On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations

Codina, Ramon; Badia, Santiago

doi:10.1016/j.cma.2013.05.004

Cited by 14 publications

(23 citation statements)

References 27 publications

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“…where = e k h , mh = x m , and C and k are constants. Substituting (26) into (25), and solving for gives repeated roots = e 2 h , meaning that k = 2, and the general solution to (25) is…”

Section: Solution Of Discrete Problem With I(2)mentioning

confidence: 99%

“…The system of equation in the reformulated problem is similar to Stokes flow, which may be discretized and solved by a variety of now classical [21][22][23] or many more advanced methods. [11][12][13]18,[24][25][26] The D-ECE formulation is a special case of a general class of error in constitutive equation (ECE) methods used for parameter estimation problems. [27][28][29] Most of the other ECE methods proposed in literature result in nonquadratic optimization problems, whose stationary conditions require solution of nonlinear systems of PDEs.…”

Section: Introductionmentioning

confidence: 99%

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Direct error in constitutive equation formulation for inverse heat conduction problem

Babaniyi

Oberai

Barbone

2018

Numerical Meth Engineering

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Summary We present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.

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“…where = e k h , mh = x m , and C and k are constants. Substituting (26) into (25), and solving for gives repeated roots = e 2 h , meaning that k = 2, and the general solution to (25) is…”

Section: Solution Of Discrete Problem With I(2)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Direct error in constitutive equation formulation for inverse heat conduction problem

Babaniyi

Oberai

Barbone

2018

Numerical Meth Engineering

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show abstract

“…29. Ω bounded is said to have the local Lipschitz property if at each point x on the boundary of Ω there is a neighborhood U x such that ∂Ω ∩ U x is the graph of a Lipschitz continuous function.…”

Section: B4 Sobolev Spacesmentioning

confidence: 99%

“…Find the velocity field u = (u(x, z), w(x, z)) : Ω d → R 2 and the pressure field p : Ω d → R, both periodic in x, satisfying Navier-Stokes equations in Ω d : 29) along with the continuity equation for incompressible fluids…”

Section: Numerical Comparison Addressing a Sinusoidal Texture Casementioning

confidence: 99%

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Modelação matemática de contatos lubrificados micro-texturizados

Palma¹

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Abstract multiscale-hybrid-mixed methods

Madureira

2014

Calcolo

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In an abstract setting, we investigate the basic ideas behind the Multiscale Hybrid Mixed (MHM) method, a Domain Decomposition scheme designed to solve multiscale partial differential equations (PDEs) in parallel. As originally proposed, the MHM method starting point is a primal hybrid formulation, which is then manipulated to result in an efficient method that is based on local independent PDEs and a global problem that is posed on the skeleton of the finite element mesh. Recasting the MHM method in a more general framework, we investigate some conditions that yield a well-posed method. We apply the general ideas to different formulations, and, in particular, come up with an interesting and fruitful connection between the Multiscale Finite Element Method and a dual hybrid method. Finally, we propose a method that combines the main ideas of the Discontinuous Enrichment Method and the MHM method. 1. THE ORIGINAL PROBLEM Multiscale problems are, by definition, difficult to solve computationally, and numerical methods with domain decomposition flavor are appealing, since they woo parallel implementations. Some important ideas have been around for a while, for instance in the seminal paper [20]. More recent attempts to solve multiscale PDEs using domain decomposition or hybrid formulations include [2, 3, 15, 19]. Several theoretical aspects and applications involving hybrid methods appear, for instance, in [4, 11-13, 26], and references therein. An interesting proposal by Harder, Paredes and Valentin [23] came out also recently. Aiming to solve heterogeneous Darcy equations using a primal hybrid method, the authors came up with an efficient formulation that is well-posed and easy to implement in parallel. They christened the scheme Multiscale Hybrid Mixed, or MHM for short.

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On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations

Cited by 14 publications

References 27 publications

Direct error in constitutive equation formulation for inverse heat conduction problem

Direct error in constitutive equation formulation for inverse heat conduction problem

Modelação matemática de contatos lubrificados micro-texturizados

Abstract multiscale-hybrid-mixed methods

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