Constrained 𝐶⁰ Finite Element Methods for Biharmonic Problem

Abstract

This paper presents some constrained 𝐶⁰ 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.

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 Babu𝑠̌ka 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

Let Ω⊂ℝ² be a bounded and convex domain with boundary 𝜕Ω. Consider the following biharmonic problem: Δ²𝑢=𝑓,inΩ,𝑢=𝜕𝑢𝜕𝑛=0,on𝜕Ω,(2.1) 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)∩𝐻²₀(Ω).

Let 𝒯_ℎ be a family of nondegenerate triangular partition of Ω into triangles. The corresponding ordered triangles are denoted by 𝐾₁,𝐾₂,…,𝐾_𝑁. Let ℎ_𝑖=diam(𝐾_𝑖),𝑖=1,…,𝑁, and ℎ=max{ℎ₁,ℎ₂,…,ℎ_𝑁}. The nondegenerate requirement is that there exists 𝜌>0 such that 𝐾_𝑖 contains a ball of radius 𝜌ℎ_𝑖 in its interior. Conventionally, the boundary of 𝐾_𝑖 is denoted by 𝜕𝐾_𝑖. We denote 𝑒_𝑖𝑗=𝜕𝐾_𝑖∩𝜕𝐾_𝑗,𝑒_𝑖=𝜕𝐾_𝑖∩𝜕Ω,ℎ_𝑒𝑖𝑗=diam𝑒_𝑖𝑗,ℎ_𝑒𝑖=diam𝑒_𝑖.(2.2)

Assume that the partition 𝒯_ℎ is quasiuniform; that is, there exists a positive constant 𝜈 such that ℎ⩽𝜈ℎ_𝑖,𝑖=1,…,𝑁.(2.3) Let ℰ_𝐼 and ℰ_𝐵 be the set of interior edges and boundary edges of 𝒯_ℎ, respectively. Let ℰ=ℰ_𝐼∪ℰ_𝐵. Denote by 𝑣^𝑖 the restriction of 𝑣 to 𝐾_𝑖. Let 𝑒=𝑒_𝑖𝑗∈ℰ_𝐼 with 𝑖>𝑗. Then we denote the jump [𝑣] and the average {𝑣} of 𝑣 on 𝑒 by [𝑣]|_𝑒=𝑣^𝑖_𝑒−𝑣^𝑗_𝑒,{𝑣}|_𝑒=12𝑣^𝑖_𝑒+𝑣^𝑗_𝑒.(2.4) If 𝑒=𝑒_𝑖∈ℰ_𝐵, we denote [𝑣] and {𝑣} of 𝑣 on 𝑒 by [𝑣]|_𝑒={𝑣}|_𝑒=𝑣^𝑖_𝑒.(2.5)

Define 𝑉 by 𝑉=𝑣∈𝐻¹₀(Ω),𝑣|_𝐾∈𝐻^𝑠(𝐾),∀𝐾∈𝒯_ℎ,𝑠⩾3(2.6) with broken 𝐻^𝑠 norm ‖|𝑣|‖_𝑠=_{𝐾∈𝒯ℎ}‖𝑣‖²_𝑠,𝐾^1/2,∀𝑣∈𝑉,(2.7) where ||⋅||_𝑠,𝐾 is the standard Sobolev norm in 𝐻^𝑠(𝐾). Define the broken 𝐻² norm by ‖𝑣‖_ℎ=_{𝐾∈𝒯ℎ}|𝑣|²_2,𝐾^1/2,∀𝑣∈𝑉,(2.8) where |⋅|_2,𝐾 is the seminorm in 𝐻²(𝐾).

For every 𝐾∈𝒯_ℎ and any 𝑣∈𝑉, we apply the integration by parts formula to obtain _𝐾Δ²𝑢𝑣𝑑𝑥=−_𝐾∇(Δ𝑢)⋅∇𝑣𝑑𝑥+_𝜕𝐾𝜕Δ𝑢=𝜕𝑛𝑣𝑑𝑠_𝐾Δ𝑢Δ𝑣𝑑𝑥−_𝜕𝐾Δ𝑢𝜕𝑣𝜕𝑛𝑑𝑠+_𝜕𝐾𝜕Δ𝑢𝜕𝑛𝑣𝑑𝑠.(2.9) Summing all 𝐾∈𝒯_ℎ, we have _{𝐾∈𝒯ℎ𝐾}Δ𝑢Δ𝑣𝑑𝑥−_{𝑒∈ℰ𝐼𝑒}{Δ𝑢}𝜕𝑣+[𝜕𝑛Δ𝑢]𝜕𝑣𝜕𝑛𝑑𝑠+_{𝑒∈ℰ𝐼𝑒}𝜕Δ𝑢𝑣}+𝜕𝑛{𝜕Δ𝑢𝑣]𝜕𝑛[𝑑𝑠−_𝜕ΩΔ𝑢𝜕𝑣𝜕𝑛𝑑𝑠=_Ω𝑓𝑣𝑑𝑥.(2.10) Since 𝑢∈𝐻⁴(Ω)∩𝐻²₀(Ω) and 𝑣∈𝑉, then [Δ𝑢]=[𝜕Δ𝑢/𝜕𝑛]=[𝑣]=0 on 𝑒∈ℰ_𝐼. Thus, the previous identity can be simplified as follows: _{𝐾∈𝒯ℎ𝐾}Δ𝑢Δ𝑣𝑑𝑥−_{𝑒∈ℰ𝐼𝑒}{Δ𝑢}𝜕𝑣𝜕𝑛𝑑𝑠−_𝜕ΩΔ𝑢𝜕𝑣𝜕𝑛𝑑𝑠=_Ω𝑓𝑣𝑑𝑥.(2.11)

