TY - JOUR

T1 - Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations

AU - Kopteva, Natalia

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2014

Y1 - 2014

N2 - We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp boundary layer. Furthermore, we give a theoretical justification of the observed numerical phenomena using a finite-difference representation of the considered finite element methods. Both standard and lumped-mass cases are addressed.

AB - We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp boundary layer. Furthermore, we give a theoretical justification of the observed numerical phenomena using a finite-difference representation of the considered finite element methods. Both standard and lumped-mass cases are addressed.

KW - Anisotropic triangulation

KW - Bakhvalov mesh

KW - Linear finite elements

KW - Maximum norm

KW - Shishkin mesh

KW - Singular perturbation

UR - http://www.scopus.com/inward/record.url?scp=84910020982&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-2014-02820-2

DO - 10.1090/S0025-5718-2014-02820-2

M3 - Article

AN - SCOPUS:84910020982

SN - 0025-5718

VL - 83

SP - 2061

EP - 2070

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 289

ER -