# ConstrainedC0 Finite Element Methods for Biharmonic Problem

Authors: Rong An; Xuehai Huang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper presents some constrainedC0 finite element approximation methods for the biharmonic problem, which include the C0 symmetric interior penalty method, the C0 nonsymmetric interior penalty method, and the C0 nonsymmetric superpenalty method. In the finite element spaces, the C1 continuity across the interelement boundaries is obtained weakly by the constrained condition. For the C0 symmetric interior penalty method, the optimal error estimates in the broken H2 norm and in the L2 norm are derived. However, for the C0 nonsymmetric interior penalty method, the error estimate in the broken H2 norm is optimal and the error estimate in the L2 norm is suboptimal because of the lack of adjoint consistency. To obtain the optimal L2 error estimate, the C0 nonsymmetric superpenalty method is introduced and the optimal L2 error estimate is derived.


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## Body

## 1. Introduction

The discontinuous Galerkin methods (DGMs) have become a popular method to deal with the partial differential equations, especially for nonlinear hyperbolic problem, which exists the discontinuous solution even when the data is well smooth, and the convection-dominated diffusion problem, and the advection-diffusion problem. For the second-order elliptic problem, according to the different numerical fluxes, there exist different discontinuous Galerkin methods, such as the interior penalty method (IP), the nonsymmetric interior penalty method (NIPG), and local discontinuous Galerkin method (LDG). A unified analysis of discontinuous Galerkin methods for the second-order elliptic problem is studied by Arnold et al. in [1].The DGM for the fourth-order elliptic problem can be traced back to 1970s. Baker in [2] used the IP method to study the biharmonic problem and obtained the optimal error estimates. Moreover, for IP method, the C0 and C1 continuity can be achieved weakly by the interior penalty. Recently, using IP method and NIPG method, Süli and Mozolevski in [3–5] studied the hp-version DGM for the biharmonic problem, where the error estimates are optimal with respect to the mesh size h and are suboptimal with respect to the degree of the piecewise polynomial approximation p. However, we observe that the bilinear forms and the norms corresponding to the IP method in [3–5] are much complicated. A method to simplify the bilinear forms and the norms is using C0 interior penalty method. C0 interior penalty method for the biharmonic problem was introduced by Babusˇka and Zlámal in [6], where they used the nonconforming element and considered the inconsistent formulation and obtained the suboptimal error estimate. Motivated by the Engel and his collaborators’ work [7], Brenner and Sung in [8] studied the C0 interior penalty method for fourth-order problem on polygonal domains. They used the C0 finite element solution to approximate C1 solution by a postprocessing procedure, and the C1 continuity can be achieved weakly by the penalty on the jump of the normal derivatives on the interelement boundaries.In this paper, thanks to Rivière et al.’s idea in [9], we will study some constrained C0 finite element approximation methods for the biharmonic problem. The C1 continuity can be weakly achieved by a constrained condition that integrating the jump of the normal derivatives over the inter-element boundaries vanish. Under this constrained condition, we discuss three C0 finite element methods which include the C0 symmetric interior penalty method based on the symmetric bilinear form, the C0 nonsymmetric interior penalty method, and C0 nonsymmetric superpenalty method based on the nonsymmetric bilinear forms. First, we study the C0 symmetric interior penalty method and obtain the optimal error estimates in the broken H2 norm and in L2 norm. However, for the C0 nonsymmetric interior penalty method, the L2 norm is suboptimal because of the lack of adjoint consistency. Finally, in order to improve the order of the L2 error estimate, we give the C0 nonsymmetric superpenalty method and show the optimal L2 error estimates.


## 2.C0 Finite Element Approximation

LetΩ⊂ℝ2 be a bounded and convex domain with boundary ∂Ω. Consider the following biharmonic problem:
(2.1)Δ2u=f,inΩ,u=∂u∂n=0,on∂Ω,
where n denotes the unit external normal vector to ∂Ω. We assume that f is sufficiently smooth such that the problem (2.1) admits a unique solution u∈H4(Ω)∩H02(Ω).Let𝒯h be a family of nondegenerate triangular partition of Ω into triangles. The corresponding ordered triangles are denoted by K1,K2,…,KN. Let hi=diam(Ki),i=1,…,N, and h=max{h1,h2,…,hN}. The nondegenerate requirement is that there exists ρ>0 such that Ki contains a ball of radius ρhi in its interior. Conventionally, the boundary of Ki is denoted by ∂Ki. We denote
(2.2)eij=∂Ki∩∂Kj,ei=∂Ki∩∂Ω,heij=diam(eij),hei=diam(ei).Assume that the partition𝒯h is quasiuniform; that is, there exists a positive constant ν such that
(2.3)h⩽νhi,i=1,…,N.
