Information on the thesis

The thesis is devoted to development and investigation of projection-mesh schemes
to solve the problems with diffusion-advection-reaction equations, in particular singularly
perturbed and/or semilinear ones.
The basis of the proposed numerical schemes is formed by both Galerkin procedure,
which permits selection of finite element method (FEM) approximation space by using, in
particular, a posteriori error estimations and h-adaptivity concept, and one step recurrent
schemes  for  integration  with  respect  to  time.  The  discretized  systems  of  equations,  if
necessary,  are  previously  linearized  by  Newton  method,  and  then  solved  by  the
generalized minimal residual (GMRES) method that is restarted and preconditioned. To
compute  guaranteed-accuracy  FEM  approximations  the  developed  software  is
supplemented by linear, bilinear and serendipity quadratic basic functions and by proposed
a  posteriori  error  estimators  and  strategies  to  control  the  local  refinement  of  nested
triangulations  that  are  generated  by  the  bisection  method.  The  quality,  efficiency  and
robustness of the numerical schemes are proved and represented in details by the analysis
of numerical results over a lot of the test and model problems, in particular nucleation and
evolution  of  spiral  waves  in  the  reaction  of  carbon  monoxide  oxidation  on  platinum
surface.
Among the results and methods that are gained in thesis, we draw attention to  the
following ones:
(і) economical, element-wise and residual-type Dirichlet and Neumann a posteriori error
estimators, that are able to calculate the lower and upper bounds of FEM approximations
error, theoretical and numerical analyses of their efficiency and robustness properties;
(іі) the efficient h-adaptive schemes that are supplemented by double sided error control
for linear and semilinear problems that calculate the convergent sequences of piecewise
linear  FEM  approximations  on  locally  refined  triangulations  that  are  generated  by  the
bisection method;
(ііі) economical one-step recurrent schemes for an integration of Cauchy problems for
semidiscrete systems of equations, where the error orders of both the time discretization
and the linearization are balanced.
Key  words:  diffusion-advection-reaction  problem,  singularly  perturbation,
nonlinearity, finite element method, one-step recurrent scheme, linearization, a posteriori
error estimator, two-sided error estimation, h-adaptive finite element scheme, criterion of
adaptivity, bisection method, GMRES, preconditioning.