• Corpus ID: 207852328

# A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients

@article{Lee2019AHD,
title={A hybridized discontinuous Galerkin method for Poisson-type problems with sign-changing coefficients},
author={Jeonghun J. Lee and Sander Rhebergen},
journal={ArXiv},
year={2019},
volume={abs/1911.01984}
}
• Published 5 November 2019
• Mathematics, Computer Science
• ArXiv
In this paper, we present a hybridized discontinuous Galerkin (HDG) method for Poisson-type problems with sign-changing coefficients. We introduce a sign-changing stabilization parameter that results in a stable HDG method independent of domain geometry and the ratio of the negative and positive coefficients. Since the Poisson-type problem with sign-changing coefficients is not elliptic, standard techniques with a duality argument to analyze the HDG method cannot be applied. Hence, we present a…
2 Citations

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## References

SHOWING 1-10 OF 22 REFERENCES

### A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number

• Computer Science
Comput. Methods Appl. Math.
• 2016
It is proved that the HDG method is absolutely stable for all wave numbers κ > 0 ${\kappa>0}$ in the sense that no mesh constraint is required for the stability.

### T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients

• Mathematics
Numerische Mathematik
• 2013
It is proved that one can use the T-coercivity theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma.

### A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

• Mathematics
Math. Comput.
• 2008
An LDG-hybridizable Galerkin method for second-order elliptic problems in several space dimensions with remarkable convergence properties is identified and thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.

### Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems

• Computer Science
SIAM J. Numer. Anal.
• 2009
A unifying framework for hybridization of finite element methods for second order elliptic problems is introduced, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.

### Error Analysis for a Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation

• Computer Science
J. Sci. Comput.
• 2011
New error estimates are derived for a hybridizable discontinuous Galerkin scheme applied to the Helmholtz equation and the condition number of the condensed hybridized system is seen to be almost independent of the wave number.

### An analysis of HDG methods for the Helmholtz equation

• Mathematics, Computer Science
• 2014
It is proved that the proposed HDG methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions are stable (hence well-posed) without any mesh constraint.

### Multigrid for an HDG method

• Computer Science
IMA Journal of Numerical Analysis
• 2013
It is proved that a non-nested multigrid V-cycle, with a single smoothing step per level, converges at a mesh independent rate, when the prolongation norm is greater than one.

### A projection-based error analysis of HDG methods for Timoshenko beams

• Mathematics
Math. Comput.
• 2010
This paper gives the first a priori error analysis of the hybridizable discontinuous Galerkin (HDG) methods for Timoshenko beams and shows that the HDG methods converge with optimal order k + 1 for all the unknowns, and that they are free from shear locking.

### T-Coercivity for the Maxwell Problem with Sign-Changing Coefficients

• Mathematics
• 2014
In this paper, we study the time-harmonic Maxwell problem with sign-changing permittivity and/or permeability, set in a domain of ℝ3. We prove, using the T-coercivity approach, that the

### A Superconvergent HDG Method for the Maxwell Equations

• Mathematics, Computer Science
J. Sci. Comput.
• 2017
From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing.