V. G. Korneev

V. G Korneev

Efficiency of the error control of numerical solutions of partial differential equations entirely depends on the two factors: accuracy of an a posteriori error majorant and the computational cost of its evaluation for some test function/vector-function plus the cost of the latter. In the paper, consistency of an a posteriori bound implies that it is the same in the order with the respective unimprovable a priori bound. Therefore, it is the basic characteristic related to the first factor. The paper is dedicated to the elliptic diffusion-reaction equations. We present a guaranteed robust a posteriori error majorant effective at any nonnegative constant reaction coefficient (r.c.). For a wide range of finite element solutions on a quasiuniform meshes the majorant is consistent. For big values of r.c. the majorant coincides with the majorant of Aubin (1972), which, as it is known, for not big r.c. ($<ch^{-2}$) is inconsistent and loses its sense at r.c. approaching zero. Our majorant improves also some other majorants derived for the Poisson and reaction-diffusion equations.

Vadim Glebovich Korneev

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].