Volume 54, Number 6, November-December 2020
|Page(s)||1797 - 1820|
|Published online||31 July 2020|
Analysis of optimal superconvergence of an ultraweak-local discontinuous Galerkin method for a time dependent fourth-order equation
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 4 April 2020
In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k + min(3, k) when piecewise ℙk polynomials with k ≥ 2 are used. We also prove a 2k-th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of (k + 2)-th and (k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.
Mathematics Subject Classification: 65M60 / 65M15
Key words: Ultraweak-local discontinuous Galerkin methods / superconvergence / fourth order derivatives
© EDP Sciences, SMAI 2020
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.