A stabilized formulation for the advection-diffusion equation using the Generalized Finite Element Method

D. Z. Turner, K. B. Nakshatrala, Keith Hjelmstad

Research output: Contribution to journalArticle

13 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)64-81
Number of pages18
JournalInternational Journal for Numerical Methods in Fluids
Volume66
Issue number1
DOIs
StatePublished - May 10 2011

Fingerprint

Generalized Finite Element Method
Advection-diffusion Equation
Advection
Finite element method
Formulation
Stabilized Methods
Extended Finite Element Method
Partition of Unity
Numerical Instability
Complex Geometry
Galerkin
One Dimension
Two Dimensions
Stabilization
Exact Solution
Polynomials
Numerical Solution
Gradient
Numerical Results
Polynomial

Keywords

  • Advection-diffusion equation
  • EXtended finite element method
  • Generalized Finite Element Method
  • Non-polynomial enrichment functions
  • Partition of unity framework
  • Stabilized methods

ASJC Scopus subject areas

  • Computer Science Applications
  • Computational Mechanics
  • Applied Mathematics
  • Mechanical Engineering
  • Mechanics of Materials

Cite this

A stabilized formulation for the advection-diffusion equation using the Generalized Finite Element Method. / Turner, D. Z.; Nakshatrala, K. B.; Hjelmstad, Keith.

In: International Journal for Numerical Methods in Fluids, Vol. 66, No. 1, 10.05.2011, p. 64-81.

Research output: Contribution to journalArticle

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