Elliptic Problems With Essential Border Singularities: Improved Solution And Qualitative Error Estimation.

Abstract

In the last twenty years two demanding problems were concurrently studied by the
chemical engineering and mathematical researchers: the modelling of chemical reactors and
other similar devices and the processes inside them and the numerical (and analytical) solutions
of the sets of differential algebraic equations that these models provide them. In this paper a
complex subproblem of those just described is considered: the appearance of singularities in
the domain or the border of the modelled process device in the case of PDE models.
The idea of this work is to isolate the possible singularities by means of simplifying the phenomenological
models as far as possible and assessing the errors in the numerical solutions
using classical methods and, in particular, the mixed mesh qualitative error estimation method
proposed by the authors. The main idea of this mixed or composite mesh method is to construct
a numerical model where two or more finite element meshes of different granularities are
superimposed over the whole domain of the problem.
The mentioned simplifications produce several 1D and 2D differential equation problems
with singularities in the domain and in its border that are solved and their errors studied.
The main conclusion of this presentation is the following: for the models the authors deal
with it is possible to numerically detect the singularities and to devise computationally cheaper
methods for their solution.