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Morley finite element method for the von Kármán obstacle problem. (English) Zbl 07420191

Summary: This paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Kármán obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Kármán obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.

MSC:

65K15 Numerical methods for variational inequalities and related problems
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] C. Bacuta, J.H. Bramble and J.E. Pasciak, Shift Theorems for the Biharmonic Dirichlet Problem. In: T.F. Chan, Y. Huang, T. Tang, J. Xu, L.A. Ying (eds.) Recent Progress in Computational and Applied PDES. Springer, Boston, MA (2002) 1-26. · Zbl 1068.65134
[2] M.S. Berger and P.C. Fife, Von Kármán equations and the buckling of a thin elastic plate, II plate with general edge conditions. Commun. Pure Appl. Math. 21 (1968) 227-241. · Zbl 0162.56501 · doi:10.1002/cpa.3160210303
[3] H. Blum and R. Rannacher, On mixed finite element methods in plate bending analysis. Comput. Mech. 6 (1990) 221-236. · Zbl 0736.73061
[4] H. Blum, R. Rannacher and R. Leis, On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980) 556-581. · Zbl 0445.35023
[5] S.C. Brenner, M. Neilan, A. Reiser and L.-Y. Sung, A # interior penalty method for a von Kármán plate. Numerische Mathematik 135 (2017) 803-832. · Zbl 1457.65181
[6] S.C. Brenner, L.-Y. Sung, H. Zhang and Y. Zhang, A quadratic # interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates. SIAM J. Numer. Anal. 50 (2012) 3329-3350. · Zbl 1263.65110
[7] S.C. Brenner, L.-Y. Sung, H. Zhang and Y. Zhang, A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates. J. Comput. Appl. Math. 254 (2013) 31-42. · Zbl 1290.65108
[8] S.C. Brenner, L.-Y. Sung and Y. Zhang, Finite element methods for the displacement obstacle problem of clamped plates. Math. Comput. 81 (2012) 1247-1262. · Zbl 1250.74023
[9] F. Brezzi, Finite element approximations of the von Kármán equations. ESAIM: M2AN 12 (1978) 303-312. · Zbl 0398.73070
[10] L.A. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 6 (1979) 151-184. · Zbl 0405.31007
[11] C. Carstensen, D. Gallistl and J. Hu, A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes. Comput. Math. Appl. 68 (2014) 2167-2181. · Zbl 1362.65123 · doi:10.1016/j.camwa.2014.07.019
[12] C. Carstensen, G. Mallik and N. Nataraj, A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations. IMA J. Numer. Anal. 39 (2019) 167-200. · Zbl 1465.65129
[13] C. Carstensen, G. Mallik and N. Nataraj, Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity. IMA J. Numer. Anal. 41 (2021) 164-205. · Zbl 1460.65145
[14] C. Carstensen and N. Nataraj, Adaptive Morley FEM for the von Kármán equations with optimal convergence rates. SIAM J. Numer. Anal. . Preprint: arXiv:1908.08013 (2020). · Zbl 1467.65107
[15] C. Carstensen and S. Puttkammer, How to prove the discrete reliability for nonconforming finite element methods. J. Comput. Math. 38 (2020) 142-175. · Zbl 1463.65362
[16] P.G. Ciarlet (ed.) The finite element method for elliptic problems, Vol. 4. Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978). · Zbl 0383.65058
[17] P.G. Ciarlet, Mathematical Elasticity: Volume II: Theory of Plates,Vol. 27. Studies in Mathematics and its Applications. Elsevier (1997). · Zbl 0888.73001
[18] J. Frehse, Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36 (1971) 140-149. · Zbl 0219.35029
[19] D. Gallistl, Morley finite element method for the eigenvalues of the biharmonic operator. IMA J. Numer. Anal. 35 (2014) 1779-1811. · Zbl 1332.65160
[20] R. Glowinski, Lectures on Numerical Methods for Non-linear Variational Problems. Springer-Verlag, Berlin-Heidelberg (2008). · Zbl 0456.65035
[21] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2002) 865-888. · Zbl 1080.90074
[22] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Vol. 88. Pure and Applied Mathematics. Academic Press (1980). · Zbl 0457.35001
[23] G.H. Knightly, An existence theorem for the von Kármán equations. Arch. Ration. Mech. Anal. 27 (1967) 233-242. · Zbl 0162.56303
[24] G. Mallik and N. Nataraj, Conforming finite element methods for the von Kármán equations. Adv. Comput. Math. 42 (2016) 1031-1054. · Zbl 1382.74121
[25] G. Mallik and N. Nataraj, A nonconforming finite element approximation for the von Kármán equations. ESAIM: M2AN 50 (2016) 433-454. · Zbl 1375.74089
[26] E. Miersemann and H.D. Mittelmann, Stability in obstacle problems for the von Kármán plate. SIAM J. Math. Anal. 23 (1992) 1099-1116. · Zbl 0758.73019
[27] T. Miyoshi, A mixed finite element method for the solution of the von Kármán equations. Numerische Mathematik 26 (1976) 255-269. · Zbl 0315.65064
[28] A.D. Muradova and G.E. Stavroulakis, A unilateral contact model with buckling in von Kármán plates. Nonlinear Anal.: Real World Appl. 8 (2007) 1261-1271. · Zbl 1114.74043
[29] K. Ohtake, J.T. Oden and N. Kikuchi, Analysis of certain unilateral problems in von Kármán plate theory by a penalty method-part 1. a variational principle with penalty. Comput. Methods Appl. Mech. Eng. 24 (1980) 187-213. · Zbl 0457.73095
[30] K. Ohtake, J.T. Oden and N. Kikuchi, Analysis of certain unilateral problems in von Kármán plate theory by a penalty method-part 2. approximation and numerical analysis. Comput. Methods Appl. Mech. Eng. 24 (1980) 317-337. · Zbl 0457.73096
[31] A. Quarteroni, Hybrid finite element methods for the von Kármán equations. Calcolo 16 (1979) 271-288. · Zbl 0443.73035 · doi:10.1007/BF02575930
[32] L. Reinhart, On the numerical analysis of the von Kármán equations: mixed finite element approximation and continuation techniques. Numerische Mathematik 39 (1982) 371-404. · Zbl 0503.73048
[33] S.-T. Yau and Y. Gao, Obstacle problem for von Kármán equations. Adv. Appl. Math. 13 (1992) 123-141. · Zbl 0766.35087
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