Overview of the discontinuous enrichment method, the ultra‐weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons

The efficient finite element discretization of the Helmholtz equation becomes challenging in the medium frequency regime because of numerical dispersion, or what is often referred to in the literature as the pollution effect. A number of FEMs with plane wave basis functions have been proposed to alleviate this effect, and improve on the unsatisfactory preasymptotic convergence of the polynomial FEM. These include the partition of unity method, the ultra‐weak variational formulation, and the discontinuous enrichment method. A previous comparative study of the performance of such methods focused on the first two aforementioned methods only. By contrast, this paper provides an overview of all three methods and compares several aspects of their performance for an acoustic scattering benchmark problem in the medium frequency regime. It is found that the discontinuous enrichment method outperforms both the partition of unity method and the ultra‐weak variational formulation by a significant margin. Copyright © 2011 John Wiley & Sons, Ltd.

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

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

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

[4]  F. Ihlenburg.  Finite Element Analysis of Acoustic Scattering , 1998 .

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

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

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

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

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

[10]  Jari P. Kaipio,et al.  The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation , 2004 .

[11]  Charbel Farhat,et al.  Higher‐order extensions of a discontinuous Galerkin method for mid‐frequency Helmholtz problems , 2004 .

[12]  Charbel Farhat,et al.  A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes , 2004 .

[13]  C. Farhat,et al.  The discontinuous enrichment method for elastic wave propagation in the medium‐frequency regime , 2006 .

[14]  Ivo Babuška,et al.  The generalized finite element method for Helmholtz equation: Theory, computation, and open problems , 2006 .

[15]  Charbel Farhat,et al.  Three‐dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid‐frequency Helmholtz problems , 2006 .

[16]  Pablo Gamallo,et al.  Comparison of two wave element methods for the Helmholtz problem , 2006 .

[17]  Charbel Farhat,et al.  A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems , 2008 .

[18]  Isaac Harari,et al.  Studies of the discontinuous enrichment method for two-dimensional acoustics , 2008 .

[19]  C. Farhat,et al.  A discontinuous enrichment method for three‐dimensional multiscale harmonic wave propagation problems in multi‐fluid and fluid–solid media , 2008 .

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

[21]  Ralf Hiptmair,et al.  PLANE WAVE DISCONTINUOUS GALERKIN METHODS: ANALYSIS OF THE h-VERSION ∗, ∗∗ , 2009 .

[22]  Charbel Farhat,et al.  A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers , 2009 .

[23]  Omar Laghrouche,et al.  Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method , 2009 .

[24]  Charbel Farhat,et al.  A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain , 2009 .

[25]  I. Harari,et al.  Three‐dimensional element configurations for the discontinuous enrichment method for acoustics , 2009 .

[26]  Charbel Farhat,et al.  A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes , 2010 .

[27]  A discontinuous enrichment method for the efficient solution of plate vibration problems in the medium‐frequency regime , 2010 .

[28]  Charbel Farhat,et al.  A discontinuous enrichment method for variable‐coefficient advection–diffusion at high Péclet number , 2011 .

[29]  G. Gabard,et al.  A comparison of wave‐based discontinuous Galerkin, ultra‐weak and least‐square methods for wave problems , 2011 .