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Publicly Available Published by De Gruyter June 17, 2014

An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators

Daniele A. Di Pietro EMAIL logo , Alexandre Ern and Simon Lemaire

Abstract

We develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (elementwise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the L2-norm of the potential for smooth solutions. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. The theoretical results are confirmed by numerical experiments up to order 4 on several polygonal meshes.

Received: 2014-4-13
Revised: 2014-5-29
Accepted: 2014-6-2
Published Online: 2014-6-17
Published in Print: 2014-10-1

© 2014 by De Gruyter

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