Corners give problems when decoupling fourth order equations into second order systems. (English) Zbl 1260.35040

Gerasimov, Tymofiy; Stylianou, Athanasios; Sweers, Guido

SIAM J. Numer. Anal. 50, No. 3, 1604-1623 (2012).

The purpose of this paper is to compare solutions coming from the decoupling of a fourth order equation from linear elasticity in planar domains with corner type singularities. In the first part of the paper under review, the authors recall some results on the existence and uniqueness for the Navier and Dirichlet bi-Laplace problems in smooth domains. In such a way, the equivalence between the original boundary value problem and its corresponding system splitting is shown. In the second part of this paper, the authors illustrate the use of piecewise affine finite elements for approximating the system solutions in both cases. In the third chapter, issues of existence of the Navier bi-Laplace problem are considered, and both an existence and a non-existence result for the Dirichlet system in domains with respectively convex and concave corners are proven. The last section addresses issues of convergence of the numerical scheme for the Dirichlet bi-Laplace system.

Reviewer: Teodora-Liliana Rădulescu (Craiova)

MSC:

35J40	Boundary value problems for higher-order elliptic equations
35J50	Variational methods for elliptic systems
74G15	Numerical approximation of solutions of equilibrium problems in solid mechanics
74K20	Plates

Keywords:

clamped plate; polygonal domain; reentrant corner; numerical approximation; mixed method