Let ℰI and ℰB be the set of interior edges and boundary edges of 𝒯h, respectively. Let ℰ=ℰI∪ℰB. Denote by vi the restriction of v to Ki. Let e=eij∈ℰI with i>j. Then we denote the jump [v] and the average {v} of v on e by
(2.4)[v]|e=vi|e-vj|e,{v}|e=12(vi|e+vj|e).
If e=ei∈ℰB, we denote [v] and {v} of v on e by
(2.5)[v]|e={v}|e=vi|e.DefineV by
(2.6)V={v∈H01(Ω),v|K∈Hs(K),∀K∈𝒯h},s⩾3
with broken Hs norm
(2.7)∥|v|∥s=(∑K∈𝒯h‍∥v∥s,K2)1/2,∀v∈V,
where ||·||s,K is the standard Sobolev norm in Hs(K). Define the broken H2 norm by
(2.8)∥v∥h=(∑K∈𝒯h‍|v|2,K2)1/2,∀v∈V,
where |·|2,K is the seminorm in H2(K).For everyK∈𝒯h and any v∈V, we apply the integration by parts formula to obtain
(2.9)∫K‍(Δ2u)vdx=-∫K‍∇(Δu)·∇vdx+∫∂K‍∂Δu∂nvds=∫K‍ΔuΔvdx-∫∂K‍Δu∂v∂nds+∫∂K‍∂Δu∂nvds.
Summing all K∈𝒯h, we have
(2.10)∑K∈𝒯h‍∫K‍ΔuΔvdx-∑e∈ℰI‍∫e‍{Δu}[∂v∂n]+[Δu]{∂v∂n}ds+∑e∈ℰI‍∫e‍[∂Δu∂n]{v}+{∂Δu∂n}[v]ds-∫∂Ω‍Δu∂v∂nds=∫Ω‍fvdx.
Since u∈H4(Ω)∩H02(Ω) and v∈V, then [Δu]=[∂Δu/∂n]=[v]=0 on e∈ℰI. Thus, the previous identity can be simplified as follows:
(2.11)∑K∈𝒯h‍∫K‍ΔuΔvdx-∑e∈ℰI‍∫e‍{Δu}[∂v∂n]ds-∫∂Ω‍Δu∂v∂nds=∫Ω‍fvdx.Now, we introduce the following two bilinear forms:(2.12)aS(u,v)=∑K∈𝒯h‍∫K‍ΔuΔvdx-∑e∈ℰ‍∫e‍{Δu}[∂v∂n]ds-∑e∈ℰ‍∫e‍{Δv}[∂u∂n]ds+∑e∈ℰ‍βhe∫e‍[∂u∂n][∂v∂n]ds,aNS(u,v)=∑K∈𝒯h‍∫K‍ΔuΔvdx-∑e∈ℰ‍∫e‍{Δu}[∂v∂n]ds+∑e∈ℰ‍∫e‍{Δv}[∂u∂n]ds+∑e∈ℰ‍1he∫e‍[∂u∂n][∂v∂n]ds.
It is clear that aS(·,·) is a symmetric bilinear form and aNS(·,·) is a nonsymmetric bilinear form. In terms of (2.11) and
(2.13)∑e∈ℰI‍∫e‍{Δv}[∂u∂n]ds+∫∂Ω‍Δv∂u∂nds=0,
the solution u to problem (2.1) satisfies the following variational problems:
(SP)aS(u,v)=(f,v),∀v∈V,(NSP)aNS(u,v)=(f,v),∀v∈V.LetPr(K) denote the space of the polynomials on K of degree at most r. Define the following constrained C0 finite element space:
(2.14)Vh={vh∈H01(Ω),vh|K∈Pr(K),r⩾4,∀K∈𝒯h,∫e‍[∂vh∂n]ds=0,∀e∈ℰ},
from which we note that the C1 continuity of vh∈Vh can be weakly achieved by the constrained condition ∫e‍[∂vh/∂n]ds=0 for all e∈ℰ. Next, we define the degrees of freedom for this finite element space. To this end, for any K∈𝒯h, denote by pi(i=1,2,3) the three vertices of K. Recall that the degrees of freedom of Lagrange element on K are v(p), for all p∈𝒞′ with (cf. [10])
(2.15)𝒞′:={p=∑j=13‍λjpj;∑j=13‍λj=1,λj∈{0,1r,…,r-1r,1},1⩽j⩽3}.
