Comparison of two wave element methods for the Helmholtz problem

In comparison with low-order finite element methods (FEMs), the use of oscillatory basis functions has been shown to reduce the computational complexity associated with the numerical approximation of Helmholtz problems at high wave numbers. We compare two different wave element methods for the 2D Helmholtz problems. The methods chosen for this study are the partition of unity FEM (PUFEM) and the ultra-weak variational formulation (UWVF). In both methods, the local approximation of wave field is computed using a set of plane waves for constructing the basis functions. However, the methods are based on different variational formulations; the PUFEM basis also includes a polynomial component, whereas the UWVF basis consists purely of plane waves. As model problems we investigate propagating and evanescent wave modes in a duct with rigid walls and singular eigenmodes in an L-shaped domain. Results show a good performance of both methods for the modes in the duct, but only a satisfactory accuracy was obtained in the case of the singular field. On the other hand, both the methods can suffer from the ill-conditioning of the resulting matrix system.

[1]  I. Babuska,et al.  The design and analysis of the Generalized Finite Element Method , 2000 .

[2]  B. Després,et al.  SUR UNE FORMULATION VARIATIONNELLE DE TYPE ULTRA-FAIBLE , 1994 .

[3]  C. Farhat,et al.  The Discontinuous Enrichment Method , 2000 .

[4]  Jari P. Kaipio,et al.  The Ultra-Weak Variational Formulation for Elastic Wave Problems , 2004, SIAM J. Sci. Comput..

[5]  R. J. Astley,et al.  The partition of unity finite element method for short wave acoustic propagation on non‐uniform potential flows , 2006 .

[6]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[7]  R. J. Astley,et al.  Special short wave elements for flow acoustics , 2005 .

[8]  Francis Collino,et al.  Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series , 2004 .

[9]  Jari P. Kaipio,et al.  An ultra-weak method for acoustic fluid-solid interaction , 2008 .

[10]  R. J. Astley,et al.  Modelling of short wave diffraction problems using approximating systems of plane waves , 2002 .

[11]  Jon Trevelyan,et al.  Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed , 2005 .

[12]  Omar Laghrouche,et al.  Short wave modelling using special finite elements , 2000 .

[13]  J. Kaipio,et al.  Computational aspects of the ultra-weak variational formulation , 2002 .

[14]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[15]  Charbel Farhat,et al.  A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime , 2003 .

[16]  O. C. Zienkiewicz,et al.  Achievements and some unsolved problems of the finite element method , 2000 .

[17]  O. Cessenat,et al.  Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem , 1998 .

[18]  H. L. Atkins,et al.  Continued Development of the Discontinuous Galerkin Method for Computational Aeroacoustic Applications , 1997 .

[19]  Bruno Després,et al.  Using Plane Waves as Base Functions for Solving Time Harmonic Equations with the Ultra Weak Variational Formulation , 2003 .

[20]  P. Ortiz,et al.  An improved partition of unity finite element model for diffraction problems , 2001 .

[21]  Olivier Cessenat,et al.  Application d'une nouvelle formulation variationnelle aux équations d'ondes harmoniques : problèmes de Helmholtz 2D et de Maxwell 3D , 1996 .

[22]  Peter Monk,et al.  A least-squares method for the Helmholtz equation , 1999 .

[23]  Pablo Gamallo,et al.  The Partition of Unity Finite Element Method for Short Wave Acoustic Propagation on Nonuniform Potential Flows , 2004 .

[24]  P. Bettess,et al.  Plane wave basis finite-elements for wave scattering in three dimensions , 2002 .