The optimal convergence rate of a \(C^{1}\) finite element method for non-smooth domains (Q847257)

The authors establish optimal (up to arbitrary \(\varepsilon >0\)) convergence rates for a finite element formulation of a model second order elliptic boundary value problem in a weighted \(H^{2}\) Sobolev space with 5th degree Argyris elements. This formulation arises while generalizing to the case of non-smooth domains an unconditionally stable scheme developed by \textit{J.-G. Liu, J. Liu} and \textit{R. L. Pego} [Commun. Pure Appl. Math. 60, No.~10, 1443--1487 (2007; Zbl 1131.35058)] for the Navier-Stokes equations. The authors prove the optimality for both quasiuniform and graded mesh refinements, and provide numerical results that agree with their theoretical predictions.