On the use of physical spline finite element method for acoustic scattering Gökhan Apaydin Applied Research and Development, University of Technology Zurich, Technoparkstrasse 1, 8005-CH Zurich, Switzerland article info Keywords: Acoustic scattering Cubic spline interpolation Error analysis Finite element method Least-squares finite element method Mixed Galerkin finite element method abstract This paper presents the comparison of physical spline finite element method (PSFEM), in which differential equations are incorporated into interpolations of basic elements, with least-squares finite element method (LSFEM) and mixed Galerkin finite element method (MGFEM) on the numerical solution of one dimensional Helmholtz equation applied to an acoustic scattering problem. Firstly, all three methods are explained in detail and then it is shown that PSFEM reaches higher precision in a shorter time with fewer nodes than the other methods. It is also observed that this method is well suited for high frequency acoustic problems. Consequently, the results of PSFEM point out better efficiency in terms of number of unknowns and accuracy level. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction The acoustic scattering problem generally requires the solution of Helmholtz equation which is solved by using either analytical or numerical methods. Analytical methods are, however, suitable for simple geometries. Hence, the numerical methods are required in most practical acoustic applications, which have received much attention over recent decades. The finite element method (FEM) is one of the popular numerical methods when studying the complex geometries and arbi- trary boundaries. Different variational formulas have been considered so as to develop FEM analysis. The Rayleigh–Ritz and the Galerkin weighted residual methods are the prevailing techniques which are based on the minimum energy of the functional and the integral of residual over the domain, respectively [1]. Ihlenburg and Babuska applied Galerkin FEM to Helmholtz equa- tion [2,3] and Bouillard studied the error estimation and adaptivity of FEM in acoustics [4,5]. While considering partition of unity method, the generalized FEM was used for high wave number Helmholtz equation [6]. Another approach to investigate FEM is the minimization of the residual in a least-squares sense, which is called the least- squares FEM (LSFEM) [7,8]. Although LSFEM has disadvantages for Helmholtz equation with high wave number, there is no need for stabilization and the matrix is positive symmetric definite. On the other hand, a particular Galerkin type weak for- mulation is the basis of mixed Galerkin FEM (MGFEM) based on the first order system of differential equations to approx- imate both primary and secondary variables while considering the unknown function with its derivative [9,10]. Better results were obtained by using MGFEM instead of LSFEM for acoustic scattering. It was found that the computation time by MGFEM is 20 times faster than the time by LSFEM when linear interpolation is used [11]. However; the linear Lagrange functions, which are generally used as basis functions for FEM, do not satisfy the smoothness requirement; because their second derivatives are zero. Therefore; higher order functions, which were considered and implemented with LSFEM and MGFEM, should be used in order to improve FEM analysis for acoustic scattering [12]. Since the derivatives of the unknown function at the nodes are still not continuous, the physical spline functions can be applied as a solution while providing smooth data [13,14]. Although they have some disadvantages, as being difficult to 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.10.054 E-mail address: gapaydin@hsz-t.ch Applied Mathematics and Computation 215 (2010) 3576–3588 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc