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

Abstract

Residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. Standard finite element approximations are considered. The error constants are independent of the diameters of mesh elements and the small perturbation parameter. In our analysis, we employ sharp bounds on the Green's function of the linearized differential operator. Numerical results are presented that support our theoretical findings.

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

“…In Figure 7.3 we plot the observed errors }u´u h } 8 ;Ω versus degrees of freedom (DOF) for fixed ε " 10´4 (left) and ε varied (right). We observe that the mesh refinement yields a very dramatic error reduction (compared with the isotropic mesh refinement [6]). While the maximum mesh aspect ratios vary between 2 and 3.35e+7, the effectivity indices do not exceed 85 in all our experiments.…”

“…We additionally assume that C f`ε 2 ě 1 (as a division by C f`ε 2 immediately reduces (1.1) to this case). Residual-type a posteriori error estimates in the maximum norm for this equation and its version in R 3 were recently proved in [6] in the case of shape-regular triangulations. In the present paper, we restrict our consideration to Ω in R 2 and linear finite elements, but our focus now shifts to more challenging anisotropic meshes, i.e.…”

“…Note that if ε " 1, then (1.3) gives a standard a posteriori error bound, similar to the results in [7,15,17], only now we prove it for anisotropic meshes. Furthermore, (1.3) is almost identical with the a posteriori error estimate in [6], where the singularly perturbed case ε P p0, 1s is handled; by contrast, now we assume no shape regularity of the mesh.…”

“…To represent the error pointwise, we employ the Green's function for a standard linearization of Lu h´L u. Most results in this section are quoted from [6].…”

“…To deal with g " G´G h inΘ, it is convenient to employ a quasi-interpolant (of Clément/Scott-Zhang type) with the property |G´G h | k,p ;T À H j´k T |G| j,p ;ω T for any 0 ď k ď j ď 2, 1 ď p ď 8; see [6] for the case of shape-regular triangulations. However, such interpolants are not readily available for general anisotropic meshes (we refer the reader to [1,Chapt.…”