53 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
Enriched discrete spaces for time domain wave equations
The second order linear wave equation is simple in representation but its numerical
approximation is challenging, especially when the system contains waves of
high frequencies. While 10 grid points per wavelength is regarded as the rule of
thumb to achieve tolerable approximation with the standard numerical approach,
high resolution or high grid density is often required at high frequency which is often
computationally demanding.
As a contribution to tackling this problem, we consider in this thesis the discretization
of the problem in the framework of the space-time discontinuous Galerkin
(DG) method while investigating the solution in a finite dimensional space whose
building blocks are waves themselves. The motivation for this approach is to reduce
the number of degrees of freedom per wavelength as well as to introduce some
analytical features of the problem into its numerical approximation.
The developed space-time DG method is able to accommodate any polynomial
bases. However, the Trefftz based space-time method proves to be efficient even
for a system operating at high frequency. Comparison with polynomial spaces of
total degree shows that equivalent orders of convergence are obtainable with fewer
degrees of freedom. Moreover, the implementation of the Trefftz based method is
cheaper as integration is restricted to the space-time mesh skeleton.
We also extend our technique to a more complicated wave problem called the
telegraph equation or the damped wave equation. The construction of the Trefftz
space for this problem is not trivial. However, the
exibility of the DG method
enables us to use a special technique of propagating polynomial initial data using
a wave-like solution (analytical) formula which gives us the required wave-like local
solutions for the construction of the space.
This thesis contains important a priori analysis as well as the convergence analysis
for the developed space-time method, and extensive numerical experiments
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
Recommended from our members
Space-time Methods for Time-dependent Partial Differential Equations
Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space.
Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations
Plane Wave Discontinuous Galerkin methods for scattering by periodic structures
This thesis explores the application of Plane Wave Discontinuous Galerkin
(PWDG) methods for the numerical simulation of electromagnetic scattering by
periodic structures. Periodic structures play a pivotal role in various
engineering and scientific applications, including antenna design, metamaterial
characterization, and photonic crystal analysis. Understanding and accurately
predicting the scattering behavior of electromagnetic waves from such
structures is crucial in optimizing their performance and advancing
technological advancements.
The thesis commences with an overview of the theoretical foundations of
electromagnetic scattering by periodic structures. This theoretical
dissertation serves as the basis for formulating the PWDG method within the
context of wave equation. The DtN operator is presented and it is used to
derive a suitable boundary condition.
The numerical implementation of PWDG methods is discussed in detail,
emphasizing key aspects such as basis function selection and boundary
conditions. The algorithm's efficiency is assessed through numerical
experiments.
We then present the DtN-PWDG method, which is discussed in detail and is used
to derive numerical solutions of the scattering problem. A comparison with the
finite element method (FEM) is presented.
In conclusion, this thesis demonstrates that PWDG methods are a powerful tool
for simulating electromagnetic scattering by periodic structures
A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation
We introduce a space–time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one-dimensional scheme of Kretzschmar et al. (IMA J Numer Anal 36:1599–1635, 2016). Test and trial discrete functions are space–time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space–time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on “tent-pitched” meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds
A space-time discontinuous Galerkin method for coupled poroelasticity-elasticity problems
This work is concerned with the analysis of a space-time finite element
discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical
discretization of wave propagation in coupled poroelastic-elastic media. The
mathematical model consists of the low-frequency Biot's equations in the
poroelastic medium and the elastodynamics equation for the elastic one. To
realize the coupling, suitable transmission conditions on the interface between
the two domains are (weakly) embedded in the formulation. The proposed PolydG
discretization in space is then coupled with a dG time integration scheme,
resulting in a full space-time dG discretization. We present the stability
analysis for both the continuous and the semidiscrete formulations, and we
derive error estimates for the semidiscrete formulation in a suitable energy
norm. The method is applied to a wide set of numerical test cases to verify the
theoretical bounds. Examples of physical interest are also presented to
investigate the capability of the proposed method in relevant geophysical
scenarios
A discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equations
International audienceTrefftz methods are known to be very efficient to reduce the numerical pollution when associated to plane wave basis. However, these local basis functions are not adapted to the computation of evanescent modes or corner singularities. In this article, we consider a two dimensional time-harmonic Maxwell system and we propose a formulation which allows to design an electromagnetic Trefftz formulation associated to local Galerkin basis computed thanks to an auxiliary Nédélec finite element method. The results are illustrated with numerous numerical examples. The considered test cases reveal that the short range and long range propagation phenomena are both well taken into account
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