Now, we introduce the following two bilinear forms: 𝑎_𝑆(𝑢,𝑣)=_{𝐾∈𝒯ℎ𝐾}Δ𝑢Δ𝑣𝑑𝑥−_{𝑒∈ℰ𝑒}{Δ𝑢}𝜕𝑣𝜕𝑛𝑑𝑠−_{𝑒∈ℰ𝑒}{Δ𝑣}𝜕𝑢𝜕𝑛𝑑𝑠+_𝑒∈ℰ𝛽ℎ_𝑒𝑒𝜕𝑢𝜕𝑛𝜕𝑣𝑎𝜕𝑛𝑑𝑠,_NS(𝑢,𝑣)=_{𝐾∈𝒯ℎ𝐾}Δ𝑢Δ𝑣𝑑𝑥−_{𝑒∈ℰ𝑒}{Δ𝑢}𝜕𝑣𝜕𝑛𝑑𝑠+_{𝑒∈ℰ𝑒}{Δ𝑣}𝜕𝑢𝜕𝑛𝑑𝑠+_𝑒∈ℰ1ℎ_𝑒𝑒𝜕𝑢𝜕𝑛𝜕𝑣𝜕𝑛𝑑𝑠.(2.12) It is clear that 𝑎_𝑆(⋅,⋅) is a symmetric bilinear form and 𝑎_NS(⋅,⋅) is a nonsymmetric bilinear form. In terms of (2.11) and _{𝑒∈ℰ𝐼𝑒}{Δ𝑣}𝜕𝑢𝜕𝑛𝑑𝑠+_𝜕ΩΔ𝑣𝜕𝑢𝜕𝑛𝑑𝑠=0,(2.13) the solution 𝑢 to problem (2.1) satisfies the following variational problems: 𝑎_𝑆(𝑢,𝑣)=𝑓,𝑣,∀𝑣∈𝑉,(SP)𝑎_𝑁𝑆(𝑢,𝑣)=𝑓,𝑣,∀𝑣∈𝑉.(NSP)

Let 𝑃_𝑟(𝐾) denote the space of the polynomials on 𝐾 of degree at most 𝑟. Define the following constrained 𝐶⁰ finite element space: 𝑉_ℎ=𝑣_ℎ∈𝐻¹₀(Ω),𝑣_ℎ𝐾∈𝑃_𝑟(𝐾),𝑟⩾4,∀𝐾∈𝒯_ℎ,_𝑒𝜕𝑣_ℎ,𝜕𝑛𝑑𝑠=0,∀𝑒∈ℰ(2.14) from which we note that the 𝐶¹ continuity of 𝑣_ℎ∈𝑉_ℎ can be weakly achieved by the constrained condition ∫_𝑒[𝜕𝑣_ℎ/𝜕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∈𝒯_ℎ, denote by 𝑝_𝑖(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]) 𝒞^′∶=𝑝=³_𝑗=1𝜆_𝑗𝑝_𝑗;³_𝑗=1𝜆_𝑗=1,𝜆_j∈10,𝑟,…,𝑟−1𝑟,1,1⩽𝑗⩽3.(2.15) Then we modify the degrees of freedom of Lagrange element to suit the constraint of normal derivatives over the edges in 𝑉_ℎ. Specifically speaking, the degrees of freedom of 𝑉_ℎ are given by 𝑣𝑝,∀𝑝∈𝒞with𝒞∶=𝒞^′⧵1𝑟𝑝₁+𝑝₂+𝑝₃+𝑟−3𝑟𝑝_𝑖,𝑖=1,2,3_𝑒,𝜕𝑣𝜕𝑛𝑑𝑠foreachedge𝑒of𝐾.(2.16)

Based on the symmetric bilinear form 𝑎_𝑆(⋅,⋅), the 𝐶⁰ symmetric interior penalty finite element approximation of (2.1) is find𝑢^𝑆_ℎ∈𝑉_ℎsuchthat,𝑎_𝑆𝑢^𝑆_ℎ,𝑣_ℎ=𝑓,𝑣_ℎ,∀𝑣_ℎ∈𝑉_ℎ.(2.17) Based on the nonsymmetric bilinear form 𝑎_NS(⋅,⋅), the 𝐶⁰ nonsymmetric interior penalty finite element approximation of (2.1) is find𝑢^NS_ℎ∈𝑉_ℎsuchthat,𝑎_NS𝑢^NS_ℎ,𝑣_ℎ=𝑓,𝑣_ℎ,∀𝑣_ℎ∈𝑉_ℎ.(2.18) Moreover, the following orthogonal equations hold: 𝑎_𝑆𝑢^𝑆_ℎ−𝑢,𝑣_ℎ=0,∀𝑣_ℎ∈𝑉_ℎ,(2.19)𝑎_NS𝑢^NS_ℎ−𝑢,𝑣_ℎ=0,∀𝑣_ℎ∈𝑉_ℎ.(2.20)