Then we modify the degrees of freedom of Lagrange element to suit the constraint of normal derivatives over the edges in Vh. Specifically speaking, the degrees of freedom of Vh are given by
(2.16)v(p),∀p∈𝒞with𝒞:=𝒞′∖{1r(p1+p2+p3)+r-3rpi,i=1,2,3},∫e‍∂v∂ndsforeachedgeeofK.Based on the symmetric bilinear formaS(·,·), the C0 symmetric interior penalty finite element approximation of (2.1) is
(2.17)finduhS∈Vhsuchthat,aS(uhS,vh)=(f,vh),∀vh∈Vh.
Based on the nonsymmetric bilinear form aNS(·,·), the C0 nonsymmetric interior penalty finite element approximation of (2.1) is
(2.18)finduhNS∈Vhsuchthat,aNS(uhNS,vh)=(f,vh),∀vh∈Vh.
Moreover, the following orthogonal equations hold:
(2.19)aS(uhS-u,vh)=0,∀vh∈Vh,(2.20)aNS(uhNS-u,vh)=0,∀vh∈Vh.In order to introduce a global interpolation operator, we first defineϕhi∈Pr(Ki) for ϕ∈Hs(Ki) and Ki∈𝒯h according to the degrees of freedom of Vh by
(2.21)ϕhi(p)=ϕ(p),ifp∈𝒞∖∂Ω,ϕhi(p)=0,ifp∈𝒞∩∂Ω,∫e‍∂ϕhi∂nds=∫e‍∂ϕ∂nds,ifedgee⊂∂Ki∖∂Ω,∫e‍∂ϕhi∂nds=0,ifedgee⊂∂Ki∩∂Ω.
Due to standard scaling argument and Sobolev embedding theorem (cf. [10]), we have that for every Ki∈𝒯h,e∈∂Ki,ϕ∈Hs(Ki) with s⩾2, i=1,…,N,
(2.22)∥ϕ-ϕhi∥q,Ki⩽chμ-q∥ϕ∥s,Ki,0⩽q⩽μ,∥ϕ-ϕhi∥m,e⩽cheμ-1/2-m∥ϕ∥s,Ki,m=0,1,2,∥[∂(ϕ-ϕhi)∂n]∥0,e⩽cheμ-3/2∥ϕ∥s,Ki,
where μ=min{s,r+1} and c>0 is independent of h. We also suppose that the following inverse inequalities hold:
(2.23)∥ϕhi∥0,e⩽che-1/2∥ϕhi∥0,Ki,∥ϕhi∥1,e⩽che-1/2∥ϕhi∥1,Ki,∥ϕhi∥1,Ki⩽ch-1∥ϕhi∥0,Ki,
where c>0 is independent of h. Then for every ϕ∈V, we define the global interpolation operator Ph:V→Vh by Phϕ|Ki=ϕhi. Moreover, from (2.22) there holds
(2.24)∥ϕ-Phϕ∥h⩽chμ-2∥|ϕ|∥s,
where c>0 is independent of h.The following lemma is useful to establish the existence and uniqueness of the finite element approximation solution.Lemma 2.1.
There exists some constantκ0>0 independent of h such that
(2.25)κ0∥vh∥h2⩽∑K∈𝒯h‍∥Δvh∥0,K2+∑e∈ℰ‍1he∥[∂vh∂n]∥0,e2,∀vh∈Vh.Proof.
Introduce aH02 conforming finite element space Zh thanks to Guzmán and Neilan [11]. The advantage of Zh is that the degrees of freedom depend only on the values of functions and their first derivatives. Denote by Lh the interpolation operator from Vh to Zh. Then there holds
(2.26)∑K∈𝒯h‍|vh-Lhvh|2,K2⩽C1∑e∈ℰ‍1he∥[∂vh∂n]∥0,e2,∀vh∈Vh,
where C1>0 is independent of h. Thus, we have
(2.27)∥vh∥h2⩽∑K∈𝒯h‍|vh-Lhvh|2,K2+∑K∈𝒯h‍|Lhvh|2,K2⩽C1∑e∈ℰ‍1he∥[∂vh∂n]∥0,e2+C2∑K∈𝒯h‍∥Δ(Lhvh)∥0,K2⩽C1∑e∈ℰ‍1he∥[∂vh∂n]∥0,e2+C2∑K∈𝒯h‍∥Δ(Lhvh-vh)∥0,K2+C2∑K∈𝒯h‍∥Δvh∥0,K2⩽C1∑e∈ℰ‍1he∥[∂vh∂n]∥0,e2+C2∑K∈𝒯h‍|Lhvh-vh|2,K2+C2∑K∈𝒯h‍∥Δvh∥0,K2⩽C1(1+C2)∑e∈ℰ‍1he∥[∂vh∂n]∥0,e2+C2∑K∈𝒯h‍∥Δvh∥0,K2,
which completes the proof of (2.25) with κ0=1/max{C1(1+C2),C2}.


