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A stabilized formulation for the advection – diffusion equation using the generalized finite element method. (English) Zbl 1429.35178
Summary: 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.

##### 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
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