239 research outputs found
A Space-Time Discontinuous Galerkin Trefftz Method for time dependent Maxwell's equations
We consider the discretization of electromagnetic wave propagation problems
by a discontinuous Galerkin Method based on Trefftz polynomials. This method
fits into an abstract framework for space-time discontinuous Galerkin methods
for which we can prove consistency, stability, and energy dissipation without
the need to completely specify the approximation spaces in detail. Any method
of such a general form results in an implicit time-stepping scheme with some
basic stability properties. For the local approximation on each space-time
element, we then consider Trefftz polynomials, i.e., the subspace of
polynomials that satisfy Maxwell's equations exactly on the respective element.
We present an explicit construction of a basis for the local Trefftz spaces in
two and three dimensions and summarize some of their basic properties. Using
local properties of the Trefftz polynomials, we can establish the
well-posedness of the resulting discontinuous Galerkin Trefftz method.
Consistency, stability, and energy dissipation then follow immediately from the
results about the abstract framework. The method proposed in this paper
therefore shares many of the advantages of more standard discontinuous Galerkin
methods, while at the same time, it yields a substantial reduction in the
number of degrees of freedom and the cost for assembling. These benefits and
the spectral convergence of the scheme are demonstrated in numerical tests
Discontinuous Galerkin Methods with Trefftz Approximation
We present a novel Discontinuous Galerkin Finite Element Method for wave
propagation problems. The method employs space-time Trefftz-type basis
functions that satisfy the underlying partial differential equations and the
respective interface boundary conditions exactly in an element-wise fashion.
The basis functions can be of arbitrary high order, and we demonstrate spectral
convergence in the \Lebesgue_2-norm. In this context, spectral convergence is
obtained with respect to the approximation error in the entire space-time
domain of interest, i.e. in space and time simultaneously. Formulating the
approximation in terms of a space-time Trefftz basis makes high order time
integration an inherent property of the method and clearly sets it apart from
methods, that employ a high order approximation in space only.Comment: 14 pages, 12 figures, preprint submitted at J Comput Phy
Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation
We describe and analyse a space-time Trefftz discontinuous Galerkin method
for the wave equation. The method is defined for unstructured meshes whose
internal faces need not be aligned to the space-time axes. We show that the
scheme is well-posed and dissipative, and we prove a priori error bounds for
general Trefftz discrete spaces. A concrete discretisation can be obtained
using piecewise polynomials that satisfy the wave equation elementwise.Comment: 8 pages, submitted to the XXIV CEDYA / XIV CMA conference, Cadiz 8-12
June 201
The FLAME-slab method for electromagnetic wave scattering in aperiodic slabs
The proposed numerical method, "FLAME-slab," solves electromagnetic wave
scattering problems for aperiodic slab structures by exploiting short-range
regularities in these structures. The computational procedure involves special
difference schemes with high accuracy even on coarse grids. These schemes are
based on Trefftz approximations, utilizing functions that locally satisfy the
governing differential equations, as is done in the Flexible Local
Approximation Method (FLAME). Radiation boundary conditions are implemented via
Fourier expansions in the air surrounding the slab. When applied to ensembles
of slab structures with identical short-range features, such as amorphous or
quasicrystalline lattices, the method is significantly more efficient, both in
runtime and in memory consumption, than traditional approaches. This efficiency
is due to the fact that the Trefftz functions need to be computed only once for
the whole ensemble.Comment: Various typos were corrected. Minor inconsistencies throughout the
manuscript were fixed. In Section II B. Additional description regarding
choice of Trefftz cell, was added. In Section III A. Detailed description
about units (used in our calculation) was adde
Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods
We present a numerical study to investigate the conditioning of the plane
wave discontinuous Galerkin discretization of the Helmholtz problem. We provide
empirical evidence that the spectral condition number of the plane wave basis
on a single element depends algebraically on the mesh size and the wave number,
and exponentially on the number of plane wave directions; we also test its
dependence on the element shape. We show that the conditioning of the global
system can be improved by orthogonalization of the local basis functions with
the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES
iterations for solving the discrete problem iteratively.Comment: Submitted as a conference proceeding; minor revisio
Trefftz Difference Schemes on Irregular Stencils
The recently developed Flexible Local Approximation MEthod (FLAME) produces
accurate difference schemes by replacing the usual Taylor expansion with
Trefftz functions -- local solutions of the underlying differential equation.
This paper advances and casts in a general form a significant modification of
FLAME proposed recently by Pinheiro & Webb: a least-squares fit instead of the
exact match of the approximate solution at the stencil nodes. As a consequence
of that, FLAME schemes can now be generated on irregular stencils with the
number of nodes substantially greater than the number of approximating
functions. The accuracy of the method is preserved but its robustness is
improved. For demonstration, the paper presents a number of numerical examples
in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering
of electromagnetic (acoustic) waves, and wave propagation in a photonic
crystal. The examples explore the role of the grid and stencil size, of the
number of approximating functions, and of the irregularity of the stencils.Comment: 28 pages, 12 figures; to be published in J Comp Phy
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