## 3.C0 Symmetric Interior Penalty Method

In this section, we will show the optimal error estimates in the brokenH2 norm and in the L2 norm between the solution u to problem (2.1) and the solution uhS to the problem (2.17). First, concerning the symmetric aS(·,·), we have the following coercive property in Vh.Lemma 3.1.
For sufficiently largeβ, there exists some constant κ1>0 such that
(3.1)aS(vh,vh)⩾κ1∥vh∥h2,∀vh∈Vh.Proof.
According to the definition ofaS(·,·), we have
(3.2)aS(vh,vh)=∑K∈𝒯h‍∫K‍|Δvh|2dx-2∑e∈ℰ‍∫e‍{Δvh}[∂vh∂n]ds+∑e∈ℰ‍βhe∫e‍|[∂vh∂n]|2ds.
Using the Hölder’s inequality and the Young’s inequality, we have
(3.3)∑e∈ℰ‍∫e‍{Δvh}[∂vh∂n]ds⩽(∑e∈ℰ‍he∥{Δvh}∥0,e2)1/2(∑e∈ℰ‍he-1∥[∂vh∂n]∥0,e2)1/2⩽c(∑K∈𝒯h‍∥Δvh∥0,K2)1/2(∑e∈ℰ‍he-1∥[∂vh∂n]∥0,e2)1/2⩽ε∑K∈𝒯h‍∥Δvh∥0,K2+cε-1∑e∈ℰ‍he-1∥[∂vh∂n]∥0,e2,
where c>0 is independent of h and ε>0 is a sufficiently small constant. Thus
(3.4)aS(vh,vh)⩾∑K∈𝒯h‍∥Δvh∥0,K2-2ε∑K∈𝒯h‍∥Δvh∥0,K2-cε-1∑e∈ℰ‍he-1∥[∂vh∂n]∥0,e2+∑e∈ℰ‍βhe∥[∂vh∂n]∥0,e2ds.
Taking ε=1/4, then, for sufficiently large β such that β>4c, using (2.25) we have
(3.5)aS(vh,vh)⩾12∑K∈𝒯h‍∥Δvh∥0,K2+(β-4c)∑e∈ℰ‍he-1∥[∂vh∂n]∥0,e2⩾κ1∥vh∥h2,
with κ1=min{1/2,β-4c}κ0.A direct result of Lemma3.1 is that the discretized problem (2.17) admits a unique solution uhS∈Vh for sufficiently large β.Lemma 3.2.
For allϕ∈V, there holds
(3.6)aS(ϕ-Phϕ,vh)⩽chμ-2∥vh∥h∥|ϕ|∥s,∀vh∈Vh,
where μ=min{s,r+1} and c>0 is independent of h.Proof.
For allϕ∈V and vh∈Vh, we have
(3.7)aS(ϕ-Phϕ,vh)=∑K∈𝒯h‍∫K‍Δ(ϕ-Phϕ)Δvhdx-∑e∈ℰ‍∫e‍{Δ(ϕ-Phϕ)}[∂vh∂n]ds-∑e∈ℰ‍∫e‍{Δvh}[∂(ϕ-Phϕ)∂n]ds+∑e∈ℰ‍βhe∫e‍[∂(ϕ-Phϕ)∂n][∂vh∂n]ds⩽∥ϕ-Phϕ∥h∥vh∥h+(∑e∈ℰ‍∥{Δ(ϕ-Phϕ)}∥0,e2)1/2(∑e∈ℰ‍∥[∂vh∂n]∥0,e2)1/2+(∑e∈ℰ‍he∥{Δvh}∥0,e2)1/2(∑e∈ℰ‍he-1∥[∂(ϕ-Phϕ)∂n]∥0,e2)1/2+c(∑e∈ℰ‍he-2∥[∂(ϕ-Phϕ)∂n]∥0,e2)1/2(∑e∈ℰ‍∥[∂vh∂n]∥0,e2)1/2,
where c>0 is independent of h. Note that ∫e‍[(∂vh)/∂n]ds=0 for e∈ℰ. Thus, for some constant Ce depending on e there holds
(3.8)∫e‍|[∂vh∂n]|2ds=∫e‍[∂vh∂n][∂vh∂n-Ce]ds⩽(∫e‍|[∂vh∂n]|2ds)1/2(∫e‍|[∂vh∂n-Ce]|2ds)1/2.
