scholarly journals Estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the Helmholtz equation

2011 ◽  
Vol 86 (10) ◽  
pp. 1197-1224 ◽  
Author(s):  
Lindaura Maria Steffens ◽  
Núria Parés ◽  
Pedro Díez
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Wave Number ◽  
Dispersion Error ◽  
Stabilized Finite Element ◽  
Finite Element Approximations ◽  
Numerical Wave

scholarly journals On the Adaptive Selection of the Parameter in Stabilized Finite Element Approximations

2013 ◽  
Vol 51 (3) ◽  
pp. 1585-1609 ◽  
Author(s):  
Mark Ainsworth ◽  
Alejandro Allendes ◽  
Gabriel R. Barrenechea ◽  
Richard Rankin
Keyword(s):  
Finite Element ◽  
Adaptive Selection ◽  
Stabilized Finite Element ◽  
Finite Element Approximations ◽  
Selection Of

A Residual Based Variational Method for Reducing Dispersion Error in Finite Element Methods

2005 ◽  
Author(s):  
Lonny L. Thompson ◽  
Prapot Kunthong
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Least Squares ◽  
High Order ◽  
High Order Accuracy ◽  
Quadrature Rules ◽  
Source Terms ◽  
Order Accuracy ◽  
Dispersion Error ◽  
Variational Framework

A difficulty of the standard Galerkin finite element method has been the ability to accurately resolve oscillating wave solutions at higher frequencies. Many alternative methods have been developed including high-order methods, stabilized Galerkin methods, multi-scale variational methods, and other wave-based discretization methods. In this work, consistent residuals, both in the form of least-squares and gradient least-squares are linearly combined and added to the Galerkin variational Helmholtz equation to form a new generalized Galerkin least-squares method (GGLS). By allowing the stabilization parameters to vary spatially within each element, we are able to select optimal parameters which reduce dispersion error for all wave directions from second-order to fourth-order in nondimensional wavenumber; a substantial improvement over standard Galerkin elements. Furthermore, the stabilization parameters are frequency independent, and thus can be used for both time-harmonic solutions to the Helmholtz equation as well as direct time-integration of the wave equation, and eigenfrequency/eigenmodes analysis. Since the variational framework preserves consistency, high-order accuracy is maintained in the presence of source terms. In the case of homogeneous source terms, we show that our consistent variational framework is equivalent to integrating the underlying stiffness and mass matrices with optimally selected numerical quadrature rules. Optimal GGLS stabilization parameters and equivalent quadrature rules are determined for several element types including: bilinear quadrilateral, linear triangle, and linear tetrahedral elements. Numerical examples on unstructured meshes validate the expected high-order accuracy.

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