This work is devoted to the study of existing and development of new hp-adaptive finite element schemes for singularly
perturbed diffusion-convection-reaction boundary-value problems. A key feature of adaptive schemes of this type is the
use of polynomial approximations of arbitrary orders on finite elements along with the spatial refinement of the finite
element mesh. Compared to the classical h-adaptive schemes, this approach allows us to construct finite-dimensional
finite element spaces with better approximation properties.
In this work, a new adaptation scheme was developed, which is based on the comparison of the norms of various
approximations to the error on each finite element. Based on this comparison, a criteria has been constructed, which
provide proper refinement of finite element mesh for the next iteration of the algorithm. The approach is proved for one-
dimensional symmetric problems. The results of numerical experiments have shown that the constructed strategy can
also be applied to nonsymmetric problems.
In the introductory part of the paper, the relevance of the research topic is described, the purpose and list of the main
tasks are formulated, the scientific novelty and the practical significance of the results of the work are highlighted.
In the first section we consider fundamental physical relations, describing the processes of diffusion, convection and
reaction in a continuous medium. On their basis, a diffusion-convection-reaction model is formulated in the form of a
boundary-value problem. A set of available numerical and analytical methods for solving the obtained boundary value
problem is analyzed. An appropriate variational problem is constructed and the theorem on its well-posedness is proved.
A class of singularly perturbed problems is described, in the sense, when the convective and reactive components
considerably dominate the diffusion component. The classification of adaptation algorithms is described. Considered 4
types of schemes: h-, r-, p-, hp-adaptive. A detailed review and comparison of known hp-adaptive algorithms has been
provided.
The second section is devoted to the construction of a posteriori error estimators for approximations of arbitrary orders. A
special explicit estimator has been constructed, which directly takes into account the order of approximation on the finite
element. Its efficiency and reliability are proved. The variational problem of error is constructed. On its basis an implicit
estimator was obtained, based on fundamental solutions. Additionally, two more implicit Dirichlet estimators have been
constructed, which allow us to estimate an error in the case of finite element bisection or increasing of its order. The
problem of effective calculation of the estimator, based on the reference solution solution is investigated and the theorem
on its elementwise decomposition is proved.
In the third section a new hp-adaptative finite element scheme is developed, which is based on comparisons of the
different error approximations. The constructed scheme is theoretically substantiated for symmetric problems. The
universal algorithm, based on the reference solution and its effective implementation is considered. Gaussian quadrature
formulas of high orders are used to calculate the component of the global system of linear equations of the finite element
method. To construct the latter, it is proposed to combine the Golub-Welsh algorithm and the QL-algorithm with
Wilkinson's shifts. To solve the system of linear equations in both hp-adaptation algorithms, the use of the method of
static condensation of internal parameters has been proposed and proved.
The fourth section contains a description of the developed software, implementing the algorithms, as well as the results
of computational experiments and their analysis for singularly perturbed diffusion-convection-reaction boundary-value
problems. The comparative analysis of the developed adaptation algorithm and algorithm, based on the reference
solution, as well as the developed algorithm with a typical h-adaptive scheme is considered. The comparison of different
estimators and adaptation criteria is given. The results of numerical experiments confirm the efficiency of the constructed
schemes.
Among the results and methods obtained in the thesis, the following should be noted:
1. program implementation and theoretical analysis of an adaptive algorithm based on the reference solution;
2. proof of elementwise decomposition of the estimator based on reference solution;
3. construction of explicit and implicit a posteriori error estimators for high-order approximations;
4. construction of the hp-adaptation strategy based on the comparison of the norms of different approximations to
local errors;
5. conducting and analysis of numerical experiments of application of constructed algorithms to singularly
perturbed problems.