That is
(3.9)∫e‍|[∂vh∂n]|2ds⩽∫e‍|[∂vh∂n-Ce]|2ds.
From [9], we have
(3.10)∥[∂vh∂n-Ce]∥0,e⩽ch1/2|vh|2,K,
where c>0 is independent of h. Thus
(3.11)(∑e∈ℰ‍∥[∂vh∂n]∥0,e2)1/2⩽ch1/2∥vh∥h.
Substituting (3.11) into (3.7) and using (2.22)-(2.23) give
(3.12)aS(ϕ-Phϕ,vh)⩽∥ϕ-Phϕ∥h∥vh∥h+ch1/2(∑e∈ℰ‍∥{Δ(ϕ-Phϕ)}∥0,e2)1/2∥vh∥h+c(∑e∈ℰ‍he-1∥[∂(ϕ-Phϕ)∂n]∥0,e2)1/2∥vh∥h+ch1/2(∑e∈ℰ‍he-2∥[∂(ϕ-Phϕ)∂n]∥0,e2)1/2∥vh∥h⩽chμ-2∥|ϕ|∥s∥vh∥h.Theorem 3.3.
Suppose thatu∈V and uhS∈Vh are the solutions to problems (SP) and (2.17), respectively; then the following optimal broken H2 error estimate holds:
(3.13)∥u-uhS∥h⩽chμ-2∥|u|∥s,
where μ=min{s,r+1} and c>0 is independent of h.Proof.
According to Lemma3.1, we have
(3.14)κ1∥Phu-uhS∥h2⩽aS(Phu-uhS,Phu-uhS)=aS(Phu-u,Phu-uhS)+aS(u-uhS,Phu-uhS)=aS(Phu-u,Phu-uhS)⩽chμ-2∥|u|∥s∥Phu-uhS∥h,
where we use the orthogonal equation (2.19) and Lemma 3.2. The previous estimate implies
(3.15)∥Phu-uhS∥h⩽chμ-2∥|u|∥s.
Finally, the triangular inequality and (2.24) yield
(3.16)∥u-uhS∥h⩽∥u-Phu∥h+∥Phu-uhS∥h⩽chμ-2∥|u|∥s.Next, we will show the optimalL2 error estimate in terms of the duality technique. Suppose g∈L2(Ω) and consider the following biharmonic problem:
(3.17)Δ2w=g,inΩ,w=∂w∂n=0,on∂Ω.
Suppose that problem (3.17) admits a unique solution w∈H02(Ω)∩H4(Ω) such that
(3.18)∥w∥4⩽c∥g∥,
where ||·||4 denotes the H4 norm in Ω and ||·|| denotes the L2 norm in Ω and c>0 is independent of h.Denote byΠh the C1 continuous interpolate operator from V to H02(Ω)∩Vh, and Πh satisfies the approximation property (2.22). Then for the solution w∈H4(Ω)∩H02(Ω) to problem (3.17), there hold
(3.19)∥w-Πhw∥2⩽ch2∥w∥4⩽ch2∥g∥,∥w-Πhw∥2,e⩽che3/2∥w∥4⩽che3/2∥g∥,∀e∈ℰ,
where c>0 is independent of h.Theorem 3.4.
Suppose thatu∈V and uhS∈Vh are the solutions to problems (SP) and (2.17), respectively; then the following optimal L2 error estimate holds:
(3.20)∥u-uhS∥⩽chμ∥|u|∥s,
where μ=min{s,r+1} and c>0 is independent of h.Proof.
