A posteriori error estimates for non-stationary non-linear convection–diffusion equations

Abstract

Motivated by stochastic convection-diffusion problems we derive a posteriori error estimates for non-stationary non-linear convection-diffusion equations acting as a deterministic paradigm. The problem considered here neither fits into the standard linear framework due to its non-linearity nor into the standard non-linear framework due to the lacking differentiability of the non-linearity. Particular attention is paid to the interplay of the various parameters controlling the relative sizes of diffusion, convection, reaction, and non-linearity (noise).(2.1) Here, Ω ⊂ R d , d ∈ {2, 3}, is a bounded polyhedral domain with Lipschitz boundary Γ. The final time T is arbitrary, but kept fixed in what follows. We assume that the data satisfy the following conditions (compare [21, §3] and [25, §6.2]):(A1) ε > 0, ν ≥ 0,

A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics

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“…We give numerical results for Problem (12), by comparing number of triangles and CPU time before and after the use of mesh self-adaptation based on a posteriori error indicators developed in Section 3. The self-adaptive mesh algorithm used in this work is based on the h-adaptation technique, the later was subject to several works like [8,14].…”

A Posteriori Error Analysis for Implicit–Explicit hp-Discontinuous Galerkin Timestepping Methods for Semilinear Parabolic Problems

“…To the best of our knowledge, however, there are no previous results on a posteriori error bounds for implicit-explicit high order fully-discrete methods involving dG-timestepping for nonlinear evolution PDEs. This is in contrast with the increasing number of interesting works on a posteriori error analyses for low order time-stepping schemes for nonlinear evolution problems; see, e.g., [8,10,12,17,36,52] and the references therein. At the same time, there exist only few works on the a posteriori error analysis of high order time-stepping schemes for time-discrete nonlinear parabolic problems [30,33,34,41,46].…”