A $C^0$ interior penalty method for a von Kármán plate

Abstract

We investigate a quadratic $C^0$ interior penalty method for the approximation of isolated solutions of a von Kármán plate. We prove that the discrete problem is uniquely solvable near an isolated solution and establish optimal order error estimates. Numerical results that illustrate the theoretical estimates are also presented.

A Proof of Lemma 3.7

Let ${H^2(\Omega ,\mathcal {T}_h)}$ be the subspace of $H^1_0(\Omega )$ whose members are piecewise $H^2$, i.e.,

$$\begin{aligned} {H^2(\Omega ,\mathcal {T}_h)}=\{v\in H^1_0(\Omega ):\,v\big |_T\in H^2(T)\quad \forall \,T\in \mathcal {T}_h\}, \end{aligned}$$

and let the norm $|\cdot |_{H^2(\Omega ,\mathcal {T}_h)}$ be defined by

$$\begin{aligned} |v|_{H^2(\Omega ,\mathcal {T}_h)}^2=\sum _{T\in \mathcal {T}_h}|v|_{H^2(T)}^2+\sum _{e\in \mathcal {E}_h}h_e^{-1}\Vert \left[ \left[ {\partial }v/{\partial }n\right] \right] \Vert _{L^2(e)}^2. \end{aligned}$$

(A.1)

We will prove the following estimates on ${H^2(\Omega ,\mathcal {T}_h)}$:

$$\begin{aligned} |u|_{H^1(\Omega )}&\le C|u|_{H^2(\Omega ,\mathcal {T}_h)}&\qquad&\forall u\in {H^2(\Omega ,\mathcal {T}_h)},\end{aligned}$$

(A.2)

$$\begin{aligned} \Vert u\Vert _{L^\infty (\Omega )}&\le C|u|_{H^2(\Omega ,\mathcal {T}_h)}&\qquad&\forall u\in {H^2(\Omega ,\mathcal {T}_h)},\end{aligned}$$

(A.3)

$$\begin{aligned} \Vert \nabla u\Vert _{L^4(\Omega )}&\le C|u|_{H^2(\Omega ,\mathcal {T}_h)}&\qquad&\forall u\in {H^2(\Omega ,\mathcal {T}_h)}, \end{aligned}$$

(A.4)

where the positive constant C depends only on the shape regularity of $\mathcal {T}_h$. The estimates (3.24)–(3.26) follow immediately since $U_h\subset {H^2(\Omega ,\mathcal {T}_h)}$ and

$$\begin{aligned} |u|_{H^2(\Omega ,\mathcal {T}_h)}\le \Vert u\Vert _h \qquad \forall \,u\in U_h. \end{aligned}$$

First we note that the estimate (3.24) was already established in [9] in a more general setting. The proofs of (3.25) and (3.26) are based on the properties of the Lagrange interpolation operator $\Pi _h:C(\bar{\Omega })\longrightarrow V_h$ and an enriching operator $E_h:V_h\longrightarrow \tilde{V}_h \subset H^2_0(\Omega )$ defined by averaging [8], where $\tilde{V}_h$ is the sixth order Argyris finite element space [2] associated with $\mathcal {T}_h$. We have two standard estimates [7, 15] for $\Pi _h$:

$$\begin{aligned} \Vert u-\Pi _hu\Vert _{L^\infty (T)}&\le Ch_T|u|_{H^2(T)}&\qquad&\forall \,u\in {H^2(\Omega ,\mathcal {T}_h)},\;T\in \mathcal {T}_h,\end{aligned}$$

(A.5)

$$\begin{aligned} \Vert \nabla (u-\Pi _hu)\Vert _{L^4(T)}&\le Ch_T|u|_{H^2(T)}&\qquad&\forall \,u\in {H^2(\Omega ,\mathcal {T}_h)},\;T\in \mathcal {T}_h, \end{aligned}$$

(A.6)

and the following estimate was established in [8, (3.16) and (3.17)]:

$$\begin{aligned} |\Pi _h u|_{H^2(\Omega ,\mathcal {T}_h)}\le C\Big (\sum _{T\in \mathcal {T}_h}|u|_{H^2(T)}^2\Big )^\frac{1}{2} \qquad \forall \,u\in {H^2(\Omega ,\mathcal {T}_h)}. \end{aligned}$$

(A.7)

The following estimates for $E_h$ were established in [8, (3.24)] and [8, (3.29)]:

$$\begin{aligned} \Vert E_hv\Vert _{H^2(\Omega )}\le C|v|_{H^2(\Omega ,\mathcal {T}_h)}\qquad \forall \,v\in V_h, \end{aligned}$$

(A.8)

and

$$\begin{aligned} \Vert v-E_hv\Vert _{L^2(T)}^2\le C \Bigg (\sum _{T'\in \mathcal {T}_{h,T}} h_{T'}^4|v|_{H^2(T')}^2+ h_T^4\sum _{e\in \mathcal {E}_{h,\mathcal {V}_T}}h_e^{-1}\Vert \left[ \left[ {\partial }v/{\partial }n\right] \right] \Vert _{L^2(e)}^2\Bigg ) \quad \forall \,v\in V_h, \end{aligned}$$

