An optimal L (L2)-error estimate for the discontinuous Galerkin approximation of a nonlinear non-stationary convection-diffusion problem

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 space-time 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 interiorpenalty discontinuous Galerkin method in space and the Crank-Nicolson scheme in time. Numerical experiments illustrate the theory.

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“…with ϑ(x, y, t) := x+y+1−t 2ε , see [10]. This problem exhibits an inner layer moving in the diagonal direction.…”

Section: Linear Diffusionmentioning

confidence: 99%

A Framework for Robust A Posteriori Error Control in Unsteady Nonlinear Advection-Diffusion Problems

Dolejší¹,

Ern²,

Vohralı́k³

2013

SIAM J. Numer. Anal.

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“…Reference 57 study a two stage IMEX RK scheme, where a symmetric stabilization is used and stabilization and convection are handled explicitly, but diffusion implicitly. For non-linear convectiondiffusion problems, dG in space and several discretizations in time, see 58,59 .…”

Section: Unsteady Problemsmentioning

confidence: 99%

Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

Roos

2012

ISRN Applied Mathematics

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“…In our recent papers [7], [10], [9] we applied the IPG methods to a scalar nonstationary convection-diffusion equation with nonlinear Lipschitz continuous convective terms. This equation represents a model problem for the solution of the system of the compressible Navier-Stokes equations which describes the flow of viscous compressible fluids.…”

Section: Introductionmentioning

confidence: 99%

“…In papers [7] and [9], in the formulation of the initial-boundary value problem the mixed Dirichlet-Neumann boundary conditions were considered. However, the reader can recognize that in the basic paper [7], the proof of the truncation error in nonlinear convective terms was carried out in the case when the Dirichlet boundary condition is prescribed on the whole boundary of the computational domain.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

2010

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An optimal L (L2)-error estimate for the discontinuous Galerkin approximation of a nonlinear non-stationary convection-diffusion problem

Cited by 25 publications

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A Framework for Robust A Posteriori Error Control in Unsteady Nonlinear Advection-Diffusion Problems

A Framework for Robust A Posteriori Error Control in Unsteady Nonlinear Advection-Diffusion Problems

Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

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