54 research outputs found
Adaptive refinement for hp-version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem
In this article we develop an hp-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h-refinement) and local basis enrichment (p-refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples
A survey of Trefftz methods for the Helmholtz equation
Trefftz methods are finite element-type schemes whose test and trial
functions are (locally) solutions of the targeted differential equation. They
are particularly popular for time-harmonic wave problems, as their trial spaces
contain oscillating basis functions and may achieve better approximation
properties than classical piecewise-polynomial spaces.
We review the construction and properties of several Trefftz variational
formulations developed for the Helmholtz equation, including least squares,
discontinuous Galerkin, ultra weak variational formulation, variational theory
of complex rays and wave based methods. The most common discrete Trefftz spaces
used for this equation employ generalised harmonic polynomials (circular and
spherical waves), plane and evanescent waves, fundamental solutions and
multipoles as basis functions; we describe theoretical and computational
aspects of these spaces, focusing in particular on their approximation
properties.
One of the most promising, but not yet well developed, features of Trefftz
methods is the use of adaptivity in the choice of the propagation directions
for the basis functions. The main difficulties encountered in the
implementation are the assembly and the ill-conditioning of linear systems, we
briefly survey some strategies that have been proposed to cope with these
problems.Comment: 41 pages, 2 figures, to appear as a chapter in Springer Lecture Notes
in Computational Science and Engineering. Differences from v1: added a few
sentences in Sections 2.1, 2.2.2 and 2.3.1; inserted small correction
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
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
Recommended from our members
Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
A quasi-optimal coarse problem and an augmented Krylov solver for the Variational Theory of Complex Rays
The Variational Theory of Complex Rays (VTCR) is an indirect Trefftz method
designed to study systems governed by Helmholtz-like equations. It uses wave
functions to represent the solution inside elements, which reduces the
dispersion error compared to classical polynomial approaches but the resulting
system is prone to be ill conditioned. This paper gives a simple and original
presentation of the VTCR using the discontinuous Galerkin framework and it
traces back the ill-conditioning to the accumulation of eigenvalues near zero
for the formulation written in terms of wave amplitude. The core of this paper
presents an efficient solving strategy that overcomes this issue. The key
element is the construction of a search subspace where the condition number is
controlled at the cost of a limited decrease of attainable precision. An
augmented LSQR solver is then proposed to solve efficiently and accurately the
complete system. The approach is successfully applied to different examples.Comment: International Journal for Numerical Methods in Engineering, Wiley,
201
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
A space-time DG method for the Schr\"odinger equation with variable potential
We present a space--time ultra-weak discontinuous Galerkin discretization of
the linear Schr\"odinger equation with variable potential. The proposed method
is well-posed and quasi-optimal in mesh-dependent norms for very general
discrete spaces. Optimal~-convergence error estimates are derived for the
method when test and trial spaces are chosen either as piecewise polynomials,
or as a novel quasi-Trefftz polynomial space. The latter allows for a
substantial reduction of the number of degrees of freedom and admits
piecewise-smooth potentials. Several numerical experiments validate the
accuracy and advantages of the proposed method
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
- …