Solutions of partial differential equations frequently exhibit corner singularities and/or sharp boundary and interior layers. To obtain reliable numerical approximations of such solutions in an efficient way, one may want to use meshes that are adapted to solution singularities. Such meshes can be constructed using a priori information on the exact solutions, however it is rarely available in real-life applications. Therefore the best hope for the future seems to be offered by the automated mesh construction by adaptive techniques. This approach requires no initial asymptotic understanding of the nature of the solutions and the solution singularity locations. Reliable adaptive algorithms are based on a posteriori error estimates, i.e. estimates of the error in terms of values obtained in the computation process: computed solution and current mesh. Such a posteriori error estimates for elliptic partial differential equations will be the subject of this talk.
Our main goal in this talk is to present residual-type a posteriori error estimates in the maximum norm for singularly perturbed semilinear reaction-diffusion equations posed in, possibly non-Lipschitz, polyhedral domains. We first consider shape-regular triangulations and describe recent joint results with Alan Demlow (Texas A&M) [1]. Then, our consideration is restricted to linear finite elements in two dimensions, but the focus shifts to more challenging anisotropic meshes, i.e. we allow mesh elements to have extremely high aspect ratios [2].
In the final part of the talk, we shall give a brief preview of the forthcoming presentation [3] at the international conference BAIL 2016, CSRC, Beijing, August 2016. In particular, some unexpected convergence properties on anisotropic meshes [4] will be reviewed.
Also, we shall discuss an extension of the approach of [2] to the estimation of the error in the energy norm [5], as well as to finite-element approximations of convection-dominated convection-diffusion equations, both on anisotropic meshes.
 
References:
[1] A. Demlow and N. Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems, Numer. Math., 133 (2016), 707-742
[2] N. Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, SIAM J. Numer. Anal., 53 (2015), 2519-2544
[3] N. Kopteva, A posteriori error estimates on anisotropic meshes, presentation at the international conference BAIL 2016, CSRC, Beijing, August 2016
[4] N. Kopteva, Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations, Math. Comp., 83 (2014), 2061-2070
[5] N. Kopteva, Energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, 2016

Prof. Natalia Kopteva received her Ph.D. from Moscow State University in 1997. She has worked in Moscow State University (Russia), National University of Ireland Cork (Ireland), Strathclyde University (UK), and University of Limerick (Ireland, current affiliation). Her research interests include numerical analysis for regular and singularly perturbed partial differential equations, a posteriori error estimation, error estimation on anisotropic meshes, numerical solution of fractional differential equations.