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Solving the biharmonic Dirichlet problem on domains with corners. (English) Zbl 1325.74090

Summary: The biharmonic Dirichlet boundary value problem on a bounded domain is the focus of the present paper. By Riesz’ representation theorem the existence and uniqueness of a weak solution is quite direct. The problem that we are interested in appears when one is looking for constructive approximations of a solution. Numerical methods using for example finite elements, prefer systems of second equations to fourth order problems. P. G. Ciarlet and P. A. Raviart [in: Proc. Symp. Math. Aspects Finite Elem. Partial Diff. Equations 33, 125–145 (1974; Zbl 0337.65058)] and P. Monk in [SIAM J. Numer. Anal. 24, 737–749 (1987; Zbl 0632.65112)] consider approaches through second order problems assuming that the domain is smooth. We discuss what happens when the domain has corners. Moreover, we suggest a setting, which is in some sense between Ciarlet-Raviart and Monk, that inherits the benefits of both settings and that give the weak solution through a system type approach.

MSC:

74K20 Plates
35Q74 PDEs in connection with mechanics of deformable solids
35D30 Weak solutions to PDEs
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[1] M.Amara and F.Dabaghi, An optimal C^0‐finite element algorithm for the 2D biharmonic problem: theoretical analysis and numerical results, Numer. Math.90(1), 19-46 (2001). · Zbl 0997.65133
[2] I.Babuška, Stabilität des Definitionsgebietes mit Rücksicht auf grundlegende Probleme der Theorie der partiellen Differentialgleichungen auch im Zusammenhang mit der Elastizitätstheorie. I, II, (Russian) Czechoslovak Math. J.11(86), 76-105 (1961). 165-203. · Zbl 0126.11401
[3] I.Babuška, J. T.Odena, and J. K.Leeb, Mixed‐hybrid finite element approximations of second‐order elliptic boundary‐value problems, Comput. Method Appl. Mech. Eng.11(2), 175-206 (1977). · Zbl 0382.65056
[4] H.Blum and R.Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci.2(4) (1980), 556-581. · Zbl 0445.35023
[5] F.Brezzi, On the existence, uniqueness and approximation of saddle‐point problems arising from Lagrangian multipliers, Rev. Française Automat. Recherche Opérationnelle Sér. Rouge8(R2), 129-151 (1974). · Zbl 0338.90047
[6] F.Brezzi and M.Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics Vol. 15 (Springer‐Verlag, New York, 1991). · Zbl 0788.73002
[7] P. G.Ciarlet and P.‐A.Raviart, A mixed finite element method for the biharmonic equation, in: Proceedings of a Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations conducted by the Mathematics Research Center, the University of Wisconsin‐Madison, 1974 (Academic Press, New York, 1974), pp. 125-145. · Zbl 0337.65058
[8] M.Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal.19(3), 613-626 (1988). · Zbl 0644.35037
[9] M.Dauge, Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics Vol. 1341, (Springer‐Verlag, Berlin, 1988). · Zbl 0668.35001
[10] M.Dauge, Stationary Stokes and Navier‐Stokes systems on two‐ or three‐dimensional domains with corners I, SIAM J. Math. Anal.20(1), 74-97 (1989). · Zbl 0681.35071
[11] T.Gerasimov, A.Stylianou, and G.Sweers, Corners give problems with decoupling fourth order equations into second order systems, SIAM J. Numer. Anal.50(3), 1604-1623 (2012). · Zbl 1260.35040
[12] P.Grisvard, Elliptic Problems in Non‐smooth Domains (Pitman, London, 1985). · Zbl 0695.35060
[13] P.Grisvard, Singularities in Boundary Value Problems (Masson‐Springer, 1992). · Zbl 0766.35001
[14] J.Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, Czechoslovak Math. J.14(89), 386-393 (1964). · Zbl 0166.37703
[15] V. A.Kondrat’ev, Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc16, 227-313 (1967). · Zbl 0194.13405
[16] V. A.Kozlov, V. G.Maz’ya, and J.Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs Vol. 52 (American Mathematical Society, Providence, RI, 1997). · Zbl 0947.35004
[17] O. A.Ladyzhenskaya, N. N.Uralt’seva, Linear and Quasilinear Elliptic Equations (Academic Press, 1968). · Zbl 0164.13002
[18] P.Lax, Functional Analysis (Wiley, New York, 2002). · Zbl 1009.47001
[19] R.Lozi, Résultats numériques de régularité du problème de Stokes et du laplacien itéré dans un polygone, RAIRO Anal. Numér.12(12), 267-282 (1978). · Zbl 0385.65025
[20] V. G.Maz’ya, S. A.Nazarov, and B. A.Plamenevskii, On the bending of a nearly polygonal plate with freely supported boundary, Sov. Math.27(8), 40-48 (1983). · Zbl 0581.73067
[21] P.Monk, A mixed finite element method for the biharmonic equation, SIAM J. Numer. Anal.24, 737-749 (1987). · Zbl 0632.65112
[22] S. A.Nazarov and G.Sweers, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners, J. Differential Equations233(1), 151-180 (2007). · Zbl 1108.35043
[23] S.Nicaise, Polygonal Interface Problems, Methoden und Verfahren der Mathematischen Physik 39 (Verlag Peter D.Lang (ed.), Frankfurt‐am‐Main, 1993). · Zbl 0794.35040
[24] R.Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Numer.12(1), 85-90 (1978). · Zbl 0382.65059
[25] J. B.Seif, On the Green’s function for the biharmonic equation on an infinite wedge, Trans. Amer. Math. Soc.182, 241-260 (1973). · Zbl 0271.31005
[26] M. L.Williams, Stress singularities resulting from various boundary conditions in angular corners of plates in extension, J. Appl. Mech.19, 526-528 (1952).
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