Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension


Authors: Yanlai Chen, Bernardo Cockburn and Bo Dong
Journal: Math. Comp. 85 (2016), 2715-2742
MSC (2010): Primary 65M60, 65N30
DOI: https://doi.org/10.1090/mcom/3091
Published electronically: March 22, 2016
MathSciNet review: 3522968
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We design and analyze the first hybridizable discontinuous
Galerkin methods for stationary, third-order linear equations in one-space dimension. The methods are defined as discrete versions of characterizations of the exact solution in terms of local problems and transmission conditions. They provide approximations to the exact solution $ u$ and its derivatives $ q:=u'$ and $ p:=u''$ which are piecewise polynomials of degree $ k_u$, $ k_q$ and $ k_p$, respectively. We consider the methods for which the difference between these polynomial degrees is at most two. We prove that all these methods have superconvergence properties which allows us to prove that their numerical traces converge at the nodes of the partition with order at least $ 2\,k+1$, where $ k$ is the minimum of $ k_u,k_q$, and $ k_p$. This allows us to use an element-by-element post-processing to obtain new approximations for $ u, q$ and $ p$ converging with order at least $ 2k+1$ uniformly. Numerical results validating our error estimates are displayed.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M60, 65N30

Retrieve articles in all journals with MSC (2010): 65M60, 65N30


Additional Information

Yanlai Chen
Affiliation: Department of Mathematics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747
Email: yanlai.chen@umassd.edu

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Bo Dong
Affiliation: Department of Mathematics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747
Email: bdong@umassd.edu

DOI: https://doi.org/10.1090/mcom/3091
Received by editor(s): May 18, 2014
Received by editor(s) in revised form: February 2, 2015
Published electronically: March 22, 2016
Additional Notes: The research of the second author was partially supported by the National Science Foundation (grant DMS-1115331).
The research of the third author was partially supported by the National Science Foundation (grant DMS-1419029)
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society