The modified method of fundamental solutions for exterior problems of the Helmholtz equation; spurious eigenvalues and their removals

For the exterior Dirichlet problems (EDP) of the Helmholtz equation in 2D, when the Sommerfeld radiation condition is satisfied, there always exists the unique solution. Denote the unbounded domain S-infinity outside of a bounded simply-connected domain, where the interior boundary Gamma(in) is non-circular. The standard MFS is first studied by using the complex Hankel function H-0((1))(kr) as the fundamental solutions (FS). However, some solutions can not be obtained correctly (e.g., the spurious eigenvalues called). This paper is devoted to explore the spurious eigenvalues and their removals. For the non-circular interior boundary Gamma(in), the error analysis is briefly made for the standard MFS, and some guidance is provided to bypass the spurious eigenvalues. Denote two nodes: P = (rho, theta) and Q = (R, phi), and r = vertical bar(PQ) over bar vertical bar = root R-2 + rho(2) - 2R rho cos(theta - phi)). To completely eliminate all spurious eigenvalues, we solicit the new combined FS: (partial derivative/partial derivative R +/- ik)H-0((1))(kr) with i = root-1 and propose the new modified MFS. The new algorithms are simple, and the strict analysis may be made. The bounds of errors are derived, and the polynomial convergence rates can be achieved. The bounds of condition numbers are derived for circular Gamma(o)(in) only, to display the exponential growth via the number of the new FS used. Numerical experiments are carried to support the analysis made. This paper is the first time for the MFS to explore the spurious eigenvalues and their removals, accompanied with strict analysis of errors and stability. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.