Γ-Convergence of external approximations in boundary value problems involving the bi-Laplacian

摘要

In addition to the various uses it was introduced for, the theory of Γ-convergence offers a rather natural setting for discussing and developing nonorthodox approximation methods for variational problems. For certain boundary value problems involving the bi-Laplacian, sequences of discrete functionals are here defined and are shown to Γ-converge to the corresponding functionals of the continuous problems. The minimizers of the discrete functionals provide converging approximations to the solution of the limit problem in question. Thus, we obtain approximation schemes that are nonconforming, but direct, and that can be treated by current algorithms for symmetric and positive definite functionals.The class of problems considered in this paper includes the Stokes problem in fluid dynamics, the loading problem of 2-D-isotropic elastostatics, and some boundary value problems of the Kirchhoff–Love theory of plates. Also discussed is an extension of the discretization method that seems suitable for treating more general boundary value problems of elastic plates, but whose convergence is conditional to a conjecture that remains to be proved. A relevant application to the so-called Babuška paradox is presented.