In order to introduce a global interpolation operator, we first define 𝜙^𝑖_ℎ∈𝑃_𝑟(𝐾_𝑖) for 𝜙∈𝐻^𝑠(𝐾_𝑖) and 𝐾_𝑖∈𝒯_ℎ according to the degrees of freedom of 𝑉_ℎ by 𝜙^𝑖_ℎ𝑝=𝜙𝑝,if𝑝∈𝒞⧵𝜕Ω,𝜙^𝑖_ℎ𝑝=0,if𝑝∈𝒞∩𝜕Ω,_𝑒𝜕𝜙^𝑖_ℎ𝜕𝑛𝑑𝑠=_𝑒𝜕𝜙𝜕𝑛𝑑𝑠,ifedge𝑒⊂𝜕𝐾_𝑖⧵𝜕Ω,_𝑒𝜕𝜙^𝑖_ℎ𝜕𝑛𝑑𝑠=0,ifedge𝑒⊂𝜕𝐾_𝑖∩𝜕Ω.(2.21) Due to standard scaling argument and Sobolev embedding theorem (cf. [10]), we have that for every 𝐾_𝑖∈𝒯_ℎ,𝑒∈𝜕𝐾_𝑖,𝜙∈𝐻^𝑠(𝐾_𝑖) with 𝑠⩾2, 𝑖=1,…,𝑁, 𝜙−𝜙^𝑖_{ℎ𝑞,𝐾𝑖}⩽𝑐ℎ^𝜇−𝑞𝜙_{𝑠,𝐾𝑖},0⩽𝑞⩽𝜇,𝜙−𝜙^𝑖_ℎ𝑚,𝑒⩽𝑐ℎ_𝑒^{𝜇−1/2−𝑚}𝜙_{𝑠,𝐾𝑖}𝜕,𝑚=0,1,2,𝜙−𝜙^𝑖_ℎ𝜕𝑛_0,𝑒⩽𝑐ℎ_𝑒^𝜇−3/2𝜙_{𝑠,𝐾𝑖},(2.22) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ. We also suppose that the following inverse inequalities hold: 𝜙^𝑖_ℎ0,𝑒⩽𝑐ℎ_𝑒^−1/2𝜙^𝑖_{ℎ0,𝐾𝑖},𝜙^𝑖_ℎ1,𝑒⩽𝑐ℎ_𝑒^−1/2𝜙^𝑖_{ℎ1,𝐾𝑖},𝜙^𝑖_{ℎ1,𝐾𝑖}⩽𝑐ℎ⁻¹𝜙^𝑖_{ℎ0,𝐾𝑖},(2.23) where 𝑐>0 is independent of ℎ. Then for every 𝜙∈𝑉, we define the global interpolation operator 𝑃_ℎ∶𝑉→𝑉_ℎ by 𝑃_ℎ𝜙|_𝐾𝑖=𝜙^𝑖_ℎ. Moreover, from (2.22) there holds 𝜙−𝑃_ℎ𝜙_ℎ⩽𝑐ℎ^𝜇−2𝜙_𝑠,(2.24) 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 constant 𝜅₀>0 independent of ℎ such that 𝜅₀𝑣_ℎ²_ℎ⩽_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾+_𝑒∈ℰ1ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒,∀𝑣_ℎ∈𝑉_ℎ.(2.25)

Proof. Introduce a 𝐻²₀ conforming finite element space 𝑍_ℎ thanks to Guzmán and Neilan [11]. The advantage of 𝑍_ℎ is that the degrees of freedom depend only on the values of functions and their first derivatives. Denote by 𝐿_ℎ the interpolation operator from 𝑉_ℎ to 𝑍_ℎ. Then there holds _{𝐾∈𝒯ℎ}𝑣_ℎ−𝐿_ℎ𝑣_ℎ²_2,𝐾⩽𝐶_1𝑒∈ℰ1ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒,∀𝑣_ℎ∈𝑉_ℎ,(2.26) where 𝐶₁>0 is independent of ℎ. Thus, we have 𝑣_ℎ²_ℎ⩽_{𝐾∈𝒯ℎ}𝑣_ℎ−𝐿_ℎ𝑣_ℎ²_2,𝐾+_{𝐾∈𝒯ℎ}𝐿_ℎ𝑣_ℎ²_2,𝐾⩽𝐶_1𝑒∈ℰ1ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒+𝐶_{2𝐾∈𝒯ℎ}Δ(𝐿_ℎ𝑣_ℎ)²_0,𝐾⩽𝐶_1𝑒∈ℰ1ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒+𝐶_{2𝐾∈𝒯ℎ}Δ𝐿_ℎ𝑣_ℎ−𝑣_ℎ²_0,𝐾+𝐶_{2𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾⩽𝐶_1𝑒∈ℰ1ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒+𝐶_{2𝐾∈𝒯ℎ}𝐿_ℎ𝑣_ℎ−𝑣_ℎ²_2,𝐾+𝐶_{2𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾⩽𝐶₁1+𝐶_2𝑒∈ℰ1ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒+𝐶_2K∈𝒯ℎΔ𝑣_ℎ²_0,𝐾,(2.27) 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 broken 𝐻² norm and in the 𝐿² norm between the solution 𝑢 to problem (2.1) and the solution 𝑢^𝑆_ℎ to the problem (2.17). First, concerning the symmetric 𝑎_𝑆(⋅,⋅), we have the following coercive property in 𝑉_ℎ.

Lemma 3.1. For sufficiently large 𝛽, there exists some constant 𝜅₁>0 such that 𝑎_𝑆𝑣_ℎ,𝑣_ℎ⩾𝜅₁𝑣_ℎ²_ℎ,∀𝑣_ℎ∈𝑉_ℎ.(3.1)

Proof. According to the definition of 𝑎_𝑆(⋅,⋅), we have 𝑎_𝑆𝑣_ℎ,𝑣_ℎ=_{𝐾∈𝒯ℎ𝐾}Δ𝑣_ℎ²𝑑𝑥−2_{𝑒∈ℰ𝑒}Δ𝑣_ℎ𝜕𝑣_ℎ𝜕𝑛𝑑𝑠+_𝑒∈ℰ𝛽ℎ_𝑒𝑒𝜕𝑣_ℎ𝜕𝑛²𝑑𝑠.(3.2) Using the Hölder’s inequality and the Young’s inequality, we have _{𝑒∈ℰ𝑒}Δ𝑣_ℎ𝜕𝑣_ℎ𝜕𝑛𝑑𝑠⩽_𝑒∈ℰℎ_𝑒Δ𝑣_ℎ²_0,𝑒^1/2_𝑒∈ℰℎ_𝑒⁻¹𝜕𝑣_ℎ𝜕𝑛²_0,𝑒^1/2⩽𝑐_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾^1/2_𝑒∈ℰℎ_𝑒⁻¹𝜕𝑣_ℎ𝜕𝑛²_0,𝑒^1/2⩽𝜀_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾+𝑐𝜀⁻¹_𝑒∈ℰℎ_𝑒⁻¹𝜕𝑣_ℎ𝜕𝑛²_0,𝑒,(3.3) where 𝑐>0 is independent of ℎ and 𝜀>0 is a sufficiently small constant. Thus 𝑎_𝑆𝑣_ℎ,𝑣_ℎ⩾_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾−2𝜀_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾−𝑐𝜀⁻¹_𝑒∈ℰℎ_𝑒⁻¹𝜕𝑣_ℎ𝜕𝑛²_0,𝑒+_𝑒∈ℰ𝛽ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒𝑑𝑠.(3.4) Taking 𝜀=1/4, then, for sufficiently large 𝛽 such that 𝛽>4𝑐, using (2.25) we have 𝑎_𝑆𝑣_ℎ,𝑣_ℎ⩾12_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾+𝛽−4𝑐_𝑒∈ℰℎ_𝑒⁻¹𝜕𝑣_ℎ𝜕𝑛²_0,𝑒⩾𝜅₁𝑣_ℎ²_ℎ,(3.5) with 𝜅₁=min{1/2,𝛽−4𝑐}𝜅₀.

