A Priori and A Posteriori Error Control of Discontinuous Galerkin Finite Element Methods for the Von K\'arm\'an Equations

Abstract

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von K\'arm\'an equations defined on a polygonal domain. A discrete inf-sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established and this allows the proof of local existence and uniqueness of a discrete solution to the non-linear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel $C^0$ interior penalty method (IPDG). Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known $C^0$-IPDG lead to complications with some non-residual type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a non convex domain.