Analysis of a Screened Harmonic System under Dirichlet and Neumann Boundary Conditions

In this paper, a screened harmonic system with Dirichlet and Neumann boundary conditions in a domain with complex geometry is considered, and a method for analyzing such a system is proposed. The development of this method is especially relevant for solving boundary value problems for the screened Poisson equation in domains with complex geometry, which are used to describe various physical systems in mechanics, hydrodynamics, electrical engineering, and heat engineering. The proposed algorithm for analyzing a screened harmonic system under these boundary conditions makes a significant contribution to this area. The proposed method includes the continuation of the screened harmonic system through boundaries with Dirichlet and Neumann conditions. Then, the continuation is discretized by a system of linear algebraic equations. An asymptotically optimal analysis of the discrete continued screened harmonic system and an algorithm implementing the method for analyzing the screened harmonic system with optimal asymptotics in the number of arithmetic operations are carried out.