Maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction-diffusion problem

A singularly perturbed semilinear reaction-diffusion equation, posed in the unit square, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio assumption is made. Numerical results are presented that support our theoretical estimat

Greenʼs function estimates for a singularly perturbed convection–diffusion problem

We consider a singularly perturbed convection-diusion problem posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Green's function and its first- and second-order derivatives are derived in the L1 norm. The dependence of these estimates on the small diusion parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem

Adaptivity in anisotropic finite element calculations

When the finite element method is used to solve boundary value problems, the corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters. There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation. Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment. In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown. Numerical experiments for the equilibrated residual method, for the hierarchical error estimator and for the adaptive algorithm confirm the theory. The adaptive algorithm shows its potential by creating the anisotropic mesh for the problem with the boundary layer starting with a very coarse isotropic mesh