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Convergence analysis of a finite element method for second order non-variational elliptic problems

Michael Neilan

Abstract

We introduce and analyze a family of finite element methods for elliptic partial differential equations in non-variational form with non-differentiable coefficients. The finite element method studied is a variant of the one recently proposed in [Lakkis & Pryer, SIAM J. Sci. Comput., 2011], where a finite element Hessian is introduced as an auxiliary unknown. We modify the definition of the finite element Hessian rendering the auxiliary variable completely local, thus resulting in a more efficient scheme. We show that the method is stable under general conditions on the coefficient matrix and derive error estimates in a discrete H2-norm provided the discretization parameter is sufficiently small. Numerical experiments are presented which verify the theoretical results.

JEL Classification: 65N30; 65N12; 35J25

Funding statement: Supported in part by NSF grant DMS–1417980 and the Alfred Sloan Foundation

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Received: 2016-3-10
Revised: 2016-3-28
Accepted: 2016-4-9
Published Online: 2016-5-20
Published in Print: 2017-9-26

© 2016 by Walter de Gruyter Berlin/Boston