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# Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Volume 27, Issue 3

# Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems

• Department of Mathematics, Birla Institute of Technology & Science, Pilani Campus, Rajasthan, 333031, India
• Other articles by this author:
/ Amiya K. Pani
• Corresponding author
• Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India
• Email
• Other articles by this author:
Published Online: 2019-09-17 | DOI: https://doi.org/10.1515/jnma-2018-0035

## Abstract

In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence $\begin{array}{}O\left(\phantom{\rule{negativethinmathspace}{0ex}}\sqrt{\mathrm{log}\left(T/{h}^{2}\right)}\phantom{\rule{thinmathspace}{0ex}}{h}^{k+2}\right)\end{array}$ in L(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems.

JEL Classification: 65M60; 65M15

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Revised: 2018-08-19

Accepted: 2018-09-29

Published Online: 2019-09-17

Published in Print: 2019-09-25

Funding: We are thankful to the referees for their valuable suggestions, which help to improve our manuscript. The first author acknowledges the financial support by DST FIST project having no. SR/FST/MSI-090/2013(C).

Citation Information: Journal of Numerical Mathematics, Volume 27, Issue 3, Pages 183–202, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820,

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