An effective numerical method for the determination of wave fields in unbounded domains (Q1385897)

The simulation of dynamic and acoustic processes in the ocean leads to the necessity of finding wave fields in domains that are not bounded in one or two dimensions. Mathematically, this problem is reduced to solving boundary value problems for elliptic equations in unbounded domains. This problem was considered by different authors. In these works, the original problem in an unbounded domain is reduced to an equivalent problem in a bounded domain by the introduction of an artificial boundary \(\Gamma_\infty\), on which an exact nonlocal boundary condition is set, which simulates the Sommerfeld radiation condition. The boundary problem for the Helmholtz equation, thus obtained, is discretized by the finite element method or finite-difference method; as a result, an algebraic system is obtained, which is ill conditioned and contains, depending on the size of the domain under consideration, \(10^6\) or more equations. Taking this into account, the basic problem in this approach is the solution of the indicated system of linear algebraic equations. In this work, an effective direct method is developed for solving the indicated system of algebraic equations and estimates are deduced, which make it possible to evaluate the influence of rounding-off errors on the accuracy of the result. The algorithm is characterized by a high reliability and stability with respect to rounding-off errors and is almost optimal (thrifty) with respect to the number of computer arithmetic operations for domains of the type of a long waveguide, whose transverse size is much less than its longitudinal size.