Maximum Norm A Posteriori Error Estimation for Parabolic Problems Using Elliptic Reconstructions

Abstract

Abstract. 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 triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter.

A posteriori error estimation for arbitrary order FEM applied to singularly perturbed one-dimensional reaction-diffusion problems

“…Furthermore, the results can be used as a building block in the a posteriori error estimation for parabolic problems in the context of [4], [7], [6]. This is one of the main motivations for the present study.…”

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

“…Elliptic reconstruction technique was developed in [12,16] for parabolic problems, and extended to [3] for non-stationary convection diffusion equations, Demlow et al [5], Kopteva and Linss [11] for maximum norm a posteriori error estimates as well as [10,18] for parabolic integro-differential equation and Schrodinger equation. The elliptic reconstruction can be viewed as an a posteriori analogue to the Ritz-elliptic projection used in a priori error analysis for parabolic equations.…”

A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

“…They have also presented a priori error estimates of the scheme. Subsequently, some a posteriori error estimates based on Green's functions and elliptic reconstructions have been introduced in [5,15].…”