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arXiv · 2004.09842

Hessian discretisation method for fourth order semi-linear elliptic equations: applications to the von K\'{a}rm\'{a}n and Navier--Stokes models

Jérome Droniou·Neela Nataraj·Devika Shylaja
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Abstract

This paper deals with the Hessian discretisation method (HDM) for fourth order semi-linear elliptic equations with a trilinear nonlinearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as, the conforming and non-conforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth order semi-linear elliptic equations with trilinear nonlinearity. Four properties namely, the coercivity, consistency, limit-conformity and compactness enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications namely, the Navier--Stokes equations in stream function vorticity formulation and the von K\'{a}rm\'{a}n equations of plate bending are discussed. Results of numerical experiments are presented for the Morley ncFEM and GR method.

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Jérome Droniou, Neela Nataraj, Devika Shylaja. 2020-04-21. Hessian discretisation method for fourth order semi-linear elliptic equations: applications to the von K\'{a}rm\'{a}n and Navier--Stokes models. https://arxiv.org/abs/2004.09842

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