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: email@example.com
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
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