TY - JOUR

T1 - On Error Control in the Numerical Solution of Reaction–Diffusion Equation

AU - Korneev, V. G.

N1 - Publisher Copyright:
© 2019, Pleiades Publishing, Ltd.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Abstract: A novel method for deriving a posteriori error bounds for approximate solutions of reaction–diffusion equations is proposed. As a model problem, the problem (Formula Presented.) with an arbitrary constant reaction coefficient σ ≥ 0 is studied. For the solutions obtained by the finite element method, bounds, which are called consistent for brevity, are proved. The order of accuracy of these bounds is the same as the order of accuracy of unimprovable a priori bounds. The consistency also assumes that the order of accuracy of such bounds is ensured by test fluxes that satisfy only the corresponding approximation requirements but are not required to satisfy the balance equations. The range of practical applicability of consistent a posteriori error bounds is very wide because the test fluxes appearing in these bounds can be calculated using numerous flux recovery procedures that were intensively developed for error indicators of the residual method. Such recovery procedures often ensure not only the standard approximation orders but also the superconvergency of the recovered fluxes. The advantages of the proposed family of a posteriori bounds are their guaranteed sharpness, no need for satisfying the balance equations in flux recovery procedures, and a much wider range of efficient applicability compared with other a posteriori bounds.

AB - Abstract: A novel method for deriving a posteriori error bounds for approximate solutions of reaction–diffusion equations is proposed. As a model problem, the problem (Formula Presented.) with an arbitrary constant reaction coefficient σ ≥ 0 is studied. For the solutions obtained by the finite element method, bounds, which are called consistent for brevity, are proved. The order of accuracy of these bounds is the same as the order of accuracy of unimprovable a priori bounds. The consistency also assumes that the order of accuracy of such bounds is ensured by test fluxes that satisfy only the corresponding approximation requirements but are not required to satisfy the balance equations. The range of practical applicability of consistent a posteriori error bounds is very wide because the test fluxes appearing in these bounds can be calculated using numerous flux recovery procedures that were intensively developed for error indicators of the residual method. Such recovery procedures often ensure not only the standard approximation orders but also the superconvergency of the recovered fluxes. The advantages of the proposed family of a posteriori bounds are their guaranteed sharpness, no need for satisfying the balance equations in flux recovery procedures, and a much wider range of efficient applicability compared with other a posteriori bounds.

KW - a posteriori error bounds

KW - finite element method

KW - flux recovery procedures

KW - reaction–diffusion equations

KW - sharp bounds

UR - http://www.scopus.com/inward/record.url?scp=85065756909&partnerID=8YFLogxK

U2 - 10.1134/S0965542519010123

DO - 10.1134/S0965542519010123

M3 - Article

AN - SCOPUS:85065756909

VL - 59

JO - Computational Mathematics and Mathematical Physics

JF - Computational Mathematics and Mathematical Physics

SN - 0965-5425

IS - 1

ER -