A stabilized formulation for the advection – diffusion equation using the generalized finite element method.

*(English)*Zbl 1429.35178Summary: 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.

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.

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

76M10 | Finite element methods applied to problems in fluid mechanics |

##### Keywords:

advection; diffusion equation; generalized finite element method; eXtended finite element method; stabilized methods; partition of unity framework; non-polynomial enrichment functions##### Software:

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\textit{D. Z. Turner} et al., Int. J. Numer. Methods Fluids 66, No. 1, 64--81 (2011; Zbl 1429.35178)

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