Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

Ayuso, Blanca; Marini, L. D.

doi:10.1137/080719583

Cited by 136 publications

(181 citation statements)

References 30 publications

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“…A,τ ) 1/2 , where the advection norm~·~A ,τ is scaled as usual with O( /2 ), e.g., see [17]. We thus obtain sharp convergence rates which have the right scaling with respect to ε, i.e., the relevant terms for advection dominated problems (ε 1) all scale as O(ε −1/2 ).…”

Section: A Priori Error Estimates For Macro Spatial Errormentioning

confidence: 81%

“…We thus obtain sharp convergence rates which have the right scaling with respect to ε, i.e., the relevant terms for advection dominated problems (ε 1) all scale as O(ε −1/2 ). The control of the error in the spatial L 2 norm in the advection dominated regime is obtained by introducing a suitable weighting function, following the ideas of [17,40]. However, the construction used in [17,40] cannot easily be extended to periodic boundary conditions and adapting it introduces the additional error term on Ω c τ .…”

Section: A Priori Error Estimates For Macro Spatial Errormentioning

confidence: 99%

“…A peculiarity of our results is, that the L 2 error at the boundary of the macro domain cannot be controlled robustly with respect to ε in the advection dominated regime. We emphasize that the same technical difficulty occurs when extending the singlescale analysis of DG-FEM given in [17,40] for Dirichlet and Neumann boundary conditions to advection-diffusion problems with periodic boundary conditions. This issue is thus not directly related to our multiscale approach.…”

Section: Introductionmentioning

confidence: 99%

“…We first mention that the main convergence result of this article is a convergence analysis for problems on rectangular spatial domains with periodically oscillating data and periodic boundary conditions, which is the setting for which rigorous homogenization is available and thus studying the convergence of the numerical solution towards an effective solution makes sense. To obtain robust convergence rates of the spatial L 2 norm in the advection dominated regime, the use of a "weighting function" which monotonically decreases along the flow has proved to be an essential tool, see [17,40] (and [7] for a multiscale context). Such a tool is however not available on periodic domains.…”

Section: Introductionmentioning

confidence: 99%

“…Such a tool is however not available on periodic domains. As a remedy, we periodize the weighting function such that the analysis of [17,40] remains valid up to a small boundary layer. A peculiarity of our results is, that the L 2 error at the boundary of the macro domain cannot be controlled robustly with respect to ε in the advection dominated regime.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Abdulle

Huber

2017

Numer. Math.

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We introduce a numerical homogenization method based on a discontinuous Galerkin finite element heterogeneous multiscale method (DG-HMM) to efficiently approximate the effective solution of parabolic advection-diffusion problems with rapidly varying coefficients, large Péclet number and compressible flows. To estimate the missing data of an effective model, numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion. For compressible flow with periodic data, we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime. Numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non-periodic data.

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Section: A Priori Error Estimates For Macro Spatial Errormentioning

confidence: 81%

Section: A Priori Error Estimates For Macro Spatial Errormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Abdulle

Huber

2017

Numer. Math.

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Exponentially fitted discontinuous Galerkin schemes for singularly perturbed problems

Lombardi

Pietra

2011

Numerical Methods Partial

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New discontinuous Galerkin schemes in mixed form are introduced for symmetric elliptic problems of second order. They exhibit reduced connectivity with respect to the standard ones. The modifications in the choice of the approximation spaces and in the stabilization term do not spoil the error estimates. These methods are then used for designing new exponentially fitted schemes for advection dominated equations. The presented numerical tests show the good performances of the proposed schemes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011

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A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

Deuring

2016

Numerical Methods Partial

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We consider a time‐dependent and a steady linear convection‐diffusion‐reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin–Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element–finite volume method: the diffusion term is discretized by Crouzeix–Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. The L 2 ( H 1 ) ‐ and the L ∞ ( L 2 ) ‐error in the unsteady case and the H1‐error in the steady one are estimated against the data, in such a way that no parameter enters exponentially into the constants involved. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1591–1621, 2016

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Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

Cited by 136 publications

References 30 publications

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Exponentially fitted discontinuous Galerkin schemes for singularly perturbed problems

A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

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