Natalia Kopteva

Natalia Kopteva

An initial-boundary value problem with a Caputo time derivative of fractional order $α\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty$ and $L_2$ norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.

Natalia Kopteva, Martin Stynes

A semilinear reaction-diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter $\eps^2$, is considered. It can have multiple solutions. An asymptotic expansion is constructed for a solution that has an interior layer. Further properties are then established for a perturbation of this expansion. These are used in\cite{KoStMain} to obtain discrete sub-solutions and super-solutions for certain finite difference methods described there, and in this way yield convergence results for those methods.

Sebastian Franz, Natalia Kopteva

Linear singularly perturbed convection-diffusion problems with characteristic layers are considered in three dimensions. We demonstrate the sharpness of our recently obtained upper bounds for the associated Green's function and its derivatives in the $L_1$ norm. For this, in this paper we establish the corresponding lower bounds. Both upper and lower bounds explicitly show any dependence on the singular perturbation parameter.

Natalia Kopteva

For linear finite element discretizations of the Laplace equation in three dimensions, we give an example of a tetrahedral mesh in the cubic domain for which the logarithmic factor cannot be removed from the standard upper bounds on the error in the maximum norm.

Natalia Kopteva

In [Kopteva, Math. Comp., 2014] a counterexample of an anisotropic triangulation was given on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. This example was given in the context of a singularly perturbed reaction-diffusion equation. In this paper, we present further examples of unanticipated pointwise convergence behaviour of Lagrange finite elements on anisotropic triangulations. In particular, we show that linear finite elements may exhibit lower than expected orders of convergence for the Laplace equation, as well as for certain singular equations, and their accuracy depends not only on the linear interpolation error, but also on the mesh topology. Furthermore, we demonstrate that pointwise convergence rates which are worse than one might expect are also observed when higher-order finite elements are employed on anisotropic meshes. A theoretical justification will be given for some of the observed numerical phenomena.

Natalia Kopteva

Fully computable a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. To deal with the latter, we employ anisotropic quadrature and explicit anisotropic flux reconstruction. Prior to the flux equilibration, divergence-free corrections are introduced for pairs of anisotropic triangles sharing a short edge. We also give an upper bound for the resulting estimator, in which the error constants are independent of the diameters and the aspect ratios of mesh elements, and of the small perturbation parameter.