The fast parametric integral equations system in an acceleration of solving polygonal potential boundary value problems Abstract   The paper presents a new way of accelerating of polygonal potential boundary value problems solving by modified parametric integral equations system (PIES). For this purpose, known from the literature, the fast multipole method (FMM) was included into conventional PIES. Therefore the fast PIES method was obtained. FMM combined with conventional PIES also allows a significant reduction of utilization of random access memory (RAM) of the computer. It gives us a possibility of solving the largescale engineering problems on a standard personal computer (PC). Finally, the paper presents the development of an algorithm for efficient solving of complex boundary value problems on a standard PC instead of supercomputers, which are usually used to accelerate calculations. The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions in a reasonable time on the PC, which was the main goal of our research.

Virtual element method for semilinear sineGordon equation over polygonal mesh using product approximation technique Abstract   In this paper, we employ the linear virtual element spaces to discretize the semilinear sinexe2x80x93Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crankxe2x80x93Nicolson scheme is used. The resulting nonlinear equations are solved using the Newtonxe2x80x99s method. We derive a priori error estimations in      L    2      and      H    1      norms. The convergence properties and the accuracy of the virtual element method for the solution of the sinexe2x80x93Gordon equation are demonstrated with academic numerical experiments.