A high-order accurate Green-Taylor element method for the numerical solution of Helmholtz’s equation with variable wavenumber

In this work, we propose a novel Green-Taylor element method (GTEM) for solving the Helmholtz equation with variable wavenumber. The method combines a boundary integral form of the problem, derived from Green’s second theorem, with a discrete element approach to incorporate the effect of domain heterogeneity. A key feature of GTEM is the treatment of the domain integrals using Taylor series expansions from the element boundaries to approximate the interior variation of the solution, thus avoiding internal variables and enabling high-order approximations without the need of additional unknowns. The method is here particularized to the one-dimensional case, studying its accuracy and convergence properties. Different tests and practical applications of the Helmholtz equation for heterogeneous media are used to demonstrate the superior accuracy of GTEM compared to other classical numerical techniques such as hp-finite elements, spectral elements, or isogeometric analysis with b-splines.