A discontinuous enrichment method for the finite element solution of high Péclet advection–diffusion problems

Kalashnikova, Irina; Farhat, Charbel; Tezaur, Radek

doi:10.1016/j.finel.2008.10.009

Cited by 33 publications

(36 citation statements)

References 21 publications

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“…Since the enrichment field contains free-space solutions of the underlying equation to be solved, it may entirely capture the homogeneous solutions rather than merely enhance the polynomial field. This observation motivates the construction of enrichment-only DGM elements in which the contribution of the standard polynomial field is dropped from the approximation entirely, resulting in improved computational efficiency without any loss of accuracy [17,29].…”

Section: Approximation Space V Hmentioning

confidence: 98%

“…DEM [16,17,29] seeks an approximate solution (u h , λ h ) ∈ (V h , W h ) of the variational problem (9). The discussion of the space of Lagrange multiplier approximations W h is held off until Section 4.…”

Section: Approximation Space V Hmentioning

confidence: 99%

“…The approximation space of the genuine DEM element R − 5 − 1 + contains five exponential enrichment functions and the polynomial field of the standard Galerkin bilinear quadrilateral element Q 1 . Both elements were shown in [29] to outperform standard Galerkin and stabilized Galerkin finite elements of comparable complexity and comparable order of convergence by a large margin. For non-trivial benchmark problems, they were shown to deliver numerical solutions with relative errors that were at least two, and in some cases many, orders of magnitude lower than those associated with the standard Galerkin solutions.…”

Section: Introductionmentioning

confidence: 98%

“…It was subsequently extended in [37,34,30] to the more general context of elastic wave propagation in fluid, solid, and fluid-solid multi-media. Discontinuous enrichment methods for the finite element solution of the one-dimensional (1D) advection-diffusion equation, two-dimensional (2D) advection-diffusion equation, and the Stokes equation were also formulated in [16,29,32]. DEM has shown tremendous potential for solving boundary value problems (BVPs) where the solutions are characterized by rapid oscillations or large gradients.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, the focus is kept on the 2D case, and the discontinuous enrichment method presented in [29] is extended to higher-order elements and unstructured meshes. This extension features a general formulation of the Lagrange multiplier approximation that is applicable to any straight-edge element.…”

Section: Introductionmentioning

confidence: 99%

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A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

Farhat

Kalashnikova

Tezaur

2009

Numerical Meth Engineering

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SUMMARYA higher-order discontinuous enrichment method (DEM) with Lagrange multipliers is proposed for the efficient finite element solution on unstructured meshes of the advection-diffusion equation in the high Péclet number regime. Following the basic DEM methodology, the usual Galerkin polynomial approximation is enriched with free-space solutions of the governing homogeneous partial differential equation (PDE). In this case, these are exponential functions that exhibit a steep gradient in a specific flow direction. Exponential Lagrange multipliers are introduced at the element interfaces to weakly enforce the continuity of the solution. The construction of several higher-order DEM elements fitting this paradigm is discussed in details. Numerical tests performed for several two-dimensional benchmark problems demonstrate their computational superiority over stabilized Galerkin counterparts, especially for high Péclet numbers.

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Section: Approximation Space V Hmentioning

confidence: 98%

Section: Approximation Space V Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

Farhat

Kalashnikova

Tezaur

2009

Numerical Meth Engineering

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

Turner

Nakshatrala

Hjelmstad

2011

Numerical Methods in Fluids

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SUMMARYThis 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.

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Numerical multiscale methods

Coutinho

Franca

Valentin

2011

Numerical Methods in Fluids

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SUMMARY We restrict the variational multiscale method to a class of methods we denote by numerical multiscale methods. Numerical multiscale methods are methods obtained by enriching the piecewise linear functions with special local functions. The enrichment provides additional stabilization via terms obtained by static condensation. The resulting methods are improvements of the coarse scale solutions by the approximations of the fine scales emanating from the enrichments.Copyright © 2011 John Wiley & Sons, Ltd.

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A discontinuous enrichment method for the finite element solution of high Péclet advection–diffusion problems

Cited by 33 publications

References 21 publications

A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

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

Numerical multiscale methods

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