TY - JOUR

T1 - A Note on a Posteriori Error Bounds for Numerical Solutions of Elliptic Equations with a Piecewise Constant Reaction Coefficient Having Large Jumps

AU - Korneev, V. G.

N1 - Publisher Copyright:
© 2020, Pleiades Publishing, Ltd.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - Abstract: We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation ΔΔu + k2u = f, where the coefficient k ≥ 0 is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to k ∈[0, ch−2] c = const, and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for k ≡ const. The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017–2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.

AB - Abstract: We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation ΔΔu + k2u = f, where the coefficient k ≥ 0 is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to k ∈[0, ch−2] c = const, and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for k ≡ const. The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017–2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.

KW - a posteriori error bounds

KW - finite element method

KW - piecewise constant reaction coefficient

KW - sharp bounds

KW - singularly perturbed fourth-order elliptic equations

KW - FINITE-ELEMENT APPROXIMATION

KW - GALERKIN APPROXIMATIONS

KW - ESTIMATOR

KW - PLATE

UR - http://www.scopus.com/inward/record.url?scp=85097312368&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/e49aa33c-18e2-30c4-bff8-053cdf5dad8d/

U2 - 10.1134/S096554252011007X

DO - 10.1134/S096554252011007X

M3 - Article

AN - SCOPUS:85097312368

VL - 60

SP - 1754

EP - 1760

JO - Computational Mathematics and Mathematical Physics

JF - Computational Mathematics and Mathematical Physics

SN - 0965-5425

IS - 11

ER -