Premium
Scaled discrete derivatives of singularly perturbed elliptic problems
Author(s) -
Gracia J. L.,
O'Riordan E.
Publication year - 2015
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21900
Subject(s) - mathematics , pointwise , piecewise , uniform convergence , norm (philosophy) , monotone polygon , bilinear form , approximations of π , tensor product , mathematical analysis , finite difference , polygon mesh , pure mathematics , geometry , computer network , bandwidth (computing) , computer science , political science , law
Numerical approximations to the solution of a singularly perturbed elliptic convection–diffusion problem in two space dimensions are generated using a monotone finite difference operator on a tensor product of piecewise‐uniform Shishkin meshes. The bilinear interpolants of these numerical approximations are parameter‐uniformly convergent to the solution of the continuous problem, in the pointwise maximum norm. In this article, discrete approximations to the first derivatives of the solution are shown to be globally first‐order (up to logarithmic factors) uniformly convergent, when the errors are scaled within the analytical layers of the continuous problem. Numerical results are presented to illustrate the theoretical error bounds established in an appropriated weighted C 1 –norm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 225–252, 2015