On the error control at numerical solution of reaction-difusion equations

Abstract

We suggest guaranteed, robust a posteriori error bounds for approximate solutions of the reaction-diffusion equations, modeled by the equation $-Δu+σu= f$ in $Ω$ with any $σ={\mathrm{const}}\ge 0$. We also term our bounds consistent due to one specific property. It assumes that their orders of accuracy in respect to mesh size $h$ are the same with the respective not improvable in the order a priori bounds. Additionally, it assumes that the pointed out equality of the orders is provided by the testing flaxes not subjected to equilibration. For any $σ\in [0,σ_*]$, the rirght part of the new general bound of the paper contains, besides the usual diffusion term, the $L_2$ norm of the residual with the factor $1/\sqrt{σ_*}$, where $σ_*$ is some critical value. For solutions by the finite element method, it is estimated as $σ_*\ge ch^{-2},\,\,c={\mathrm{const}}$, if $\partial Ω$ is sufficiently smooth and the finite element space is of the 1$^{\mathrm{st}}$ order of accuracy at least. In general, at the derivation of a posteriori bounds, consistency is achieved by taking adequately into account the difference of the orders of the $L_2$ and $H^1$ error norms, that can be done in various ways with accordingly introduced $σ_*$. Two advantages of the obtained consistent a posteriori error bounds deserve attention. They are better accuracy and the possibility to avoid the use of the equilibration in the flax recovery procedures, that may greatly simplify these procedures and make them much more universal. The technique of obtaining the consistent a posteriori bounds was briefly exposed by the author in [arXiv:1702.00433v1 [math.NA], 1 Feb 2017] and [$Doklady Mathematics$, ${\mathbf{96}}$ (1), 2017, 380-383].