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

Susanne C. Brenner; Michael Neilan; Armin Reiser; Li-Yeng Sung

doi:10.1007/s00211-016-0817-y

Numerische Mathematik

March 2017, Volume 135, Issue 3, pp 803–832 | Cite as

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

Authors
Authors and affiliations

Susanne C. BrennerEmail author
Michael Neilan
Armin Reiser
Li-Yeng Sung

Article

First Online: 12 July 2016

Received: 10 March 2015

Revised: 10 May 2016

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.

Mathematics Subject Classification

Primary 65N30 65N15 Secondary 74K20

This is a preview of subscription content, to check access.

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]).

References

1.
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)MATHGoogle Scholar
2.
Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc. 72, 701–709 (1968)Google Scholar
3.
Berger, M.S.: On von Kármán’s equations and the buckling of a thin elastic plate. I. The clamped plate. Comm. Pure Appl. Math. 20, 687–719 (1967)MathSciNetCrossRefMATHGoogle Scholar
4.
Bjørstad, P.E., Tjøstheim, B.P.: High precision solutions of two fourth order eigenvalue problems. Computing 63, 97–107 (1999)MathSciNetCrossRefMATHGoogle Scholar
5.
Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2, 556–581 (1980)MathSciNetCrossRefMATHGoogle Scholar
6.
Brenner, S.C., Neilan, M., Sung, L.-Y.: Isoparametric ${C^0}$ interior penalty methods for plate bending problems on smooth domains. Calcolo 49, 35–67 (2013)MathSciNetCrossRefMATHGoogle Scholar
7.
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)CrossRefMATHGoogle Scholar
8.
Brenner, S.C., Sung, L.-Y.: $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005)MathSciNetCrossRefMATHGoogle Scholar
9.
Brenner, S.C., Wang, K., Zhao, J.: Poincaré-Friedrichs inequalities for piecewise $H^2$ functions. Numer. Funct. Anal. Optim. 25, 463–478 (2004)MathSciNetCrossRefMATHGoogle Scholar
10.
Brezzi, F.: Finite element approximations of the von Kármán equations. RAIRO Anal. Numér. 12, 303–312 (1978)MathSciNetMATHGoogle Scholar
11.
Brezzi, F., Rappaz, J., Raviart, P.-A.: Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36, 1–25 (1980/81)Google Scholar
12.
Brezzi, F., Rappaz, J., Raviart, P.-A.: Finite-dimensional approximation of nonlinear problems. III. Simple bifurcation points. Numer. Math. 38, 1–30 (1981/82)Google Scholar
13.
Cheng, M., Warren, J.A.: An efficient algorithm for solving the phase field crystal model. J. Comput. Phys. 227, 6241–6248 (2008)MathSciNetCrossRefMATHGoogle Scholar
14.
Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations. Springer, New York (2010)CrossRefMATHGoogle Scholar
15.
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
16.
Ciarlet, P.G.: Mathematical Elasticity Volume II: Theory of Plates. North-Holland, Amsterdam (1997)Google Scholar
17.
Ciarlet, P.G., Rabier, P.: Les équations de von Kármán. Lecture Notes in Mathematics, vol. 826. Springer, Berlin (1980)Google Scholar
18.
Crouzeix, M., Rappaz, J.: On numerical approximation in bifurcation theory, vol. 13 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris. Springer, Berlin (1990)Google Scholar
19.
Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg. 191, 3669–3750 (2002)MathSciNetCrossRefMATHGoogle Scholar
20.
Evans, L.C.: Partial differential equations, vol. 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
21.
Gomez, H., Nogueira, X.: An unconditionally energy-stable method for the phase field crystal equation. Comput. Methods Appl. Mech. Engrg. 249/252, 52–61 (2012)MathSciNetCrossRefMATHGoogle Scholar
22.
Gudi, T., Neilan, M.: An interior penalty method for a sixth-order elliptic equation. IMA J. Numer. Anal. 31, 1734–1753 (2011)MathSciNetCrossRefMATHGoogle Scholar
23.
Knightly, G.H.: An existence theorem for the von Kármán equations. Arch. Ration. Mech. Anal. 27, 233–242 (1967)CrossRefMATHGoogle Scholar
24.
Miyoshi, T.: A mixed finite element method for the solution of the von Kármán equations. Numer. Math. 26, 255–269 (1976)MathSciNetCrossRefGoogle Scholar
25.
Nelson, M.R., King, J.R., Jensen, O.E.: Buckling of a growing tissue and the emergence of two-dimensional patterns. Math. Biosci. 246, 229–241 (2013)MathSciNetCrossRefMATHGoogle Scholar
26.
Reinhart, L.: On the numerical analysis of the von Kármán equations: mixed finite element approximation and continuation techniques. Numer. Math. 39, 371–404 (1982)MathSciNetCrossRefMATHGoogle Scholar
27.
Reiser, A.K.: A $C^0$ Interior Penalty Method for the von Kármán Equations. PhD thesis, Louisiana State University (2011)Google Scholar
28.
Schatz, A.: An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp. 28, 959–962 (1974)MathSciNetCrossRefMATHGoogle Scholar
29.
von Kármán, Th: Festigkeitsprobleme im maschinenbau. Encyklopädie der Mathematischen Wissenschaften. vol. IV, pp. 348–352. Teubner, Leipzig (1910)Google Scholar
30.
Wise, S.M., Wang, C., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47, 2269–2288 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

Authors and Affiliations

Susanne C. Brenner
- 1
Email author
Michael Neilan
- 2
Armin Reiser
- 3
Li-Yeng Sung
- 1

1.Department of Mathematics and Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
2.Department of MathematicsUniversity of PittsburghPittsburghUSA
3.Walther-Groz-SchuleAlbstadtGermany

A \(C^0\) interior penalty method for a von Kármán plate

Abstract

Mathematics Subject Classification

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article