Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations
The mechanical quadrature method for solving the Helmholtz equation with nonlinear boundary conditions is presented. The advantage of this method is its high accuracy (of order \(\mathcal{O}(h^3)\)), its low computational complexity and the fact that no singular integration is needed. The authors prove stability and convergence for the resulting nonlinear system using the Stepleman theorem. The nonlinear system is solved with the Newton method, and the theory is based on the Ostrowski fixed point theorem. In addition, the accuracy is improved to \(\mathcal{O}(h^5)\) using the \(h^3\)-Richardson extrapolation algorithm. Two numerical examples are presented which are in agreement with the theoretical results.
scientific article; zbMATH DE number 6014569
scientific article