(A.9)

where $\mathcal {T}_{h,T}$ is the set of triangles in $\mathcal {T}_h$ that share a vertex with T (including T itself) and $\mathcal {E}_{h,\mathcal {V}_T}$ is the set of the edges in $\mathcal {T}_h$ emanating from the vertices of T. Let $v\in V_h$ be arbitrary. By a standard Sobolev inequality [1], we have

$$\begin{aligned} \Vert E_h v\Vert _{L^\infty (\Omega )}\le C_\Omega \Vert E_hv\Vert _{H^2(\Omega )}. \end{aligned}$$

(A.10)

Furthermore, by (A.1), (A.9) and a standard inverse estimate [7, 15], we also have

$$\begin{aligned} \Vert v-E_hv\Vert _{L^\infty (\Omega )}\le Ch|v|_{H^2(\Omega ,\mathcal {T}_h)}. \end{aligned}$$

(A.11)

In view of (A.8), (A.10), (A.11) and the triangle inequality, we have

$$\begin{aligned} \Vert v\Vert _{L^\infty (\Omega )}\le C|v|_{H^2(\Omega ,\mathcal {T}_h)}\qquad \forall \,v\in V_h. \end{aligned}$$

(A.12)

Using (A.1), (A.5), (A.7), and (A.12), we can now finish the proof of (A.3):

$$\begin{aligned} \Vert u\Vert _{L^\infty (\Omega )}&\le \Vert u-\Pi _hu\Vert _{L^\infty (\Omega )}+\Vert \Pi _hu\Vert _{L^\infty (\Omega )}\\&\le C\big (h|u|_{H^2(\Omega ,\mathcal {T}_h)}+|\Pi _hu|_{H^2(\Omega ,\mathcal {T}_h)}\big )\le C|u|_{H^2(\Omega ,\mathcal {T}_h)}\qquad \forall \,u\in U_h. \end{aligned}$$

The proof of (A.4) is similar. Again, by a standard Sobolev inequality, we have

$$\begin{aligned} \Vert \nabla (E_hv)\Vert _{L^4(\Omega )} \le C_\Omega \Vert E_hv\Vert _{H^2(\Omega )}. \end{aligned}$$

(A.13)

On the other hand, we have

$$\begin{aligned} \Vert \nabla (v-E_hv)\Vert _{L^\infty (T)}&\le Ch_T^{-2}\Vert v-E_hv\Vert _{L^2(T)} \qquad \forall \,T\in \mathcal {T}_h, \end{aligned}$$

(A.14)

$$\begin{aligned} \Vert \nabla (v-E_hv)\Vert _{L^2(T)}&\le Ch_T^{-1}\Vert v-E_hv\Vert _{L^2(T)} \qquad \forall \,T\in \mathcal {T}_h, \end{aligned}$$

(A.15)

by standard inverse estimates. It follows from (A.1), (A.9), (A.14) and (A.15) that

$$\begin{aligned} \Vert \nabla (v-E_hv)\Vert _{L^4(\Omega )}^4&\le \sum _{T\in \mathcal {T}_h}\Vert \nabla (v-E_hv)\Vert _{L^2(T)}^2 \Vert \nabla (v-E_hv)\Vert _{L^\infty (T)}^2\\&\le |v|_{H^2(\Omega ,\mathcal {T}_h)}^2\sum _{T\in \mathcal {T}_h}\Vert \nabla (v-E_hv)\Vert _{L^2(T)}^2 \le Ch^2|v|_{H^2(\Omega ,\mathcal {T}_h)}^4, \end{aligned}$$

and hence

$$\begin{aligned} \Vert \nabla (v-E_hv)\Vert _{L^4(\Omega )}\le Ch|v|_{H^2(\Omega ,\mathcal {T}_h)}\qquad \forall \,v\in V_h. \end{aligned}$$

(A.16)

Using (A.8), (A.13), (A.16) and the triangle inequality, we find

$$\begin{aligned} \Vert \nabla v\Vert _{L^4(\Omega )}\le C|v|_{H^2(\Omega ,\mathcal {T}_h)}\qquad \forall \,v\in V_h. \end{aligned}$$

(A.17)

The estimate (A.4) now follows from (A.1), (A.6), (A.7) and (A.17):

$$\begin{aligned} \Vert \nabla u\Vert _{L^4(\Omega )}&\le \Vert \nabla (u-\Pi _hu)\Vert _{L^4(\Omega )}+\Vert \nabla \Pi _hu\Vert _{L^4(\Omega )}\\&\le C\big (h|u|_{H^2(\Omega ,\mathcal {T}_h)}+|\Pi _hu|_{H^2(\Omega ,\mathcal {T}_h)}\big )\le C|u|_{H^2(\Omega ,\mathcal {T}_h)}\qquad \forall \,u\in U_h. \end{aligned}$$

Remark A.1

The estimates (A.2)–(A.4) are also valid for general partitions (cf. [9]).