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Superconvergence of the direct discontinuous Galerkin method for convection‐diffusion equations
Author(s) -
Cao Waixiang,
Liu Hailiang,
Zhang Zhimin
Publication year - 2017
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22087
Subject(s) - superconvergence , mathematics , discontinuous galerkin method , convection–diffusion equation , discretization , partial derivative , order (exchange) , galerkin method , polynomial , mathematical analysis , degree of a polynomial , finite element method , physics , finance , economics , thermodynamics
This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2 k ‐th and ( k + 2 ) ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ( k + 1 ) ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017