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Finite element methods for second order linear elliptic partial differential equations in non-divergence form


Authors: Xiaobing Feng, Lauren Hennings and Michael Neilan
Journal: Math. Comp. 86 (2017), 2025-2051
MSC (2010): Primary 65N30, 65N12, 35J25
DOI: https://doi.org/10.1090/mcom/3168
Published electronically: February 13, 2017
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Abstract: This paper is concerned with finite element approximations of $ W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. A non-standard (primal) finite element method, which uses finite-dimensional subspaces consisting of globally continuous piecewise polynomial functions, is proposed and analyzed. The main novelty of the finite element method is to introduce an interior penalty term, which penalizes the jump of the flux across the interior element edges/faces, to augment a non-symmetric piecewise defined and PDE-induced bilinear form. Existence, uniqueness and error estimate in a discrete $ W^{2,p}$ energy norm are proved for the proposed finite element method. This is achieved by establishing a discrete Calderon-Zygmund-type estimate and mimicking strong solution PDE techniques at the discrete level. Numerical experiments are provided to test the performance of proposed finite element methods and to validate the convergence theory.


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Additional Information

Xiaobing Feng
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email: xfeng@math.utk.edu

Lauren Hennings
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: LNH31@pitt.edu

Michael Neilan
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: neilan@pitt.edu

DOI: https://doi.org/10.1090/mcom/3168
Keywords: Finite element methods, non-divergence form, convergence analysis
Received by editor(s): May 11, 2015
Received by editor(s) in revised form: February 1, 2016, and April 1, 2016
Published electronically: February 13, 2017
Additional Notes: The first author was partially supported in part by NSF grants DMS-1016173 and DMS-1318486.
This work was partially supported by the NSF through grants DMS-1016173 and DMS-1318486 (Feng), and DMS-1417980 (Neilan) and the Alfred Sloan Foundation (Neilan).
The third author was partially supported in part by NSF grant DMS-1417980 and the Alfred Sloan Foundation.
Article copyright: © Copyright 2017 American Mathematical Society