Maximum-Norm A Posteriori Error Estimates for Singularly Perturbed Reaction-Diffusion Problems on Anisotropic Meshes

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.

Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems

“…Remark 2 Similar Green's function bounds for the case ε << 1 and C f ∼ 1, but on significantly simpler tensor-product domains are given in [7,24]. An inspection of the proofs in these papers reveals that in this case, all bounds of Theorem 1 are sharp with respect to ε, ρ and ρ.…”

“…The maximum norm, by contrast, is sufficiently strong to capture sharp layers in the exact solution, so it appears more suitable for such problems. A posteriori estimates in the maximum norm for equations of type (1.1) are given in [24,7]; the results are independent of the mesh aspect ratios, but apply only to tensor-product meshes. The situation with a priori error estimates in the maximum norm for such equations is much more satisfactory.…”

“…The situation with a priori error estimates in the maximum norm for such equations is much more satisfactory. In [37,30], such bounds are given for finite element methods on globally quasiuniform meshes, while for a priori bounds in the maximum norm on locally-anisotropic layer-adapted meshes (for both finite element and finite difference methods) we refer the reader to [4,5,9,24,39] and references therein.…”

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

“…Note that most of our analysis applies to more general node types that were considered in [6,7]; see Remarks 3.2 and 4.3 for details.…”

“…The presence of such matching functions in the estimator is clearly undesirable. It is entirely avoided in the more recent papers [6,7,8], as well as here.…”