Beshley A. Numerical solution of planar boundary value problems for an elliptic equation with variable coefficients by integral equations approach

The thesis is devoted to the numerical solution of planar problems for a second-order elliptic equation with variable coefficients (EEVC). A brief overview 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 problems 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 Dirichlet and Neumann boundary value problems for EEVC has been developed. Using the concept of a parametrix and indirect integral 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 Nyström 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 boundary 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 approximate 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 Nyström scheme in combination with Tikhonov regularization and L-curve method for regularization 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 presented 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 domains and conductivities, using exact as well as noisy Cauchy data, showing that a stable solution can be obtained with good accuracy and small computational cost.