Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the Discontinuous Galerkin time discretization

Author (s): Parés, N., Díez, P. and Huerta, A..
Journal: Computer Methods in Applied Mechanics and Engineering

Volume: 197, Issues 19-20
Pages: 1641 – 1660
Date: 2008

Classical implicit residual type error estimators require using an underlying spatial finer
mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained
are only guaranteed asymptotically that is with respect to the reference solution computed
with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution,
are needed to properly certify the accuracy of the results, especially if the meshes
are coarse. The paper introduces a procedure to compute strict upper and lower bounds of
the error in linear functional outputs of parabolic problems. In this first part, the bounds
account for the error associated with the spatial discretization. The error coming from the
time marching scheme is therefore assumed to be negligible in front of the spatial error.
The time discretization is performed using the discontinuous Galerkin method, both for
the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at
interelement edges are calculated using hybridization techniques.



author = "Parés, N.; Díez, P. and Huerta, A.“,
title = "Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous Galerkin time discretization ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "197",
number = "19–20",
pages = "1641 - 1660",
year = "2008",
note = "Computational Methods in Fluid–Structure Interaction ",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2007.08.025",
url = "http://www.sciencedirect.com/science/article/pii/S0045782507003453",