Information on the thesis

The thesis is devoted to the numerical solution of planar problems for a
second-order elliptic equation with variable coefficients (EEVC). A brief over-
view of its applications in different areas together with existing approaches
for numerical solving have been provided.
There have been considered Dirichlet and Neumann boundary value prob-
lems in a bounded simply connected domain, mixed boundary value problems
and Cauchy problem in a bounded doubly connected domain in current work.
A numerical approximation involving integral equations technique for the
solution of the Dirihlet and Neumann boundary value problems for EEVC
has been developed. Using the concept of a parametrix and indirect inte-
gral equations approach that represents the solution as a sum of potentials,
the problems are reduced to a system of boundary-domain integral equations
(BDIEs) to be solved for two unknown densities.
Via a change of variables based on shrinkage of the boundary curve of the
solution domain a parameterized system of BDIEs is obtained. The strong
and logarithmic singularities in kernels have been examined. It is shown
how to write these singularities in the system in an explicit form for further
discretization. An effective full discretization by the Nystr¨om method is given.
Solving the system of linear algebraic equations, the approximate values
of unknown densities over boundary and domain are calculated. The formulas
of the approximate numerical solution in the domain are provided for both
boundary value problems. The numerical experiments for different input data
and domains are showing that the proposed approach can be turned into a
practical working method.
As mixed boundary value problems, Dirichlet-Neumann and Neumann-
Dirichlet boundary value problems in a doubly connected domain have been
considered. Similarly to the Dirichlet and Neumann problems, a solution is
represented as a sum of single layer potentials over the domain and over two
boundary curves with unknown densities and Levi function (parametrix) as a
kernel. Making the change of variables based on shrinkage of the outer bound-
ary curve, the system of integral equations is rewritten in the parameterized
form. Using the same steps including singularities exploring and rewriting
them explicitly, quadratures application with collocation at specific points,
solving the system of linear equations to get densities values, the approxi-
mate solution in the domain is obtained.
Separately, the numerical solution of the mixed boundary value problem
for an arbitrary doubly connected domain is examined, where the change of
variables in the system of BDIEs happens via the parametric representation
of inner and outer boundaries.
For the numerical solution of the ill-posed Cauchy problem an indirect
integral equations method with Tikhonov regularization and two iterative
methods (alternating method and Landweber method) are considered.
For the integral based method for numerical solving the Cauchy problem,
the solution is represented as a sum of parametrix-potentials with unknown
densities to be identified. The densities are calculated from the system of
BDIEs for the numerical solution of which an efficient Nystr¨om scheme incombination with Tikhonov regularization and L-curve method for regulariza-
tion parameter choosing is proposed. Having approximate values of densities
it is possible to find approximate Cauchy data on the inner boundary using
appropriate formulas based on the view of solution representation.
A numerical implementation of the alternating iterative method is pre-
sented for the Cauchy problem. On each step of the iterative procedure, two
well-posed mixed problems investigated in the thesis are being solved. The
convergence analysis of this method is also provided. An iterative Landweber
method is considered at each iteration step of which two Dirichlet-Neumann
problems are being solved.
Numerical results are presented for all three approaches, for different do-
mains and conductivities, using exact as well as noisy Cauchy data, showing
that a stable solution can be obtained with good accuracy and small compu-
tational cost.
Key words: elliptic equation with variable coefficients, parametrix (Levi
function), planar boundary value problems, Cauchy problem, indirect integral
equations approach, boundary-domain integral equation, domain parameteri-
zation, Nystr¨om method.