrete -norm, for any positive integers and . Roughly speaking, we have

 ∥uh∥m,h≤ C∥f∥H−1(Ω), ∥uh−u∥m,h≤ Chmin(r+1−m,s−m)∥u∥Hs(Ω),

where . 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

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

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

2. C0 interior penalty method

In this section, firstly we give notations to define the interior penalty method for (1.1). Then in section 2.1, we derive the interior penalty method for any . Finally in section 2.2, we provide concrete examples of the method for .

Let be a quasi-uniform conforming simplicial mesh of . Here we define where is the diameter of the element . We denote by , and the collections of all -dimensional faces, interior faces and boundary faces of , respectively. Obviously, . For any positive integer , we define , where .

We introduce some trace operators. For any interior face , let be two elements sharing . We denote by and the outward unit normal vectors along and , respectively. For scalar function and vector field , which may be discontinuous across , we define the following quantities. For , , and , we define

 {{v}}=12(v−|F+v+|F),{{ϕ}}=12(ϕ−|F+ϕ+|F), ⟦v⟧=v−ν−|F+v+ν+|F,⟦ϕ⟧=ϕ−⋅ν−|F+ϕ+⋅ν+|F;

if , we define

 {{v}}=v+|F,{{ϕ}}=ϕ+|F,⟦v⟧=v+ν|F,⟦ϕ⟧=ϕ+⋅ν|F.

2.1. Derivation of C0 interior penalty method

We assume the exact solution . For any , via -times integrating by parts,

 ((−1)mΔmu,vh)Ω = ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(Δ~mu,Δ~mvh)Th−Σ~m−1i=0⟨Δ~m+iu,∂νΔ~m−i−1vh⟩∂Th+Σ~m−2i=0⟨∂νΔ~m+iu,Δ~m−i−1vh⟩∂Th, if m=2~m(m is an even number);(∇Δ~mu,∇Δ~mvh)Th+Σ~m−1i=0⟨Δ~m+i+1u,∂νΔ~m−i−1vh⟩∂Th−Σ~m−1i=0⟨∂νΔ~m+iu,Δ~m−ivh⟩∂Th, if m=2~m+1(m is an odd number).

Since , for any ,

 ((−1)mΔmu,vh)Ω (2.1) =

(2.1) inspires us to define the coupling term in Definition 2.1.

Definition 2.1.

For any , we define the coupling term along mesh interface by

 Ch(wh,vh) = ⎧⎪⎨⎪⎩−Σ~m−1i=0⟨{{Δ~m+iwh}},⟦∇Δ~m−i−1vh⟧⟩Fh+Σ~m−2i=0⟨{{∇Δ~m+iwh}},⟦Δ~m−i−1vh⟧⟩Fh, if m=2~m;Σ~m−1i=0⟨{{Δ~m+i+1wh}},⟦∇Δ~m−i−1vh⟧⟩Fh−Σ~m−1i=0⟨{{∇Δ~m+iwh}},⟦Δ~m−ivh⟧⟩Fh, if m=2~m+1.

In order to define interior penalty method, we need the stabilization term in Definition 2.2.

Definition 2.2.

For any , we define the stabilization term along mesh interface by

 Sh(wh,vh) = ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Σ~m−1i=0h−(4i+1)⟨⟦∇Δ~m−i−1wh⟧,⟦∇Δ~m−i−1vh⟧⟩Fh+Σ~m−2i=0h−(4i+3)⟨⟦Δ~m−i−1wh⟧,⟦Δ~m−i−1vh⟧⟩Fh, if m=2~m(m is an even number);Σ~m−1i=0h−(4i+3)⟨⟦∇Δ~m−i−1wh⟧,⟦∇Δ~m−i−1vh⟧⟩Fh+Σ~m−1i=0h−(4i+1)⟨⟦Δ~m−iwh⟧,⟦Δ~m−ivh⟧⟩Fh, if m=2~m+1(m is an odd number).

We would like to point out that if .

The interior penalty method is to find , such that for any ,

 ⎧⎪⎨⎪⎩(Δ~muh,Δ~mvh)Th+Ch(uh,vh)+Ch(vh,uh)+τSh(uh,vh)=(f,vh)Ω, if m=2~m;(∇Δ~muh,∇Δ~mvh)Th+Ch(uh,vh)+Ch(vh,uh)+τSh(uh,vh)=(f,vh)Ω, if m=2~m+1. (2.2)

Here the parameter shall be large enough but independent of .

