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A stabilized formulation for the advection–diffusion equation using the Generalized Finite Element Method
Author(s) -
Turner D. Z.,
Nakshatrala K. B.,
Hjelmstad K. D.
Publication year - 2011
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.2248
Subject(s) - mathematics , finite element method , partition of unity , galerkin method , advection , partition (number theory) , stability (learning theory) , convection–diffusion equation , numerical analysis , work (physics) , exponential function , discontinuous galerkin method , diffusion , mathematical analysis , computer science , physics , combinatorics , machine learning , thermodynamics
This paper presents a stable formulation for the advection–diffusion equation based on the Generalized (or eXtended) Finite Element Method, GFEM (or X‐FEM). Using enrichment functions that represent the exponential character of the exact solution, smooth numerical solutions are obtained for problems with steep gradients and high Péclet numbers in one‐ and two‐dimensions. In contrast with traditional stabilized methods that require the construction of stability parameters and stabilization terms, the present work avoids numerical instabilities by improving the classical Galerkin solution with enrichment functions (that need not be polynomials) using GFEM, which is an instance of the partition of unity framework . This work also presents a strategy for constructing enrichment functions for problems involving complex geometries by employing a global–local‐type approach. Representative numerical results are presented to illustrate the performance of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.