Discontinuous Galerkin methods through the lens of variational multiscale analysis

Stoter, Stein K. F.; Cockburn, Bernardo; Hughes, Thomas J.R.; Schillinger, Dominik

doi:10.1016/j.cma.2021.114220

Cited by 12 publications

(5 citation statements)

References 68 publications

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“…Additionally, the mapping P IP is idempotent, and is a linear and bounded operator on the space Ṽ × Q × Q. As a consequence, P IP is a projector, and we refer to it as the interior penalty projector [40,41]. The penalty parameter η penalizes mismatches of interface jumps.…”

Section: Projection Operatorsmentioning

confidence: 99%

Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

Eikelder¹,

Stoter²,

Bazilevs³

et al. 2023

Preprint

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One of the major challenges in finite element methods is the mitigation of spurious oscillations near sharp layers and discontinuities known as the Gibbs phenomenon. In this article, we propose a set of functionals to identify spurious oscillations in best approximation problems in finite element spaces. Subsequently, we adopt these functionals in the formulation of constraints in an effort to eliminate the Gibbs phenomenon. By enforcing these constraints in best approximation problems, we can entirely eliminate over-and undershoot in one dimensional continuous approximations, and significantly suppress them in one-and higher-dimensional discontinuous approximations.

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Section: Projection Operatorsmentioning

confidence: 99%

Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

Eikelder¹,

Stoter²,

Bazilevs³

et al. 2023

Preprint

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“…( 8). However, the particular structures of the coarse-scale problem and the Nitsche projector allow for a more direct inversion of (part of) the fine scales in the coarse-scale equation [33,34]. First, we recognize that the fine-scale terms in eq.…”

Section: Variational Multiscale Weak Formulationmentioning

confidence: 99%

“…Additionally, the fine-scale solution does, by design, not vanish on the Dirichlet boundary, which violates one of the key assumptions on which traditional residual-based fine-scale models are built. In previous work, we focused on discontinuous Galerkin methods, where the discontinuities between elements give rise to similar issues [32][33][34]. The goal of this article is to completely eliminate these issues for weak boundary imposition in Nitsche-type formulations.…”

Section: Introductionmentioning

confidence: 99%

Unification of variational multiscale analysis and Nitsche's method, and a resulting boundary layer fine-scale model

Stoter,

Eikelder,

de Prenter

et al. 2020

Preprint

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We show that in the variational multiscale framework, the weak enforcement of essential boundary conditions via Nitsche's method corresponds directly to a particular choice of projection operator. The consistency, symmetry and penalty terms of Nitsche's method all originate from the fine-scale closure dictated by the corresponding scale decomposition. As a result of this formalism, we are able to determine the exact fine-scale contributions in Nitsche-type formulations. In the context of the advection-diffusion equation, we develop a residual-based model that incorporates the non-vanishing fine scales at the Dirichlet boundaries. This results in an additional boundary term with a new model parameter. We then propose a parameter estimation strategy for all parameters involved that is also consistent for higher-order basis functions. We illustrate with numerical experiments that our new augmented model mitigates the overly diffusive behavior that the classical residual-based fine-scale model exhibits in boundary layers at boundaries with weakly enforced essential conditions.

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“…Its stability is induced by the numerical fluxes imposed on internal element interfaces, leading to discretely stable solutions with local conservation. dG is related to residual-based stabilization methods in [52][53][54] and is used for new stabilized methods based on VMS, including the Multiscale Discontinuous Galerkin method (MSDG) [55][56][57], the Discontinuous residual-free bubble method [58], among other recent approaches [59,60]. Alternative stabilization techniques that generalize dG ideas are the Interior Penalty methods that use continuous functions and can handle difficulties encountered by continuous finite element methods in advection-diffusion problems [61,62].…”

Section: Introductionmentioning

confidence: 99%

An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations

Giraldo

Calo

2023

MCA

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We construct a stabilized finite element method for linear and nonlinear unsteady advection–diffusion–reaction equations using the method of lines. We propose a residual minimization strategy that uses an ad-hoc modified discrete system that couples a time-marching schema and a semi-discrete discontinuous Galerkin formulation in space. This combination delivers a stable continuous solution and an on-the-fly error estimate that robustly guides adaptivity at every discrete time. We show the performance of advection-dominated problems to demonstrate stability in the solution and efficiency in the adaptivity strategy. We also present the method’s robustness in the nonlinear Bratu equation in two dimensions.

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Discontinuous Galerkin methods through the lens of variational multiscale analysis

Cited by 12 publications

References 68 publications

Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

Unification of variational multiscale analysis and Nitsche's method, and a resulting boundary layer fine-scale model

An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations

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