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

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

doi:10.1002/fld.2248

Cited by 20 publications

(6 citation statements)

References 43 publications

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“…Remark 3. Due to the well-known instabilities associated with a high ratio of advection to diffusion in the transport model [32,33], some diffusion was necessary to include for numerical reasons, even though the primary quantities of interest are related to pure advection. Also a constant scalar diffusivity was chosen to avoid instabilities associated with violations of the discrete maximumminimum principle [34,35,36].…”

Section: Representative Numerical Resultsmentioning

confidence: 99%

On the performance of the variational multiscale formulation for subsurface flow and transport in heterogeneous porous media

Turner¹,

Nakshatrala²,

Notz³

2010

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The following work compares two popular mixed finite elements used to model subsurface flow and transport in heterogeneous porous media; the lowest order Raviart-Thomas element and the variational multiscale stabilized element. Comparison is made based on performance for several problems of engineering relevance that involve highly heterogenous material properties (permeability ratios of up to 1×10 5 ), open flow boundary conditions (pressure driven flows), and large scale domains in two dimensions. Numerical experiments are performed to show the degree to which mass conservation is violated when a flow field computed using either element is used as the advection velocity in a transport model. The results reveal that the variational multiscale element shows considerable mass production or loss for problems that involve flow tangential to layers of differing permeability, but marginal violation of local mass balance for problems of less orthogonality in the permeability. The results are useful in establishing rudimentary estimates of the error produced by using the variational mutliscale element for several different types of problems.

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Section: Representative Numerical Resultsmentioning

confidence: 99%

On the performance of the variational multiscale formulation for subsurface flow and transport in heterogeneous porous media

Turner¹,

Nakshatrala²,

Notz³

2010

Preprint

Self Cite

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“…The most natural way to build an enrichment function is to take something that looks similar to an exact solution of a problem similar to the one we are trying to solve. Of course, this idea is not new and is already used in and even in the DEM .…”

Section: Numerical Scheme: An Enriched Stabilized Finite Element Methodsmentioning

confidence: 99%

“…The adaptive variational multiscale method is a try to avoid this drawback by a local, numerical resolution of the small scales but is designed for elliptic problems. In , the authors also propose an exponential based enrichment in the variational multiscale framework, but it is only used to improve the stability of the scheme. They also propose an adaptive algorithm where the solution at a low Péclet number is used to enrich the space at a higher Péclet.…”

Section: Introductionmentioning

confidence: 99%

An adaptive enrichment algorithm for advection‐dominated problems

Abgrall

Krust

2012

Numerical Methods in Fluids

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SUMMARY We are interested in developing a numerical framework well suited for advection–diffusion problems when the advection part is dominant. In that case, given Dirichlet type boundary condition, it is well known that a boundary layer develops. To resolve correctly this layer, standard methods consist in increasing the mesh resolution and possibly increasing the formal accuracy of the numerical method. In this paper, we follow another path: we do not seek to increase the formal accuracy of the scheme but, by a careful choice of finite element, to lower the mesh resolution in the layer. Indeed the finite element representation we choose is locally the sum of a standard one plus an enrichment. This paper proposes such a method and with several numerical examples, we show the potential of this approach. In particular, we show that the method is not very sensitive to the choice of the enrichment and develop an adaptive algorithm to automatically choose the enrichment functions.Copyright © 2012 John Wiley & Sons, Ltd.

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“…More recently, in Farhat et al [4], an approach consisting on a higher-order discontinuous enrichment method was proposed for the finite element solution on unstructured meshes of the 384 M. RAMIREZ AND M. A. MORELES advection-diffusion equation in the high Péclet number regime. Also in Turner et al [5], a generalized (extended) finite element formulation for the advection-diffusion equation is presented. Using enrichment functions that represent the exponential nature of the exact solution, smooth numerical solutions are obtained for problems with steep gradients and high Péclet numbers in one and two-dimensions.…”

Section: Introductionmentioning

confidence: 99%

On the Finite Increment Calculus method for stabilizing advection‐diffusion equations, analysis and computation of the stabilization parameter

Ramírez

Moreles

2011

Numerical Methods in Fluids

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SUMMARY In this work we are concerned with the finite increment calculus (FIC) method. The method has been developed for efficient approximation of advection‐diffusion equations with high Péclet numbers. Since the natural application of FIC is within the framework of the FEM, we consider the BVP in a weak sense on finite dimensional spaces. Here we provide a result on existence and uniqueness of the solution as well as an error analysis. Also we propose a choice of the stabilization parameter. We test the method on some troublesome 2D problems. Copyright © 2011 John Wiley & Sons, Ltd.

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

Cited by 20 publications

References 43 publications

On the performance of the variational multiscale formulation for subsurface flow and transport in heterogeneous porous media

On the performance of the variational multiscale formulation for subsurface flow and transport in heterogeneous porous media

An adaptive enrichment algorithm for advection‐dominated problems

On the Finite Increment Calculus method for stabilizing advection‐diffusion equations, analysis and computation of the stabilization parameter

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