A FRAMEWORK FOR ROBUST A POSTERIORI ERROR CONTROL IN UNSTEADY NONLINEAR ADVECTION-DIFFUSION PROBLEMS

We derive a framework for a posteriori error estimates in unsteady, nonlinear, possibly degenerate, advection-diffusion problems. Our estimators are based on a space-time equilibrated flux reconstruction and are locally computable. They are derived for the error measured in a spacetime mesh-dependen...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2013-01, Vol.51 (2), p.773-793
Main Authors: DOLEJŠÍ, VÍT, ERN, ALEXANDRE, VOHRALÍK, MARTIN
Format: Article
Language:English
Subjects:
A posteriori knowledge
Advection
Approximation
Cauchy Schwarz inequality
Conservation
Enhanced oil recovery
Error analysis
Error rates
Errors
Estimates
Estimators
Galerkin methods
Mathematical problems
Mathematics
Nonlinearity
Numerical Analysis
Polynomials
Spacetime
Unsteady
Upper bounds
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Summary:We derive a framework for a posteriori error estimates in unsteady, nonlinear, possibly degenerate, advection-diffusion problems. Our estimators are based on a space-time equilibrated flux reconstruction and are locally computable. They are derived for the error measured in a spacetime mesh-dependent dual norm stemming from the problem and meshes at hand augmented by a jump seminorm measuring possible nonconformities in space. Owing to this choice, a guaranteed and globally efficient upper bound is achieved, as well as robustness with respect to nonlinearities, advection dominance, domain size, final time, and absolute and relative size of space and time steps. Local-in-time and in-space efficiency is also shown for a localized upper bound of the error measure. In order to apply the framework to a given numerical method, two simple conditions, local space-time mass conservation and an approximation property of the reconstructed fluxes, need to be verified. We show how to do this for the interior-penalty discontinuous Galerkin method in space and the Crank-Nicolson scheme in time. Numerical experiments illustrate the theory.
ISSN:0036-1429
1095-7170
DOI:10.1137/110859282