638 research outputs found
A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods
We present here a domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by a discontinuous Galerkin method. In order to allow the treatment of irregularly shaped geometries, the discontinuous Galerkin method is formulated on unstructured tetrahedral meshes. The domain decomposition strategy takes the form of a Schwarz-type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains. A multifrontal sparse direct solver is used at the subdomain level. The resulting domain decomposition strategy can be viewed as a hybrid iterative/direct solution method for the large, sparse and complex coefficients algebraic system resulting from the discretization of the time-harmonic Maxwell equations by a discontinuous Galerkin method
Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell equations
We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell equations using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach does not lead at convergence of the Schwarz method to the mono-domain DG discretization, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present an alternative discretization of the transmission conditions in the framework of a DG weak formulation, and prove that for this discretization the multidomain and mono-domain solutions for the Maxwell's equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media
Optimized Schwarz method for solving time-harmonic Maxwell's equations discretized by a discontinuous Galerkin method
4 pages.International audienceThe numerical solution of the three-dimensional time-harmonic Maxwell equations using high order methods such as discontinuous Galerkin formulations require efficient solvers. A domain decomposition strategy is introduced for this purpose. This strategy is based on optimized Schwarz methods applied to the first order form of the Maxwell system and leads to the best possible convergence of these algorithms. The principles are explained for a 2D model problem and numerical simulations confirm the predicted theoretical behavior. The efficiency is further demonstrated on more realistic 3D geometries including a bioelectromagnetism application
A Coupled Hybridizable Discontinuous Galerkin and Boundary Integral Method for Analyzing Electromagnetic Scattering
A coupled hybridizable discontinuous Galerkin (HDG) and boundary integral
(BI) method is proposed to efficiently analyze electromagnetic scattering from
inhomogeneous/composite objects. The coupling between the HDG and the BI
equations is realized using the numerical flux operating on the equivalent
current and the global unknown of the HDG. This approach yields sparse coupling
matrices upon discretization. Inclusion of the BI equation ensures that the
only error in enforcing the radiation conditions is the discretization.
However, the discretization of this equation yields a dense matrix, which
prohibits the use of a direct matrix solver on the overall coupled system as
often done with traditional HDG schemes. To overcome this bottleneck, a
"hybrid" method is developed. This method uses an iterative scheme to solve the
overall coupled system but within the matrix-vector multiplication subroutine
of the iterations, the inverse of the HDG matrix is efficiently accounted for
using a sparse direct matrix solver. The same subroutine also uses the
multilevel fast multipole algorithm to accelerate the multiplication of the
guess vector with the dense BI matrix. The numerical results demonstrate the
accuracy, the efficiency, and the applicability of the proposed HDG-BI solver
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
A nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures
In this paper, we develop a nested hybridizable discontinuous Galerkin (HDG)
method to numerically solve the Maxwell's equations coupled with the
hydrodynamic model for the conduction-band electrons in metals. By means of a
static condensation to eliminate the degrees of freedom of the approximate
solution defined in the elements, the HDG method yields a linear system in
terms of the degrees of freedom of the approximate trace defined on the element
boundaries. Furthermore, we propose to reorder these degrees of freedom so that
the linear system accommodates a second static condensation to eliminate a
large portion of the degrees of freedom of the approximate trace, thereby
yielding a much smaller linear system. For the particular metallic structures
considered in this paper, the resulting linear system obtained by means of
nested static condensations is a block tridiagonal system, which can be solved
efficiently. We apply the nested HDG method to compute the second harmonic
generation (SHG) on a triangular coaxial periodic nanogap structure. This
nonlinear optics phenomenon features rapid field variations and extreme
boundary-layer structures that span multiple length scales. Numerical results
show that the ability to identify structures which exhibit resonances at
and is paramount to excite the second harmonic response.Comment: 31 pages, 7 figure
Computational Electromagnetism and Acoustics
It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems
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