Settingg=u-uhS in (3.17), multiplying (3.17) by u-uhS, and integrating over Ω, we have
(3.21)∥u-uhS∥2=∫Ω‍(Δ2w)(u-uhS)dx=aS(w,u-uhS)=aS(w-Πhw,u-uhS)=aS(w-Πhw,Phu-uhS)+aS(w-Πhw,u-Phu),
where we use the orthogonal equation (2.19). We estimate two terms in the right-hand side of (3.21) as follows:
(3.22)aS(w-Πhw,Phu-uhS)=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(Phu-uhS)dx-∑e∈ℰ‍∫e‍{Δ(w-Πhw)}[∂(Phu-uhS)∂n]ds-∑e∈ℰ‍∫e‍{Δ(Phu-uhS)}[∂(w-Πhw)∂n]ds+∑e∈ℰ‍βhe∫e‍[∂(w-Πhw)∂n][∂(Phu-uhS)∂n]ds=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(Phu-uhS)dx-∑e∈ℰ‍∫e‍{Δ(w-Πhw)}[∂(Phu-uhS)∂n]ds⩽∥w-Πhw∥2∥Phu-uhS∥h+(∑e∈ℰ‍∥w-Πhw∥2,e2)1/2(∑e∈ℰ‍∥Phu-uhS∥1,e2)1/2⩽ch2∥Phu-uhS∥h∥u-uhS∥⩽chμ∥|u|∥s∥u-uhS∥,
where we use the estimate (3.15). In terms of the inequalities (2.22)-(2.23), we have
(3.23)aS(w-Πhw,u-Phu)=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(u-Phu)dx-∑e∈ℰ‍∫e‍{Δ(w-Πhw)}[∂(u-Phu)∂n]ds-∑e∈ℰ‍∫e‍{Δ(u-Phu)}[∂(w-Πhw)∂n]ds+∑e∈ℰ‍βhe∫e‍[∂(w-Πhw)∂n][∂(u-Phu)∂n]ds=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(u-Phu)dx-∑e∈ℰ‍∫e‍{Δ(w-Πhw)}[∂(u-Phu)∂n]ds⩽∥w-Phw∥2∥u-Phu∥h+(∑e∈ℰ‍∥w-Πhw∥2,e2)1/2(∑e∈ℰ‍∥[∂(u-Phu)∂n]∥0,e2)1/2⩽chμ∥|u|∥s∥u-uhS∥.
Substituting the estimates (3.22)-(3.23) into (3.21) yields
(3.24)∥u-uhS∥⩽chμ∥|u|∥s.


## 4.C0 Nonsymmetric Interior Penalty Method

In this section, we will show the error estimates in the brokenH2 norm and in L2 norm between the solution u to problem (2.1) and the solution uhNS to the problem (2.18). The optimal broken H2 error estimate is derived. However, the L2 error estimate is suboptimal because of the lack of adjoint consistency. According to Lemma 2.1, we have
(4.1)aNS(vh,vh)⩾κ0∥vh∥h2,∀vh∈Vh.
Moreover, for the nonsymmetric bilinear form aNS(·,·), proceeding as in the proof of Lemma 3.2, we have the following lemma.Lemma 4.1.
For allϕ∈V, there holds
(4.2)aNS(ϕ-Phϕ,vh)⩽chμ-2∥vh∥h∥|ϕ|∥s,∀vh∈Vh,
where μ=min{s,r+1} and c>0 is independent of h.Theorem 4.2.
Suppose thatu∈V and uhNS∈Vh are the solutions to problems (NSP) and (2.18), respectively; then there holds
(4.3)∥u-uhNS∥h⩽chμ-2∥|u|∥s,
where μ=min{s,r+1} and c>0 is independent of h.Proof.
According to (2.20), (4.1), and Lemma 4.1, we have
(4.4)κ0∥Phu-uhNS∥h2⩽aNS(Phu-uhNS,Phu-uhNS)=aNS(Phu-u,Phu-uhNS)+aNS(u-uhNS,Phu-uhNS)=aNS(Phu-u,Phu-uhNS)⩽chμ-2∥|u|∥s∥Phu-uhNS∥h,
where c>0 is independent of h. That is
(4.5)∥Phu-uhNS∥h⩽chμ-2∥|u|∥s.
Using the triangular inequality yields
(4.6)∥u-uhNS∥h⩽∥u-Phu∥h+∥Phu-uhNS∥h⩽chμ-2∥|u|∥s.Theorem 4.3.
Suppose thatu∈V and uhNS∈Vh are the solutions to problems (NSP) and (2.18), respectively; then there holds
(4.7)∥u-uhNS∥⩽chμ-1∥|u|∥s,
where μ=min{s,r+1} and c>0 is independent of h.Proof.
Settingg=u-uhNS in (3.17), multiplying (3.17) by u-uhNS, and integrating over Ω, we have
(4.8)∥u-uhNS∥2=∫Ω‍(Δ2w)(u-uhNS)dx=aNS(w,u-uhNS)=aNS(w-Πhw,u-uhNS)+2∑e∈ℰ‍∫e‍{ΔΠhw}[∂(u-uhNS)∂n]ds-2∑e∈ℰ‍∫e‍{Δ(u-uhNS)}[∂Πhw∂n]ds=aNS(w-Πhw,u-uhNS)+2∑e∈ℰ‍∫e‍{ΔΠhw}[∂(u-uhNS)∂n]ds=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(u-uhNS)dx+3∑e∈ℰ‍∫e‍{Δ(Πhw-w)}[∂(u-uhNS)∂n]ds+2∑e∈ℰ‍∫e‍{Δw}[∂(u-uhNS)∂n]ds=I1+I2+I3,
where we use Πhw∈H02(Ω). We estimate I1 as follows:
(4.9)I1=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(u-uhNS)dx⩽∥u-uhNS∥h∥w-Πhw∥2⩽chμ∥|u|∥s∥u-uhNS∥.