A direct result of Lemma 3.1 is that the discretized problem (2.17) admits a unique solution 𝑢^𝑆_ℎ∈𝑉_ℎ for sufficiently large 𝛽.

Lemma 3.2. For all 𝜙∈𝑉, there holds 𝑎_𝑆𝜙−𝑃_ℎ𝜙,𝑣_ℎ⩽𝑐ℎ^𝜇−2𝑣_ℎℎ𝜙_𝑠,∀𝑣_ℎ∈𝑉_ℎ,(3.6) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

Proof. For all 𝜙∈𝑉 and 𝑣_ℎ∈𝑉_ℎ, we have 𝑎_𝑆𝜙−𝑃_ℎ𝜙,𝑣_ℎ=_{𝐾∈𝒯ℎ𝐾}Δ𝜙−𝑃_ℎ𝜙Δ𝑣_ℎ𝑑𝑥−_{𝑒∈ℰ𝑒}Δ𝜙−𝑃_ℎ𝜙𝜕𝑣_ℎ𝜕𝑛𝑑𝑠−_{𝑒∈ℰ𝑒}Δ𝑣_ℎ𝜕𝜙−𝑃_ℎ𝜙𝜕𝑛𝑑𝑠+_𝑒∈ℰ𝛽ℎ_𝑒𝑒𝜕𝜙−𝑃_ℎ𝜙𝜕𝑛𝜕𝑣_ℎ⩽𝜕𝑛𝑑𝑠𝜙−𝑃_ℎ𝜙_ℎ𝑣_ℎℎ+_𝑒∈ℰΔ𝜙−𝑃_ℎ𝜙²_0,𝑒^1/2_𝑒∈ℰ𝜕𝑣_ℎ𝜕𝑛²_0,𝑒^1/2+_𝑒∈ℰℎ_𝑒Δ𝑣_ℎ²_0,𝑒^1/2_𝑒∈ℰℎ_𝑒⁻¹𝜕𝜙−𝑃_ℎ𝜙𝜕𝑛²_0,𝑒^1/2+𝑐_𝑒∈ℰℎ_𝑒⁻²𝜕𝜙−𝑃_ℎ𝜙𝜕𝑛²_0,𝑒^1/2_𝑒∈ℰ𝜕𝑣_ℎ𝜕𝑛²_0,𝑒^1/2,(3.7) where 𝑐>0 is independent of ℎ. Note that ∫_𝑒[(𝜕𝑣_ℎ)/𝜕𝑛]𝑑𝑠=0 for 𝑒∈ℰ. Thus, for some constant 𝐶^𝑒 depending on 𝑒 there holds _𝑒𝜕𝑣_ℎ𝜕𝑛²𝑑𝑠=_𝑒𝜕𝑣_ℎ𝜕𝑛𝜕𝑣_ℎ𝜕𝑛−𝐶^𝑒𝑑𝑠⩽_𝑒𝜕𝑣_ℎ𝜕𝑛²𝑑𝑠^1/2_𝑒𝜕𝑣_ℎ𝜕𝑛−𝐶^𝑒2𝑑𝑠^1/2.(3.8) That is _𝑒𝜕𝑣_ℎ𝜕𝑛²𝑑𝑠⩽_𝑒𝜕𝑣_ℎ𝜕𝑛−𝐶^𝑒2𝑑𝑠.(3.9) From [9], we have 𝜕𝑣_ℎ𝜕𝑛−𝐶^𝑒_0,𝑒⩽𝑐ℎ^1/2𝑣_ℎ2,𝐾,(3.10) where 𝑐>0 is independent of ℎ. Thus _𝑒∈ℰ𝜕𝑣_ℎ𝜕𝑛²_0,𝑒^1/2⩽𝑐ℎ^1/2𝑣_ℎℎ.(3.11) Substituting (3.11) into (3.7) and using (2.22)-(2.23) give 𝑎_𝑆𝜙−𝑃_ℎ𝜙,𝑣_ℎ⩽𝜙−𝑃_ℎ𝜙_ℎ𝑣_ℎℎ+𝑐ℎ^1/2_𝑒∈ℰΔ𝜙−𝑃_ℎ𝜙²_0,𝑒^1/2𝑣_ℎℎ+𝑐_𝑒∈ℰℎ_𝑒⁻¹𝜕𝜙−𝑃_ℎ𝜙𝜕𝑛²_0,𝑒^1/2𝑣_ℎℎ+𝑐ℎ^1/2_𝑒∈ℰℎ_𝑒⁻²𝜕𝜙−𝑃_ℎ𝜙𝜕𝑛²_0,𝑒^1/2𝑣_ℎℎ⩽𝑐ℎ^𝜇−2𝜙_𝑠𝑣_ℎℎ.(3.12)

