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Title:
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Stabilized Petrov-Galerkin methods for the convection-diffusion-reaction and the Helmholtz equations
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Author:
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Nadukandi, Prashanth
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Other authors:
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Agència de Gestió d'Ajuts Universitaris i de Recerca; Centro Internacional de Métodos Numéricos en Ingeniería |
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Resum:
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We present two new stabilized high-resolution numerical methods for the convection–diffusion–reaction (CDR) and the
Helmholtz equations respectively. The work embarks upon a priori analysis of some consistency recovery procedures for
some stabilization methods belonging to the Petrov–Galerkin framework. It was found that the use of some standard
practices (e.g. M-Matrices theory) for the design of essentially non-oscillatory numerical methods is not feasible when
consistency recovery methods are employed. Hence, with respect to convective stabilization, such recovery methods are
not preferred. Next, we present the design of a high-resolution Petrov–Galerkin (HRPG) method for the 1D CDR problem.
The problem is studied from a fresh point of view, including practical implications on the formulation of the maximum
principle, M-Matrices theory, monotonicity and total variation diminishing (TVD) finite volume schemes. The current
method is next in line to earlier methods that may be viewed as an upwinding plus a discontinuity-capturing operator.
Finally, some remarks are made on the extension of the HRPG method to multidimensions. Next, we present a new
numerical scheme for the Helmholtz equation resulting in quasi-exact solutions. The focus is on the approximation of the
solution to the Helmholtz equation in the interior of the domain using compact stencils. Piecewise linear/bilinear
polynomial interpolation are considered on a structured mesh/grid. The only a priori requirement is to provide a mesh/grid
resolution of at least eight elements per wavelength. No stabilization parameters are involved in the definition of the
scheme. The scheme consists of taking the average of the equation stencils obtained by the standard Galerkin finite
element method and the classical finite difference method. Dispersion analysis in 1D and 2D illustrate the quasi-exact
properties of this scheme. Finally, some remarks are made on the extension of the scheme to unstructured meshes by
designing a method within the Petrov–Galerkin framework. |
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Publication date:
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2010-11-30 |
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Subject (UDC):
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51 - Matemàtiques |
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Subject(s):
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Galerkin, Mètodes de
Elements finits, Mètode dels |
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Rights:
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Aquest document està subjecte a una llicència d'ús de Creative Commons, amb la qual es permet copiar, distribuir i comunicar públicament l'obra sempre que se'n citin l'autor original i l’Agència i no se'n faci cap ús comercial ni obra derivada, tal com queda estipulat en la llicència d'ús (http://creativecommons.org/licenses/by-nc-nd/2.5/es/) |
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Pages:
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8 p. |
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Document type:
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Report |
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Collection:
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Els ajuts de l'AGAUR;2006FI00836
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