2.2. Examples of C0 interior penalty method

• The interior penalty method for is to find satisfying

 (∇uh,∇vh)Ω=(f,vh)Ω,∀vh∈Vh. (2.3)
• The interior penalty method for is to find satisfying

 (2.4) +τh−1⟨⟦∇uh⟧,⟦∇vh⟧⟩Fh=(f,vh)Ω,∀vh∈Vh.
• The interior penalty method for is to find satisfying

 (∇Δuh,∇Δvh)Th+(⟨{{Δ2uh}},⟦∇vh⟧⟩Fh−⟨{{∇Δuh}},⟦Δvh⟧⟩Fh) (2.5) +τ(h−3⟨⟦∇uh⟧,⟦∇vh⟧⟩Fh+h−1⟨⟦Δuh⟧,⟦Δvh⟧⟩Fh)=(f,vh)Ω,∀vh∈Vh.

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

• The interior penalty method for is to find satisfying

 (Δ2uh,Δ2vh)Th (2.6) +τ(h−5⟨⟦∇uh⟧,⟦∇vh⟧⟩Fh+h−3⟨⟦Δuh⟧,⟦Δvh⟧⟩Fh+h−1⟨⟦∇Δuh⟧,⟦∇Δvh⟧⟩Fh) =(f,vh)Ω,∀vh∈Vh.

3. Analysis

In this section, firstly we prove Theorem 3.4, which states the discrete -norm (see Definition 3.1) bounded by the natural energy semi-norm associated with the 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 in the discrete -norm.

Definition 3.1.

For any integers , we define the discrete -norm by

 ∥v∥2m,h=Σmi=0∥Div∥2L2(Th)+Σm−1j=1h−(2m−2j−1)∥⟦Djv⟧J∥2L2(Fh), := Σmi=0ΣK∈Th∥Div∥2L2(K)+Σm−1j=1ΣF∈Fhh−(2m−2j−1)∥⟦Djv⟧J∥2L2(F),∀v∈H10(Ω)∩Hm(Th).

For any , there are two elements sharing the common face . We denote by and . We define

 ∥⟦Djv⟧J∥2L2(F) = ∑1≤k1,⋯,kj≤d∥⟦∂xk1∂xk2⋯∂xkjv⟧J∥2L2(F) := ∑1≤k1,⋯,kj≤d∥(∂xk1∂xk2⋯∂xkjv−)|F−(∂xk1∂xk2⋯∂xkjv+)|F∥2L2(F).

For any , we define

 ∥⟦Djv⟧J∥2L2(F) = ∑1≤k1,⋯,kj≤d∥⟦∂xk1∂xk2⋯∂xkjv⟧J∥2L2(F):=∑1≤k1,⋯,kj≤d∥(∂xk1∂xk2⋯∂xkjv)|F∥2L2(F).

3.1. Theorem 3.4 : discrete Hm-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 , there is a constant such that

 Σm−1j=1h−(2m−2j−1)∥⟦Djvh⟧J∥2L2(Fh)≤CSh(vh,vh),∀vh∈Vh. (3.1)
Proof.

We choose arbitrarily. There is an orthonormal coordinate system such that the -axis is parallel to normal vector along . Therefore -axis, , -axis are all parallel to .

We claim that for any , there is a positive integer such that

 ∥⟦Dl~vh⟧J∥2L2(F)≤C′(Σlj=0h−(2l−2j)∥⟦∂jyd~vh⟧J∥2L2(F)),∀~vh∈Pr(Th). (3.2)

When , it is easy to see

 ∥⟦D~vh⟧J∥2L2(F)=Σdk=1∥⟦∂xk~vh⟧J∥2L2(F)=Σdk=1∥⟦∂yk~vh⟧J∥2L2(F).

By discrete inverse inequality and the fact that -axis, , -axis are all parallel to , we have that

 Σd−1k=1∥⟦∂yk~vh⟧J∥2L2(F)≤Ch−2∥⟦~vh⟧J∥2L2(F).

Therefore we have

 ∥⟦D~vh⟧J∥2L2(F)≤C(h−2∥⟦~vh⟧J∥2L2(F)+∥⟦∂yd~vh⟧J∥2L2(F)).