Theorem 3.3. Suppose that 𝑢∈𝑉 and 𝑢^𝑆_ℎ∈𝑉_ℎ are the solutions to problems (SP) and (2.17), respectively; then the following optimal broken 𝐻² error estimate holds: 𝑢−𝑢^𝑆_ℎℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠,(3.13) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

Proof. According to Lemma 3.1, we have 𝜅₁𝑃_ℎ𝑢−𝑢^𝑆_ℎ²_ℎ⩽𝑎_𝑆𝑃_ℎ𝑢−𝑢^𝑆_ℎ,𝑃_ℎ𝑢−𝑢^𝑆_ℎ=𝑎_𝑆𝑃_ℎ𝑢−𝑢,𝑃_ℎ𝑢−𝑢^𝑆_ℎ+𝑎_𝑆𝑢−𝑢^𝑆_ℎ,𝑃_ℎ𝑢−𝑢^𝑆_ℎ=𝑎_𝑆𝑃_ℎ𝑢−𝑢,𝑃_ℎ𝑢−𝑢^𝑆_ℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠𝑃_ℎ𝑢−𝑢^𝑆_ℎℎ,(3.14) where we use the orthogonal equation (2.19) and Lemma 3.2. The previous estimate implies 𝑃_ℎ𝑢−𝑢^𝑆_ℎℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠.(3.15) Finally, the triangular inequality and (2.24) yield 𝑢−𝑢^𝑆_ℎℎ⩽𝑢−𝑃_ℎ𝑢_ℎ+𝑃_ℎ𝑢−𝑢^𝑆_ℎℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠.(3.16)

Next, we will show the optimal 𝐿² error estimate in terms of the duality technique. Suppose 𝑔∈𝐿²(Ω) and consider the following biharmonic problem: Δ²𝑤=𝑔,inΩ,𝑤=𝜕𝑤𝜕𝑛=0,on𝜕Ω.(3.17) Suppose that problem (3.17) admits a unique solution 𝑤∈𝐻²₀(Ω)∩𝐻⁴(Ω) such that ‖𝑤‖₄⩽𝑐𝑔,(3.18) where ||⋅||₄ denotes the 𝐻⁴ norm in Ω and ||⋅|| denotes the 𝐿² norm in Ω and 𝑐>0 is independent of ℎ.

Denote by Π_ℎ the 𝐶¹ continuous interpolate operator from 𝑉 to 𝐻²₀(Ω)∩𝑉_ℎ, and Π_ℎ satisfies the approximation property (2.22). Then for the solution 𝑤∈𝐻⁴(Ω)∩𝐻²₀(Ω) to problem (3.17), there hold 𝑤−Π_ℎ𝑤₂⩽𝑐ℎ²‖𝑤‖₄⩽𝑐ℎ²𝑔,𝑤−Π_ℎ𝑤_2,𝑒⩽𝑐ℎ_𝑒^3/2‖𝑤‖₄⩽𝑐ℎ_𝑒^3/2𝑔,∀𝑒∈ℰ,(3.19) where 𝑐>0 is independent of ℎ.

Theorem 3.4. Suppose that 𝑢∈𝑉 and 𝑢^𝑆_ℎ∈𝑉_ℎ are the solutions to problems (SP) and (2.17), respectively; then the following optimal 𝐿² error estimate holds: 𝑢−𝑢^𝑆_ℎ⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠,(3.20) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

Proof. Setting 𝑔=𝑢−𝑢^𝑆_ℎ in (3.17), multiplying (3.17) by 𝑢−𝑢^𝑆_ℎ, and integrating over Ω, we have 𝑢−𝑢^𝑆_ℎ²=_ΩΔ²𝑤𝑢−𝑢^𝑆_ℎ𝑑𝑥=𝑎_𝑆𝑤,𝑢−𝑢^𝑆_ℎ=𝑎_𝑆𝑤−Π_ℎ𝑤,𝑢−𝑢^𝑆_ℎ=𝑎_𝑆𝑤−Π_ℎ𝑤,𝑃_ℎ𝑢−𝑢^𝑆_ℎ+𝑎_𝑆𝑤−Π_ℎ𝑤,𝑢−𝑃_ℎ𝑢,(3.21) where we use the orthogonal equation (2.19). We estimate two terms in the right-hand side of (3.21) as follows: 𝑎_𝑆𝑤−Π_ℎ𝑤,𝑃_ℎ𝑢−𝑢^𝑆_ℎ=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑃_ℎ𝑢−𝑢^𝑆_ℎ𝑑𝑥−_{𝑒∈ℰ𝑒}Δ𝑤−Π_ℎ𝑤𝜕𝑃_ℎ𝑢−𝑢^𝑆_ℎ−𝜕𝑛𝑑𝑠_{𝑒∈ℰ𝑒}Δ𝑃_ℎ𝑢−𝑢^𝑆_ℎ𝜕𝑤−Π_ℎ𝑤𝜕𝑛𝑑𝑠+_𝑒∈ℰ𝛽ℎ_𝑒𝑒𝜕𝑤−Π_ℎ𝑤𝜕𝑃𝜕𝑛^ℎ𝑢−𝑢^𝑆_ℎ𝜕𝑛𝑑𝑠=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑃_ℎ𝑢−𝑢^𝑆_ℎ𝑑𝑥−_{𝑒∈ℰ𝑒}Δ𝑤−Π_ℎ𝑤𝜕𝑃_ℎ𝑢−𝑢^𝑆_ℎ⩽𝜕𝑛𝑑𝑠𝑤−Π_ℎ𝑤₂𝑃_ℎ𝑢−𝑢^𝑆_ℎℎ+_𝑒∈ℰ𝑤−Π_ℎ𝑤²_2,𝑒^1/2_𝑒∈ℰ𝑃_ℎ𝑢−𝑢^𝑆_ℎ²_1,𝑒^1/2⩽𝑐ℎ²𝑃_ℎ𝑢−𝑢^𝑆_ℎℎ𝑢−𝑢^𝑆_ℎ⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠𝑢−𝑢^𝑆_ℎ,(3.22) where we use the estimate (3.15). In terms of the inequalities (2.22)-(2.23), we have 𝑎_𝑆𝑤−Π_ℎ𝑤,𝑢−𝑃_ℎ𝑢=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑢−𝑃_ℎ𝑢𝑑𝑥−_{𝑒∈ℰ𝑒}Δ𝑤−Π_ℎ𝑤𝜕𝑢−𝑃_ℎ𝑢𝜕𝑛𝑑𝑠−_{𝑒∈ℰ𝑒}Δ𝑢−𝑃_ℎ𝑢𝜕𝑤−Π_ℎ𝑤𝜕𝑛𝑑𝑠+_𝑒∈ℰ𝛽ℎ_𝑒𝑒𝜕𝑤−Π_ℎ𝑤𝜕𝜕𝑛𝑢−𝑃_ℎ𝑢𝜕𝑛𝑑𝑠=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑢−𝑃_ℎ𝑢𝑑𝑥−_{𝑒∈ℰ𝑒}Δ𝑤−Π_ℎ𝑤𝜕𝑢−𝑃_ℎ𝑢⩽𝜕𝑛𝑑𝑠𝑤−𝑃_ℎ𝑤₂𝑢−𝑃_ℎ𝑢_ℎ+_𝑒∈ℰ𝑤−Π_ℎ𝑤²_2,𝑒^1/2_𝑒∈ℰ𝜕𝑢−𝑃_ℎ𝑢𝜕𝑛²_0,𝑒^1/2⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠𝑢−𝑢^𝑆_ℎ.(3.23) Substituting the estimates (3.22)-(3.23) into (3.21) yields 𝑢−𝑢^𝑆_ℎ⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠.(3.24)

