A stabilized formulation for the advection–diffusion equation using the Generalized Finite Element Method |
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Authors: | D. Z. Turner K. B. Nakshatrala K. D. Hjelmstad |
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Affiliation: | 1. Computational Thermal and Fluid Mechanics Department, Sandia National Laboratories, Mail Stop 0836, PO Box 5800, Albuquerque 87112, New Mexico;2. Department of Mechanical Engineering, 216 Engineering and Physics Building, Texas A&M University, College Station, TX 77843, U.S.A.;3. College of Technology and Innovation, 7231 E. Sonoran Arroyo Mall, Arizona State University, Mesa, AZ 85212, U.S.A. |
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Abstract: | 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. |
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Keywords: | advection– diffusion equation Generalized Finite Element Method eXtended finite element method stabilized methods partition of unity framework non‐polynomial enrichment functions |
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