We estimate I2 as follows:
(4.10)I2=3∑e∈ℰ‍∫e‍{Δ(Πhw-w)}[∂(u-uhNS)∂n]ds=3∑e∈ℰ‍∫e‍{Δ(Πhw-w)}[∂(u-Phu)∂n]ds+3∑e∈ℰ‍∫e‍{Δ(Πhw-w)}[∂(Phu-uhNS)∂n]ds⩽c(∑e∈ℰ‍∥w-Πhw∥2,e2)1/2(∑e∈ℰ‍∥[∂(u-Phu)∂n]∥0,e2)1/2+(∑e∈ℰ‍∥w-Πhw∥2,e2)1/2(∑e∈ℰ‍∥[∂(Phu-uhNS)∂n]∥0,e2)1/2⩽ch3/2(hμ-3/2∥|u|∥s+ch1/2∥Phu-uhNS∥h)∥u-uhNS∥⩽chμ∥|u|∥s∥u-uhNS∥.
We estimate I3 as follows:
(4.11)I3=2∑e∈ℰ‍∫e‍{Δw}[∂(u-uhNS)∂n]ds=2∑e∈ℰ‍∫e‍{Δw-Ce}[∂(u-uhNS)∂n]ds=2∑e∈ℰ‍∫e‍{Δw-Ce}[∂(u-Phu)∂n]ds+2∑e∈ℰ‍∫e‍{Δw-Ce}[∂(Phu-uhNS)∂n]ds⩽c(∑e∈ℰ‍∥Δw-Ce∥0,e2)1/2(∑e∈ℰ‍∥u-Phu∥1,e2)1/2+(∑e∈ℰ‍∥Δw-Ce∥0,e2)1/2(∑e∈ℰ‍∥[∂(Phu-uhNS)∂n]∥0,e2)1/2⩽c(∑e∈ℰ‍∥Δw-Ce∥0,e2)1/2(hμ-3/2∥|u|∥s+ch1/2∥Phu-uhNS∥h)⩽chμ-3/2(∑e∈ℰ‍∥Δw-Ce∥0,e2)1/2∥|u|∥s,
where Ce is some positive constant. Substituting the following estimate
(4.12)∑e∈ℰ‍∥Δw-Ce∥0,e2⩽c∑e∈ℰ‍he∥w∥3,e2⩽ch∥w∥42⩽ch∥u-uhNS∥2
into (4.11) gives
(4.13)I3⩽chμ-1∥|u|∥s∥u-uhNS∥.
Finally, substituting (4.9)–(4.13) into (4.8), we obtain
(4.14)∥u-uhNS∥⩽chμ-1∥|u|∥s.


## 5.C0 Superpenalty Nonsymmetric Method

In order to obtain the optimalL2 error estimate for the nonsymmetric method, in this section we will consider the C0 superpenalty nonsymmetric method. First, we introduce a new nonsymmetric bilinear form:
(5.1)aSNS(u,v)=∑K∈𝒯h‍∫K‍ΔuΔvdx-∑e∈ℰ‍∫e‍{Δu}[∂v∂n]ds+∑e∈ℰ‍∫e‍{Δv}[∂u∂n]ds+∑e∈ℰ‍1he3∫e‍[∂u∂n][∂v∂n].
The broken H2 norm is modified to
(5.2)∥|vh|∥h=(∑K∈𝒯h‍|vh|2,Ki2+∑e∈ℰ‍1he3∥[∂vh∂n]∥0,e2)1/2,∀vh∈Vh.
From Lemma 2.1, it is easy to show that there exists some constant κ2>0 such that
(5.3)aSNS(vh,vh)⩾κ2∥|vh|∥h2,∀vh∈Vh.