4. 𝐶⁰ Nonsymmetric Interior Penalty Method

In this section, we will show the error estimates in the broken 𝐻² norm and in 𝐿² norm between the solution 𝑢 to problem (2.1) and the solution 𝑢^NS_ℎ 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 𝑎_NS𝑣_ℎ,𝑣_ℎ⩾𝜅₀𝑣_ℎ²_ℎ,∀𝑣_ℎ∈𝑉_ℎ.(4.1) Moreover, for the nonsymmetric bilinear form 𝑎_NS(⋅,⋅), proceeding as in the proof of Lemma 3.2, we have the following lemma.

Lemma 4.1. For all 𝜙∈𝑉, there holds 𝑎_NS𝜙−𝑃_ℎ𝜙,𝑣_ℎ⩽𝑐ℎ^𝜇−2𝑣_ℎℎ𝜙_𝑠,∀𝑣_ℎ∈𝑉_ℎ,(4.2) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

Theorem 4.2. Suppose that 𝑢∈𝑉 and 𝑢^NS_ℎ∈𝑉_ℎ are the solutions to problems (NSP) and (2.18), respectively; then there holds 𝑢−𝑢^NS_ℎℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠,(4.3) 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 𝜅₀𝑃_ℎ𝑢−𝑢^NS_ℎ²_ℎ⩽𝑎_NSP_ℎ𝑢−𝑢^NS_ℎ,𝑃_ℎ𝑢−𝑢^NS_ℎ=𝑎_NS𝑃_ℎ𝑢−𝑢,𝑃_ℎ𝑢−𝑢^NS_ℎ+𝑎_NS𝑢−𝑢^NS_ℎ,𝑃_ℎ𝑢−𝑢^NS_ℎ=𝑎_NS𝑃_ℎ𝑢−𝑢,𝑃_ℎ𝑢−𝑢^NS_ℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠𝑃_ℎ𝑢−𝑢^NS_ℎℎ,(4.4) where 𝑐>0 is independent of ℎ. That is 𝑃_ℎ𝑢−𝑢^NS_ℎℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠.(4.5) Using the triangular inequality yields 𝑢−𝑢^NS_ℎℎ⩽𝑢−𝑃_ℎ𝑢_ℎ+𝑃_ℎ𝑢−𝑢^NS_ℎℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠.(4.6)

Theorem 4.3. Suppose that 𝑢∈𝑉 and 𝑢^NS_ℎ∈𝑉_ℎ are the solutions to problems (NSP) and (2.18), respectively; then there holds 𝑢−𝑢^NS_ℎ⩽𝑐ℎ^𝜇−1‖|𝑢|‖_𝑠,(4.7) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

