Energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes

Abstract

In the recent article [7] the author obtained residual-type a posteriori error estimates in the energy norm for singularly perturbed semilinear reaction-diffusion equations on unstructured anisotropic triangulations. The error constants in these estimates are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter. The purpose of this note is to improve the weights in the jump residual part of the estimator. This is attained by using a novel sharper version of the scaled trace theorem for anisotropic elements, in which the hat basis functions are involved as weights.

A simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems

“…Note that this lemma holds true for any flux τ ∈ H(div, Ω) given by (24). Its local components τ n ∈ W (ω n ) are not required to minimize the quadratic functional E n defined in (18).…”

“…Locally efficient and robust guaranteed error bounds for the vertex-centred finite volume discretization of (1) were proposed in [9]. Robust reliability estimate for singularly perturbed problem on anisotropic meshes is proved in [18]. A similar result for a guaranteed and fully computable error bound is provided in [19].…”

Lower a posteriori error estimates on anisotropic meshes

“…The presence of such matching functions in the estimator is clearly undesirable. It is entirely avoided in the more recent papers [2,3,4], where upper a posteriori error estimates on anisotropic meshes were obtained for singularly perturbed semilinear reaction-diffusion equations in the energy norm and in the maximum norm.…”

“…Interestingly, the efficiency of the estimators in [2,3,4] cannot be established using the standard bubble function approach, employed in [5,6,7]. To be more precise, this approach (which will be reviewed in §2) leads to lower error bounds with significantly smaller weights at the short-edge jump residual terms than those in the upper bounds.…”

“…Compared to [2,3,4], to simplify the presentation, we shall restrict consideration to the simpler Laplace equation and consider the problem ´△u " f px, yq for px, yq P Ω, u " 0 on BΩ,…”