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

De Coster, Colette; Nicaise, Serge; Sweers, Guido

Math. Nachr. 288, No. 8-9, 854-871 (2015).

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

Keywords:

biharmonic operator; corner domains

Citations:

Zbl 0337.65058; Zbl 0632.65112

References:

[1]	Amara, An optimal C0-finite element algorithm for the 2D biharmonic problem: theoretical analysis and numerical results, Numer. Math. 90 (1) pp 19– (2001) · Zbl 0997.65133
[2]	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) pp 76– (1961)
[3]	Babuška, Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems, Comput. Method Appl. Mech. Eng. 11 (2) pp 175– (1977) · Zbl 0382.65056
[4]	Blum, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci. 2 (4) pp 556– (1980) · Zbl 0445.35023
[5]	Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat, Recherche Opérationnelle Sér. Rouge 8 (R2) pp 129– (1974)
[6]	Brezzi, Springer Series in Computational Mathematics Vol. 15 (1991)
[7]	Ciarlet, 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 pp 125– (1974) · Zbl 0337.65058
[8]	Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (3) pp 613– (1988) · Zbl 0644.35037
[9]	Dauge, Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics Vol. 1341 (1988) · Zbl 0668.35001
[10]	Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners I, SIAM J. Math. Anal. 20 (1) pp 74– (1989) · Zbl 0681.35071
[11]	Gerasimov, Corners give problems with decoupling fourth order equations into second order systems, SIAM J. Numer. Anal. 50 (3) pp 1604– (2012) · Zbl 1260.35040
[12]	Grisvard, Elliptic Problems in Non-smooth Domains (1985) · Zbl 0695.35060
[13]	Grisvard, Singularities in Boundary Value Problems (1992) · Zbl 0766.35001
[14]	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) pp 386– (1964) · Zbl 0166.37703
[15]	Kondrat’ev, Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc 16 pp 227– (1967)
[16]	Kozlov, Mathematical Surveys and Monographs Vol. 52 (1997)
[17]	Ladyzhenskaya, Linear and Quasilinear Elliptic Equations (1968)
[18]	Lax, Functional Analysis (2002)
[19]	Lozi, Résultats numériques de régularité du problème de Stokes et du laplacien itéré dans un polygone, RAIRO Anal. Numér. 12 (12) pp 267– (1978)
[20]	Maz’ya, On the bending of a nearly polygonal plate with freely supported boundary, Sov. Math. 27 (8) pp 40– (1983)
[21]	Monk, A mixed finite element method for the biharmonic equation, SIAM J. Numer. Anal. 24 pp 737– (1987) · Zbl 0632.65112
[22]	Nazarov, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners, J. Differential Equations 233 (1) pp 151– (2007) · Zbl 1108.35043
[23]	Nicaise, Methoden und Verfahren der Mathematischen Physik 39 (1993)
[24]	Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Numer. 12 (1) pp 85– (1978)
[25]	Seif, On the Green’s function for the biharmonic equation on an infinite wedge, Trans. Amer. Math. Soc. 182 pp 241– (1973) · Zbl 0271.31005
[26]	Williams, Stress singularities resulting from various boundary conditions in angular corners of plates in extension, J. Appl. Mech. 19 pp 526– (1952)