A Discontinuous Petrov--Galerkin Method with Lagrangian Multipliers for Second Order Elliptic Problems

Causin, Paola; Sacco, Riccardo

doi:10.1137/s0036142903427871

Cited by 27 publications

(30 citation statements)

References 28 publications

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“…We refer to [6] and [12] for a presentation of the DPG method and its convergence and stability analysis in the case of diffusion problems.…”

Section: Dpg Formulation Of the Advection-diffusion Problemmentioning

confidence: 99%

“…Remark 2.1. The finite element space adopted in this work for the approximation of the hybrid variable λ is not the same as the one used in previous presentations and analyses of the DPG method [6,[10][11][12]. As a matter of fact, in these references the space Λ h was chosen to be the set of piecewise constant (single-valued) functions over E h (suitably modified to account for Dirichlet boundary conditions).…”

Section: Dpg Formulation Of the Advection-diffusion Problemmentioning

confidence: 99%

“…As a matter of fact, in these references the space Λ h was chosen to be the set of piecewise constant (single-valued) functions over E h (suitably modified to account for Dirichlet boundary conditions). The analysis in [12] for the case of the DPG method applied to the solution of a diffusive boundary value problem shows that λ h is actually the trace of a function belonging to P CR 1 (T h ). Thus, from the point of view of the formulation for a pure diffusive problem nothing changes with each of the two choices.…”

Section: Dpg Formulation Of the Advection-diffusion Problemmentioning

confidence: 99%

“…is the usual (note, not the harmonic) average of ε on K. It is well known that in the presence of rough (or strongly varying) coefficients, the use of harmonic averaging provides superior accuracy and stability than standard averaging (see [4] for the 1D case, and [9,12,24] for applications in 2D).…”

Section: The Single-field Problem Associated With the Dpg Formulationmentioning

confidence: 99%

“…The method here presented is based on the Discontinuous Petrov-Galerkin (DPG) formulation, discussed and analyzed in references [6,7,[10][11][12]. The DPG method is a dual-primal mixed-hybrid formulation that, after static condensation, reduces to a nonconforming single-field method.…”

Section: Introductionmentioning

confidence: 99%

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Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems

2005

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Abstract. In this work we consider the dual-primal Discontinuous Petrov-Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis is developed, proving first-order accuracy of the method in a discrete H 1 -norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.Mathematics Subject Classification. 65N99.

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“…We refer to [6] and [12] for a presentation of the DPG method and its convergence and stability analysis in the case of diffusion problems.…”

Section: Dpg Formulation Of the Advection-diffusion Problemmentioning

confidence: 99%

Section: Dpg Formulation Of the Advection-diffusion Problemmentioning

confidence: 99%

Section: Dpg Formulation Of the Advection-diffusion Problemmentioning

confidence: 99%

Section: The Single-field Problem Associated With the Dpg Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems

2005

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Discontinuous P etrov– G alerkin ( DPG ) Method

Demkowicz

Gopalakrishnan

2017

Encyclopedia of Computational Mechanics Second Edition

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This chapter reviews the fundamentals of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. The main idea admits three different interpretations: a Petrov–Galerkin method with (optimal) test functions that realize the supremum in the inf–sup condition; a minimum‐residual method with residual measured in a dual norm; and a mixed formulation where one solves simultaneously for the Riesz representation of the residual. The methodology can be applied to any well‐posed variational problem, but it is especially effective in context of discontinuous (broken) test spaces. We discuss how one can “break” test functions in any variational formulation, and use the convection‐dominated diffusion model problem to illustrate challenges related to the choice of an appropriate test norm.

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The ocular mathematical virtual simulator: A validated multiscale model for hemodynamics and biomechanics in the human eye

Sala,

Prud'homme,

Guidoboni

et al. 2023

Numer Methods Biomed Eng

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We present our continuous efforts from a modeling and numerical viewpoint to develop a powerful and flexible mathematical and computational framework called Ocular Mathematical Virtual Simulator (OMVS). The OMVS aims to solve problems arising in biomechanics and hemodynamics within the human eye. We discuss our contribution towards improving the reliability and reproducibility of computational studies by performing a thorough validation of the numerical predictions against experimental data. The OMVS proved capable of simulating complex multiphysics and multiscale scenarios motivated by the study of glaucoma. Furthermore, its modular design allows the continuous integration of new models and methods as the research moves forward, and supports the utilization of the OMVS as a promising non‐invasive clinical investigation tool for personalized research in ophthalmology.

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A Discontinuous Petrov--Galerkin Method with Lagrangian Multipliers for Second Order Elliptic Problems

Cited by 27 publications

References 28 publications

Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems

Flux-upwind stabilization of the discontinuous Petrov–Galerkin formulation with Lagrange multipliers for advection-diffusion problems

Discontinuous P etrov– G alerkin ( DPG ) Method

The ocular mathematical virtual simulator: A validated multiscale model for hemodynamics and biomechanics in the human eye

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