Proof. Setting 𝑔=𝑢−𝑢^NS_ℎ in (3.17), multiplying (3.17) by 𝑢−𝑢^NS_ℎ, and integrating over Ω, we have 𝑢−𝑢^NS_ℎ²=_ΩΔ²𝑤𝑢−𝑢^NS_ℎ𝑑𝑥=𝑎_NS𝑤,𝑢−𝑢^NS_ℎ=𝑎_NS𝑤−Π_ℎ𝑤,𝑢−𝑢^NS_ℎ+2_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝑤𝜕𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠−2_{𝑒∈ℰ𝑒}Δ𝑢−𝑢^NS_ℎ𝜕Π_ℎ𝑤𝜕𝑛𝑑𝑠=𝑎_NS𝑤−Π_ℎ𝑤,𝑢−𝑢^NS_ℎ+2_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝑤𝜕𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑢−𝑢^NS_ℎ𝑑𝑥+3_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝜕𝑤−𝑤𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠+2_{𝑒∈ℰ𝑒}{𝜕Δ𝑤}𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠=𝐼₁+𝐼₂+𝐼₃,(4.8) where we use Π_ℎ𝑤∈𝐻²₀(Ω). We estimate 𝐼₁ as follows: 𝐼₁=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑢−𝑢^NS_ℎ𝑑𝑥⩽𝑢−𝑢^NS_ℎℎ𝑤−Π_ℎ𝑤₂⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠𝑢−𝑢^NS_ℎ.(4.9) We estimate 𝐼₂ as follows: 𝐼₂=3_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝜕𝑤−𝑤𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠=3_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝜕𝑤−𝑤𝑢−𝑃_ℎ𝑢𝜕𝑛𝑑𝑠+3_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝜕𝑃𝑤−𝑤^ℎ𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠⩽𝑐_𝑒∈ℰ𝑤−Π_ℎ𝑤²_2,𝑒^1/2_𝑒∈ℰ𝜕𝑢−𝑃_ℎ𝑢𝜕𝑛²_0,𝑒^1/2+_𝑒∈ℰ𝑤−Π_ℎ𝑤²_2,𝑒^1/2_𝑒∈ℰ𝜕𝑃_ℎ𝑢−𝑢^NS_ℎ𝜕𝑛²_0,𝑒^1/2⩽𝑐ℎ^3/2ℎ^𝜇−3/2‖|𝑢|‖_𝑠+𝑐ℎ^1/2𝑃_ℎ𝑢−𝑢^NS_ℎℎ𝑢−𝑢^NS_ℎ⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠𝑢−𝑢^NS_ℎ.(4.10) We estimate 𝐼₃ as follows: 𝐼₃=2_{𝑒∈ℰ𝑒}{𝜕Δ𝑤}𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠=2_{𝑒∈ℰ𝑒}Δ𝑤−𝐶^𝑒𝜕𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠=2_{𝑒∈ℰ𝑒}Δ𝑤−𝐶^𝑒𝜕u−𝑃_ℎ𝑢𝜕𝑛𝑑𝑠+2_{𝑒∈ℰ𝑒}Δ𝑤−𝐶^𝑒𝜕𝑃_ℎ𝑢−𝑢^NS_ℎ𝜕𝑛𝑑𝑠⩽𝑐_𝑒∈ℰΔ𝑤−𝐶^𝑒2_0,𝑒^1/2_𝑒∈ℰ𝑢−𝑃_ℎ𝑢²_1,𝑒^1/2+_𝑒∈ℰΔ𝑤−𝐶^𝑒2_0,𝑒^1/2_𝑒∈ℰ𝜕𝑃_ℎ𝑢−𝑢^NS_ℎ𝜕𝑛²_0,𝑒^1/2⩽𝑐_𝑒∈ℰΔ𝑤−𝐶^𝑒2_0,𝑒^1/2ℎ^𝜇−3/2‖|𝑢|‖_𝑠+𝑐ℎ^1/2𝑃_ℎ𝑢−𝑢^NS_ℎℎ⩽𝑐ℎ^𝜇−3/2_𝑒∈ℰΔ𝑤−𝐶^𝑒2_0,𝑒^1/2‖|𝑢|‖_𝑠,(4.11) where 𝐶^𝑒 is some positive constant. Substituting the following estimate _𝑒∈ℰΔ𝑤−𝐶^𝑒2_0,𝑒⩽𝑐_𝑒∈ℰℎ_𝑒‖𝑤‖²_3,𝑒⩽𝑐ℎ‖𝑤‖²₄⩽𝑐ℎ𝑢−𝑢^NS_ℎ²(4.12) into (4.11) gives 𝐼₃⩽𝑐ℎ^𝜇−1‖|𝑢|‖_𝑠𝑢−𝑢^NS_ℎ.(4.13) Finally, substituting (4.9)–(4.13) into (4.8), we obtain 𝑢−𝑢^NS_ℎ⩽𝑐ℎ^𝜇−1‖|𝑢|‖_𝑠.(4.14)

5. 𝐶⁰ Superpenalty Nonsymmetric Method

In order to obtain the optimal 𝐿² 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: 𝑎_SNS(u,𝑣)=_{𝐾∈𝒯ℎ𝐾}Δ𝑢Δ𝑣𝑑𝑥−_{𝑒∈ℰ𝑒}{Δ𝑢}𝜕𝑣𝜕𝑛𝑑𝑠+_{𝑒∈ℰ𝑒}{Δ𝑣}𝜕𝑢𝜕𝑛𝑑𝑠+_𝑒∈ℰ1ℎ³_𝑒^𝑒𝜕𝑢𝜕𝑛𝜕𝑣.𝜕𝑛(5.1) The broken 𝐻² norm is modified to 𝑣_ℎℎ=_{𝐾∈𝒯ℎ}𝑣_ℎ²_2,𝐾𝑖+_𝑒∈ℰ1ℎ³_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒^1/2,∀𝑣_ℎ∈𝑉_ℎ.(5.2) From Lemma 2.1, it is easy to show that there exists some constant 𝜅₂>0 such that 𝑎_SNS𝑣_ℎ,𝑣_ℎ⩾𝜅₂𝑣_ℎ²_ℎ,∀𝑣_ℎ∈𝑉_ℎ.(5.3) In fact, we have 𝑎_SNS𝑣_ℎ,𝑣_ℎ=_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾+_𝑒∈ℰ1ℎ³_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒⩾12_𝑒∈ℰ1ℎ³_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒+12_{𝐾∈𝒯ℎ}Δ𝑣_ℎ²_0,𝐾+_𝑒∈ℰ1ℎ_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒⩾12_𝑒∈ℰ1ℎ³_𝑒𝜕𝑣_ℎ𝜕𝑛²_0,𝑒+𝜅₀2_{𝐾∈𝒯ℎ}𝑣_ℎ²_2,𝐾⩾𝜅₂𝑣_ℎ²_ℎ(5.4) for 𝜅₂=min{1/2,𝜅₀/2}. Since the solution 𝑢 to problem (2.1) belongs to 𝐻⁴(Ω)∩𝐻²₀(Ω), then it satisfies 𝑎_SNS(𝑢,𝑣)=𝑓,𝑣,∀𝑣∈𝑉.(SNSP) In this case, the the 𝐶⁰ superpenalty nonsymmetric finite element approximation of (2.1) is find𝑢^SNS_ℎ∈𝑉_ℎsuchthat,𝑎_SNS𝑢^SNS_ℎ,𝑣_ℎ=𝑓,𝑣_ℎ,∀𝑣_ℎ∈𝑉_ℎ.(5.5) Then, we have the following orthogonal equation: 𝑎_SNS𝑢^SNS_ℎ−𝑢,𝑣_ℎ=0,∀𝑣_ℎ∈𝑉_ℎ.(5.6)

