Geodesic
A posteriori upper and lower error bound of the high-order discontinuous Galerkin method for the heat conduction equation
Applications of Mathematics, Tome 59 (2014) no. 2, pp. 121-144.

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We deal with the numerical solution of the nonstationary heat conduction equation with mixed Dirichlet/Neumann boundary conditions. The backward Euler method is employed for the time discretization and the interior penalty discontinuous Galerkin method for the space discretization. Assuming shape regularity, local quasi-uniformity, and transition conditions, we derive both a posteriori upper and lower error bounds. The analysis is based on the Helmholtz decomposition, the averaging interpolation operator, and on the use of cut-off functions. Numerical experiments are presented.
DOI : 10.1007/s10492-014-0045-7
Classification : 65M15, 65M60
Keywords: discontinuous Galerkin method; Helmholtz decomposition; averaging interpolation operator; Euler backward scheme; residual-based a posteriori error estimate; local cut-off function 
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     title = {A posteriori upper and lower error bound of the high-order discontinuous {Galerkin} method for the heat conduction equation},
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Šebestová, Ivana. A posteriori upper and lower error bound of the high-order discontinuous Galerkin method for the heat conduction equation. Applications of Mathematics, Tome 59 (2014) no. 2, pp. 121-144. doi : 10.1007/s10492-014-0045-7. http://geodesic.mathdoc.fr/articles/10.1007/s10492-014-0045-7/

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