Analysis of an Interior Penalty Method for Fourth Order Problems on Polygonal Domains

Abstract

Error analysis for a stable C ⁰ interior penalty method is derived for general fourth order problems on polygonal domains under minimal regularity assumptions on the exact solution. We prove that this method exhibits quasi-optimal order of convergence in the discrete H ², H ¹ and L ₂ norms. L _∞ norm error estimates are also discussed. Theoretical results are demonstrated by numerical experiments.

Appendix

In this section, we present the detailed derivation of the USIP method. In the derivation, we assume that the coefficient $\mathbb{A}$ and the solution u of (1.4) are sufficiently smooth. It is important to point out here that this smoothness assumption is only used to motivate the interior penalty method. However, abstract and concrete error estimates that are obtained in Sects. 3 and 4 are derived under minimal regularity assumptions.

Note that the solution u of (1.4) satisfies

(6.1)

(6.2)

where $(\nabla \cdot \varPhi)_{i} = \sum_{j=1}^{2} \partial_{j} \phi_{ij}$ ∀Φ∈W.

A key for the derivation is to split the fourth order equation (6.1) into a system of second order equations by introducing the auxiliary variables p and Ψ as follows:

(6.3)

(6.4)

(6.5)

(6.6)

Remark 5.1

In aniso -/ortho -/ isotropic / biharmonic plate bending problems, u denotes the deflection of the bent plate, p=(u _,ij )(i,j=1,2) denote the components of the change in curvature tensor and Ψ=(ψ _ij )=(a _ijkl u _,kl ) (i,j=1,2) denote the bending and twisting moments in the plate.

Remark 5.2

The USIP method can also be derived by splitting the fourth order equation into a system of first order equations [18, 34].

We will now use the following Green’s formula: $\forall T \in \mathcal{T}_{h}$,

(6.7)

where t=(t ₁,t ₂) denotes the unit tangent vector along ∂T, (Φ⋅n) _i =ϕ _i1 n ₁+ϕ _i2 n ₂ (i=1,2) for Φ∈W and n=(n ₁,n ₂).

Multiply (6.5) with v _h ∈V _h , use the Green’s formula (6.7) and the fact that $[\hskip -2.2pt[(\nabla \cdot \varPsi) \cdot n]\hskip -2.2pt]=[\hskip -2.2pt[n\cdot \varPsi \cdot n]\hskip -2.2pt]=[\hskip -2.2pt[n\cdot \varPsi \cdot t]\hskip -2.2pt]=0$ for all $e\in \mathcal{E}_{h}^{i}$ to obtain

Multiply (6.3) with q _h ∈W _h , integrate over Ω, add a zero term using the fact that $[\hskip -2.2pt[\partial u/\partial n]\hskip -2.2pt]=0$ for all $e\in \mathcal{E}_{h}$ to obtain:

(6.8)

Note that though the second term on the right hand side of (6.8) is not zero when u is substituted by u _h .

For $v\in H^{2}(\varOmega, \mathcal{T}_{h})$ and q∈W, define the bilinear form B _h (⋅,⋅) by:

(6.9)

Note that if u,p and Ψ are sufficiently smooth, then they satisfy the following equations:

(6.10)

(6.11)

(6.12)

where the bilinear form J is defined by (2.3).

Based on (6.10)–(6.12), we formulate the interior penalty method defined by:

Find (u _h ,p _h ,Ψ _h )∈V _h ×W _h ×W _h such that

(6.13)

(6.14)

(6.15)

Primal Formulation

With the help of a lifting operator $\mathcal{R}_{h}:H^{2}(\varOmega,\mathcal{T}_{h}) \rightarrow \mathbf{W}_{h}$ defined in (2.2), we derive the primal formulation (2.4) involving only one unknown u _h which was defined in Sect. 2. However since the global mass matrix is block diagonal, the equations (6.13)–(6.15) can be used to compute the global stiffness matrix for the primal formulation without computing the lifting operators $\mathcal{R}_{h}$, see [21] for more details.

From (6.13), (6.9) and (2.2),

$$ \mathbf{p}_h=D^2_hu_h+\mathcal{R}_h(u_h). $$

(6.16)

Use (6.16) in (6.14) to obtain

(6.17)

where Π _h :[L ₂(Ω)]_2×2→W _h is the L ₂-projection defined by

(6.18)

Using (2.2), we rewrite (6.15) as

which together with (6.17) implies,

$$ \bigl(\bigl(\mathbb{A}\oplus \bigl(D^2 u_h +\mathcal{R}_h(u_h) \bigr),\,D^2 v_h +\mathcal{R}_h(v_h) \bigr)\bigr) +J(u_h,v_h)=f(v_h) \quad \forall v_h \in V_h. $$

(6.19)

Below, we derive error estimates for the auxiliary variables.

Theorem 5.3

For p and p _h satisfying (6.3) and (6.16) respectively, ∃ a positive constant C such that, for v _h ∈V _h , the following estimate holds true:

Proof

From (6.3) and (6.16), we note that $\mathbf{p}-\mathbf{p}_{h}= D^{2}_{h}(u-u_{h})-\mathcal{R}_{h}(u_{h})$. Since $\mathcal{R}_{h}(u)=0$, we have $\mathbf{p}-\mathbf{p}_{h}= D^{2}_{h}(u-u_{h})+\mathcal{R}_{h}(u-u_{h})$. A use of triangle inequality, Lemma 2.1 and Theorem 3.1 completes the proof. □

Theorem 5.4

For Ψ and Ψ _h satisfying (6.4) and (6.17) respectively, ∃ a positive constant C such that

where Π _h is the L ₂ projection onto W _h defined by (6.18) and v _h ∈V _h .

Proof

We note from (6.17) that

Since , we find for any w _h ∈W _h that

Therefore,

The proof now follows from triangle inequality and Theorem 6.3. □

The following error estimates can be deduced easily from Theorem 6.3 and Theorem 6.4.

Theorem 5.5

For the solutions p and p _h of (6.3) and (6.16) respectively, and Ψ and Ψ _h of (6.4) and (6.17) respectively, the following estimates hold true: