Electrical field calculation is particularly important when designing electrical apparatus. This is even more pronounced when designing the insulation of the transformer because the insulation must be able to withstand the various electrical stresses it is subjected during the transformer operation. Therefore, it is very important to use modelling techniques to analyze important device model features during the design phase and before the production phase. Results of the computation of electrical characteristics should be obtained as quickly as possible, with the greatest possible precision. The dissertation deals with the development of a methodology for an advanced and fast electrostatic transformer winding computation based on the boundary element method. The two-dimensional computation using linear isoparametric elements is developed. The computation is based on known geometrical parameters and materials of the apparatus. The computation is made for each turn of the winding. Axially symmetric turns are approximated using infinite conductors in the Cartesian coordinate system. This kind of approach is proved to be satisfactorily accurate. Due to the complexity of the computation, and thus the requirements on memory resources and computational costs, the methods for computation acceleration and matrix compression are studied. In order to accelerate the computation and compress the system matrix, the adaptive cross approximation is implemented and presented in the dissertation. Examples of transformer models are defined, and electrostatic computation is employed using the developed methodology and another numerical method. The analysis of the results has shown the satisfactory accuracy of the presented methodology for the computation of the electric field and the transformer winding capacitance matrix.