Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems

2012 ◽  
Vol 124 (1) ◽  
pp. 151-182 ◽  
Author(s):  
Thomas Richter ◽  
Andreas Springer ◽  
Boris Vexler
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Numerical Realization ◽  
Parabolic Problems ◽  
Galerkin Methods ◽  
Temporal Discretization

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

2019 ◽  
Vol 27 (3) ◽  
pp. 183-202 ◽  
Author(s):  
Sangita Yadav ◽  
Amiya K. Pani
Keyword(s):  
Discontinuous Galerkin ◽  
Order Of Convergence ◽  
Discontinuous Galerkin Methods ◽  
Time Variable ◽  
Parabolic Problems ◽  
Galerkin Methods ◽  
Post Processing ◽  
Spatial Direction ◽  
Piecewise Polynomials ◽  
Nonlinear Parabolic

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 $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ 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.

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