A comparison of two Trefftz-type methods: the ultraweak variational formulation and the least-squares method, for solving shortwave 2-D Helmholtz problems

Gamallo, P. and Astley, R.J. (2007) A comparison of two Trefftz-type methods: the ultraweak variational formulation and the least-squares method, for solving shortwave 2-D Helmholtz problems. International Journal for Numerical Methods in Engineering, 71 (4), 406-432. (doi:10.1002/nme.1948).

Abstract

Trefftz methods for the numerical solution of partial differential equations (PDEs) on a given domain involve trial functions which are defined in subdomains, are generally discontinuous, and are solutions of the governing PDE (or its adjoint) within each subdomain. The boundary conditions and matching conditions between subdomains must be enforced separately. An interesting novel result presented in this paper is that the least-squares method (LSM) and the ultraweak variational formulation, two methods already established for solving the Helmholtz equation, can be derived in the framework of the Trefftz-type methods. In the first case, the boundary conditions and interelement continuity are enforced by means of a least-squares procedure. In the second, a Galerkin-type weighted residual method is used. Another goal of the work is to assess the relative efficiency of each method for solving shortwave problems in acoustics and to study the stability of each method. The numerical performance of each scheme is assessed with reference to two 2-D test problems; acoustic propagation in an uniform soft-walled duct, and propagation in an L-shaped domain, the latter involving singular behaviour at a sharp corner.

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