In fact, we have
(5.4)aSNS(vh,vh)=∑K∈𝒯h‍∥Δvh∥0,K2+∑e∈ℰ‍1he3∥[∂vh∂n]∥0,e2⩾12∑e∈ℰ‍1he3∥[∂vh∂n]∥0,e2+12(∑K∈𝒯h‍∥Δvh∥0,K2+∑e∈ℰ‍1he∥[∂vh∂n]∥0,e2)⩾12∑e∈ℰ‍1he3∥[∂vh∂n]∥0,e2+κ02∑K∈𝒯h‍|vh|2,K2⩾κ2∥|vh|∥h2
for κ2=min{1/2,κ0/2}. Since the solution u to problem (2.1) belongs to H4(Ω)∩H02(Ω), then it satisfies
(SNSP)aSNS(u,v)=(f,v),∀v∈V.
In this case, the the C0 superpenalty nonsymmetric finite element approximation of (2.1) is
(5.5)finduhSNS∈Vhsuchthat,aSNS(uhSNS,vh)=(f,vh),∀vh∈Vh.
Then, we have the following orthogonal equation:
(5.6)aSNS(uhSNS-u,vh)=0,∀vh∈Vh.LetΠh be the C1 continuous interpolated operator defined in Section 3. Observe that [(∂u-Πhu)/∂n]=0 on every e∈ℰ, and proceeding as in the proof of Lemmas 3.2 and 4.1, we have the following.Lemma 5.1.
For allϕ∈V, there holds
(5.7)aNS(ϕ-Πhϕ,vh)⩽chμ-2∥|vh|∥h∥|ϕ|∥s,∀vh∈Vh,
where μ=min{s,r+1} and c>0 is independent of h.Using Lemma5.1, the following optimal broken H2 error estimate holds.Theorem 5.2.
Suppose thatu∈V and uhSNS∈Vh are the solutions to problems (SNSP) and (5.5), respectively; then there holds
(5.8)∥|u-uhSNS|∥h⩽chμ-2∥|u|∥s,
where μ=min{s,r+1} and c>0 is independent of h.The main result in this section is the following optimalL2 error estimate.Theorem 5.3.
Suppose thatu∈V and uhSNS∈Vh are the solutions to problems (SNSP) and (5.5), respectively; then there holds
(5.9)∥u-uhSNS∥⩽chμ∥|u|∥s,
where μ=min{s,r+1} and c>0 is independent of h.Proof.
Letg=u-uhSNS. Multiplying (3.17) by u-uhSNS and integrating over Ω, we have
(5.10)∥u-uhSNS∥2=∫Ω‍(Δ2w)(u-uhSNS)dx=aSNS(w,u-uhSNS)=aSNS(w-Πhw,u-uhSNS)+2∑e∈ℰ‍∫e‍{ΔΠhw}[∂(u-uhSNS)∂n]ds-2∑e∈ℰ‍∫e‍{Δ(u-uhSNS)}[∂Πhw∂n]ds=aSNS(w-Πhw,u-uhSNS)+2∑e∈ℰ‍∫e‍{ΔΠhw}[∂(u-uhSNS)∂n]ds=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(u-uhSNS)dx+3∑e∈ℰ‍∫e‍{Δ(Πhw-w)}[∂(u-uhSNS)∂n]ds+2∑e∈ℰ‍∫e‍{Δw}[∂(u-uhSNS)∂n]ds=I4+I5+I6,
where we use Πhw∈C1(Ω)∩H02(Ω). Proceeding as in the proof of Theorem 4.2, we can estimate I4 and I5 as follows:
(5.11)I4=∑K∈𝒯h‍∫K‍Δ(w-Πhw)Δ(u-uhSNS)dx⩽chμ∥|u|∥s∥u-uhSNS∥,(5.12)I5=3∑e∈ℰ‍∫e‍{Δ(Πhw-w)}[∂(u-uhSNS)∂n]ds⩽chμ∥|u|∥s∥u-uhSNS∥.
The different estimate compared to Theorem 4.3 is the estimate of I6. Under the new norm |||·|||h, we have
(5.13)I6=2∑e∈ℰ‍∫e‍{Δw}[∂(u-uhSNS)∂n]ds=2∑e∈ℰ‍∫e‍{Δw-c}[∂(u-uhSNS)∂n]ds⩽c(∑e∈ℰ‍he3∥Δw-C∥0,e2)1/2(∑e∈ℰ‍1he3∥[∂(u-uhSNS)∂n]∥0,e2)1/2⩽ch2∥u-uhSNS∥·∥|u-uhSNS|∥h⩽chμ∥|u|∥s∥u-uhSNS∥.
Substituting (5.11)–(5.13) into (5.10) yields the following optimal L2 error estimate:
(5.14)∥u-uhSNS∥⩽chμ∥|u|∥s.



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*Source: 863125-2012-12-13.xml*