Information on the thesis

This thesis is devoted to the study of the application of hp-adaptive schemes of
the finite element method to singularly perturbed diffusion-convection-reaction
boundary value problems.
In the introduction the relevance of the thesis topic is substantiated, the
scientific novelty and practical value of the work are defined, the subject and the goal
of the research are formulated. Besides, the list of the applicant`s publications on the
thesis topic is given here.
Chapter 1 is devoted to introduction into modeling of diffusion-convection-
reaction phenomena. Derived formulation of diffusion-convection-reaction boundary
value problem is reformulated as a corresponding variational problem. The theorem
which provides sufficient conditions of derived problem’s well-posedness is proved.
The problem is analyzed in the terms of Péclet and Strouhal criteria. In the end of the
chapter, the overview of existing hp-adaptive schemes is provided.
Chapter 2 is dedicated to construction of a posteriori error estimates for high-
order approximations. Special explicit error estimator is and also two implicit
estimators are constructed. In addition the reference solution estimator is studied and
the possibility of its elementwise decomposition is proved. All constructed estimators
are used in the next chapter to construct corresponding adaptation strategy.
In Chapter 3 hp-adaptation schemes are considered. New strategy is
constructed, which is based on the comparison of different error estimates on finite
elements. Its correctness is proved for symmetric problems. Optimization of
numerical procedures of constructed algorithm is discussed. Special algorithms
constructed for solving the system of linear equations and generation of Gaussian
quadratures of arbitrary order.
Chapter 4 is the last and it is devoted to numerical solution of some test
singular perturbed problems using algorithm, constructed in chapter 3. The
comparison of this algorithm with algorithm based on reference solution is provided.
In addition different variations of algorithm produced by using different estimators
and adaptation criteria are compared. In the end comparison with h-adaptive scheme
is considered.
The results of the theoretical research have been confirmed by the series of
numerical experiments, which were conducted using developed software.