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== Abstract == | == Abstract == | ||
| − | A new Petrov–Galerkin (PG) method involving two parameters, namely | + | A new Petrov–Galerkin (PG) method involving two parameters, namely <math>\alpha_1</math> and <math>\alpha_2</math>, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is, <math>\alpha_1=\alpha_2=\alpha</math>; and (ii) the nonstandard compact stencil presented in (''Int. J. Numer. Meth. Engng'' 2011; 86:18–46) for the Helmholtz equation if the parameters are distinct, that is, <math>\alpha_1\ne\alpha_2</math>. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by <math>\alpha_1</math>) and the mass terms (specified by <math>\alpha_2</math>) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (''Int. J. Numer. Meth. Engng'' 2011; 86:18–46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions. |
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| + | ==Full document== | ||
| + | <pdf>Media:Nadukandi_et_al_2012b_5328_IJNME_2012.pdf</pdf> | ||
Latest revision as of 20:37, 5 October 2019
Abstract
A new Petrov–Galerkin (PG) method involving two parameters, namely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha_1
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha_2
, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha_1=\alpha_2=\alpha
- and (ii) the nonstandard compact stencil presented in (Int. J. Numer. Meth. Engng 2011; 86
- 18–46) for the Helmholtz equation if the parameters are distinct, that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha_1\ne\alpha_2
. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha_1 ) and the mass terms (specified by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha_2 ) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (Int. J. Numer. Meth. Engng 2011; 86:18–46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions.
