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 -