Let Π_ℎ be the 𝐶¹ continuous interpolated operator defined in Section 3. Observe that [(𝜕𝑢−Π_ℎ𝑢)/𝜕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 all 𝜙∈𝑉, there holds 𝑎_NS𝜙−Π_ℎ𝜙,𝑣_ℎ⩽𝑐ℎ^𝜇−2𝑣_ℎℎ𝜙_𝑠,∀𝑣_ℎ∈𝑉_ℎ,(5.7) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

Using Lemma 5.1, the following optimal broken 𝐻² error estimate holds.

Theorem 5.2. Suppose that 𝑢∈𝑉 and 𝑢^SNS_ℎ∈𝑉_ℎ are the solutions to problems (SNSP) and (5.5), respectively; then there holds 𝑢−𝑢^SNS_ℎℎ⩽𝑐ℎ^𝜇−2‖|𝑢|‖_𝑠,(5.8) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

The main result in this section is the following optimal 𝐿² error estimate.

Theorem 5.3. Suppose that 𝑢∈𝑉 and 𝑢^SNS_ℎ∈𝑉_ℎ are the solutions to problems (SNSP) and (5.5), respectively; then there holds 𝑢−𝑢^SNS_ℎ⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠,(5.9) where 𝜇=min{𝑠,𝑟+1} and 𝑐>0 is independent of ℎ.

Proof. Let 𝑔=𝑢−𝑢^SNS_ℎ. Multiplying (3.17) by 𝑢−𝑢^SNS_ℎ and integrating over Ω, we have 𝑢−𝑢^SNS_ℎ²=_ΩΔ²𝑤𝑢−𝑢^SNS_ℎ𝑑𝑥=𝑎_SNS𝑤,𝑢−𝑢^SNS_ℎ=𝑎_SNS𝑤−Π_ℎ𝑤,𝑢−𝑢^SNS_ℎ+2_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝑤𝜕𝑢−𝑢^SNS_ℎ𝜕𝑛𝑑𝑠−2_{𝑒∈ℰ𝑒}Δ𝑢−𝑢^SNS_ℎ𝜕Π_ℎ𝑤𝜕𝑛𝑑𝑠=𝑎_SNS𝑤−Π_ℎ𝑤,𝑢−𝑢^SNS_ℎ+2_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝑤𝜕𝑢−𝑢^SNS_ℎ𝜕𝑛𝑑𝑠=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑢−𝑢^SNS_ℎ𝑑𝑥+3_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝜕𝑤−𝑤𝑢−𝑢^SNS_ℎ𝜕𝑛𝑑𝑠+2_{𝑒∈ℰ𝑒}{𝜕Δ𝑤}𝑢−𝑢^SNS_ℎ𝜕𝑛𝑑𝑠=𝐼₄+𝐼₅+𝐼₆,(5.10) where we use Π_ℎ𝑤∈𝐶¹(Ω)∩𝐻²₀(Ω). Proceeding as in the proof of Theorem 4.2, we can estimate 𝐼₄ and 𝐼₅ as follows: 𝐼₄=_{𝐾∈𝒯ℎ𝐾}Δ𝑤−Π_ℎ𝑤Δ𝑢−𝑢^SNS_ℎ𝑑𝑥⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠𝑢−𝑢^SNS_ℎ,(5.11)𝐼₅=3_{𝑒∈ℰ𝑒}ΔΠ_ℎ𝜕𝑤−𝑤𝑢−𝑢^SNS_ℎ𝜕𝑛𝑑𝑠⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠𝑢−𝑢^SNS_ℎ.(5.12) The different estimate compared to Theorem 4.3 is the estimate of 𝐼₆. Under the new norm |||⋅|||_ℎ, we have 𝐼₆=2_{𝑒∈ℰ𝑒}{𝜕Δ𝑤}𝑢−𝑢^SNS_ℎ𝜕𝑛𝑑𝑠=2_{𝑒∈ℰ𝑒}{𝜕Δ𝑤−𝑐}𝑢−𝑢^SNS_ℎ𝜕𝑛𝑑𝑠⩽𝑐_𝑒∈ℰℎ³_𝑒‖Δ𝑤−𝐶‖²_0,𝑒^1/2_𝑒∈ℰ1ℎ³_𝑒𝜕𝑢−𝑢^SNS_ℎ𝜕𝑛²_0,𝑒^1/2⩽𝑐ℎ²𝑢−𝑢^SNS_ℎ⋅𝑢−𝑢^SNS_ℎℎ⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠𝑢−𝑢^SNS_ℎ.(5.13) Substituting (5.11)–(5.13) into (5.10) yields the following optimal 𝐿² error estimate: 𝑢−𝑢^SNS_ℎ⩽𝑐ℎ^𝜇‖|𝑢|‖_𝑠.(5.14)

Acknowledgments

The authors would like to thank the anonymous reviewers for their careful reviews and comments to improve this paper. This material is based upon work funded by the National Natural Science Foundation of China under Grants no. 10901122, no. 11001205, and no. 11126226 and by Zhejiang Provincial Natural Science Foundation of China under Grants no. LY12A01015 and no. Y6110240.