Publication:
Numerische Mathematik;146, pp. 159-179

Publication type:
Article

Abstract:

Lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes. A numerical example is given that clearly demonstrates that the short-edge jump residual terms in such bounds are not sharp. Hence, for linear ﬁnite element approximations of the Laplace equation in polygonal domains, a new approach is employed to obtain essentially sharper lower a posteriori error bounds and thus to show that the upper error estimator in the recent paper [3] is eﬃcient on partially structured anisotropic meshes.

Kopteva, Natalia; O'Riordan, Eugene(Institute for Scientific Computing and Informaton, 2010)

This article reviews some of the salient features of the piecewise-uniform Shishkin mesh. The central analytical techniques involved in the associated numerical analysis are explained via a particular class of singularly ...

Kopteva, Natalia; Linss, Torsten(SIAM:Society for Industrial and Applied Mathematics, 2013)

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully ...

Kopteva, Natalia(SIAM: Society for Industrial and Applied Mathematics, 2015)

Residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic ...

A two-point boundary value problem is considered on the interval , where the leading term in the differential operator is a Caputo fractional-order derivative of order with . The problem is reformulated as a Volterra ...

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 ...