Thus (3.2) holds when . We assume that (3.2) holds for any . Then by discrete inverse inequality and the fact that -axis, , -axis are all parallel to ,

 ∥⟦Dl+1~vh⟧J∥2L2(F)=Σdk=1∥⟦∂xkDl~vh⟧J∥2L2(F)=Σdk=1∥⟦∂ykDl~vh⟧J∥2L2(F) ≤ C(h−2∥⟦Dl~vh⟧J∥2L2(F)+∥Dl(∂yd~vh)∥2L2(F)).

Since , then . Since we assume (3.2) holds for , we have

 ∥Dl(∂yd~vh)∥2L2(F)≤C(Σlj=0h−(2l−2j)∥⟦∂jyd(∂yd~vh)⟧J∥2L2(F)), h−2∥⟦Dl~vh⟧J∥2L2(F)≤C(Σlj=0h−(2(l+1)−2j)∥⟦∂jyd~vh⟧J∥2L2(F)).

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

Since , (3.2) and the fact imply

 ∥⟦Dvh⟧J∥L2(F)=∥⟦∇vh⟧∥L2(F). (3.3)

Applying (3.2) with , we have

 ∥⟦D2vh⟧J∥2L2(F) (3.4) ≤ C(h−4∥⟦vh⟧J∥2L2(F)+h−2∥⟦∂ydvh⟧J∥2L2(F)+∥⟦∂2ydvh⟧J∥2L2(F)) = C(h−2∥⟦∇vh⟧∥2L2(F)+∥⟦∂2ydvh⟧J∥2L2(F)).

The last equality in (3.4) holds since and . We notice that

 Δvh=(∂2y1vh+⋯+∂2yd−1vh)+∂2ydvh. (3.5)

Since -axis, , -axis are all parallel to , discrete inverse inequality implies

 ∥⟦(∂2y1vh+⋯+∂2yd−1vh)⟧∥2L2(F) = ∥⟦(∂2y1vh+⋯+∂2yd−1vh)⟧J∥2L2(F) ≤ Ch−2∥⟦Dvh⟧J∥2L2(F)=Ch−2∥⟦∇vh⟧∥2L2(F).

By (3.5) and the above inequality, we have

 ∥⟦∂2ydvh⟧J∥2L2(F)≤C(h−2∥⟦∇vh⟧∥2L2(F)+∥⟦Δvh⟧∥2L2(F)). (3.6)

By (3.4, 3.6), we have

 (3.7)

Since is chosen arbitrarily, (3.3, 3.7) imply that (3.1) holds when .

We assume that is an odd number, and

 ∥⟦Dlvh⟧J∥2L2(F) (3.8) ≤ C(Σ~li=0h−4i∥⟦∇Δ~l−ivh⟧∥2L2(F)+Σ~l−1i=0h−(4i+2)∥⟦Δ~l−ivh⟧∥2L2(F)).

Then by applying (3.8) for each , we have

 ∥⟦Dl+1vh⟧J∥2L2(F)=Σdk=1∥⟦Dl(∂xkvh)⟧J∥2L2(F)=Σdk=1∥⟦Dl(∂ykvh)⟧J∥2L2(F) ≤ CΣdk=1(Σ~li=0h−4i∥⟦∇Δ~l−i(∂ykvh)⟧∥2L2(F)+Σ~l−1i=0h−(4i+2)∥⟦Δ~l−i(∂ykvh)⟧∥2L2(F)).

Here . Since -axis, , -axis are all parallel to , discrete inverse inequality implies

 ∥⟦Dl+1vh⟧J∥2L2(F) ≤ Ch−2(Σ~li=0h−4i∥⟦∇Δ~l−ivh⟧∥2L2(F)+Σ~l−1i=0h−(4i+2)∥⟦Δ~l−ivh⟧∥2L2(F)) +C(Σ~li=0h−4i∥⟦∇Δ~l−i(∂ydvh)⟧∥2L2(F)+Σ~l−1i=0h−(4i+2)∥⟦Δ~l−i(∂ydvh)⟧∥2L2(F)).

Again by the fact that -axis, , -axis are all parallel to , we have that for any ,

 ∥⟦∇Δ~l−i(∂ydvh)⟧∥2L2(F)=∥⟦∇(∂ydΔ~l−ivh)⟧J∥2L2(F) = (∥⟦∂y1(∂yd