Constrained C0 Finite Element Methods for Biharmonic Problem

and Applied Analysis 3 Let EI and EB be the set of interior edges and boundary edges of Th, respectively. Let E EI ∪ EB. Denote by v 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 v |e v ∣ ∣ ∣ e −v ∣ ∣ ∣ e , {v}|e 1 2 ( v ∣ ∣ ∣ e v ∣ ∣ ∣ e ) . 2.4 If e ei ∈ EB, we denote v and {v} of v on e by v |e {v}|e v ∣ ∣ ∣ e . 2.5


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 C 0 and C 1 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 C 0 interior penalty method. C 0 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 C 0 interior penalty method for fourth-order problem on polygonal domains. They used the C 0 finite element solution to approximate C 1 solution by a postprocessing procedure, and the C 1 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 C 0 finite element approximation methods for the biharmonic problem. The C 1 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 C 0 finite element methods which include the C 0 symmetric interior penalty method based on the symmetric bilinear form, the C 0 nonsymmetric interior penalty method, and C 0 nonsymmetric superpenalty method based on the nonsymmetric bilinear forms. First, we study the C 0 symmetric interior penalty method and obtain the optimal error estimates in the broken H 2 norm and in L 2 norm. However, for the C 0 nonsymmetric interior penalty method, the L 2 norm is suboptimal because of the lack of adjoint consistency. Finally, in order to improve the order of the L 2 error estimate, we give the C 0 nonsymmetric superpenalty method and show the optimal L 2 error estimates.

C 0 Finite Element Approximation
Let Ω ⊂ R 2 be a bounded and convex domain with boundary ∂Ω. Consider the following biharmonic problem:

2.12
It is clear that a S ·, · is a symmetric bilinear form and a NS ·, · is a nonsymmetric bilinear form. the solution u to problem 2.1 satisfies the following variational problems: Let P r K denote the space of the polynomials on K of degree at most r. Define the following constrained C 0 finite element space: 14 from which we note that the C 1 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 Abstract and Applied Analysis 5 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 : 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 ∂n ds for each edge e of K.

2.16
Based on the symmetric bilinear form a S ·, · , the C 0 symmetric interior penalty finite element approximation of 2.1 is

2.17
Based on the nonsymmetric bilinear form a NS ·, · , the C 0 nonsymmetric interior penalty finite element approximation of 2.1 is

2.18
Moreover, the following orthogonal equations hold: 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 where μ min{s, r 1} and c > 0 is independent of h. We also suppose that the following inverse inequalities hold: The following lemma is useful to establish the existence and uniqueness of the finite element approximation solution.
Proof. Introduce a H 2 0 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 Abstract and Applied Analysis

C 0 Symmetric Interior Penalty Method
In this section, we will show the optimal error estimates in the broken H 2 norm and in the L 2 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 .
Proof. According to the definition of a S ·, · , we have 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 where μ min{s, r 1} and c > 0 is independent of h.
Proof. According to Lemma 3.1, we have 3.14 where we use the orthogonal equation 2.19 and Lemma 3.2. The previous estimate implies Finally, the triangular inequality and 2.24 yield Next, we will show the optimal L 2 error estimate in terms of the duality technique. Suppose g ∈ L 2 Ω and consider the following biharmonic problem: 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 L 2 error estimate holds: 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

3.21
where we use the orthogonal equation 2.19 . We estimate two terms in the right-hand side of 3.21 as follows: where we use the estimate 3.15 . In terms of the inequalities 2.22 -2.23 , we have

C 0 Nonsymmetric Interior Penalty Method
In this section, we will show the error estimates in the broken H 2 norm and in L 2 norm between the solution u to problem 2.1 and the solution u NS h to the problem 2.18 . The optimal broken H 2 error estimate is derived. However, the L 2 error estimate is suboptimal because of the lack of adjoint consistency. According to Lemma 2.1, we have Moreover, for the nonsymmetric bilinear form a NS ·, · , proceeding as in the proof of Lemma 3.2, we have the following lemma.
where μ min{s, r 1} and c > 0 is independent of h. 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 where c > 0 is independent of h. That is Using the triangular inequality yields 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 4.8 where we use Π h w ∈ H 2 0 Ω . We estimate I 1 as follows:

4.9
We estimate I 2 as follows:

4.10
We estimate I 3 as follows:

C 0 Superpenalty Nonsymmetric Method
In order to obtain the optimal L 2 error estimate for the nonsymmetric method, in this section we will consider the C 0 superpenalty nonsymmetric method. First, we introduce a new nonsymmetric bilinear form:

5.5
Then, we have the following orthogonal equation:

5.6
Let Π h be the C 1 continuous interpolated operator defined in Section 3. Observe that ∂u − Π h u /∂n 0 on every e ∈ E, and proceeding as in the proof of Lemmas 3.2 and 4.1, we have the following. where μ min{s, r 1} and c > 0 is independent of h.
Using Lemma 5.1, the following optimal broken H 2 error estimate holds. where μ min{s, r 1} and c > 0 is independent of h.
The main result in this section is the following optimal L 2 error estimate. where μ min{s, r 1} and c > 0 is independent of h.