94 research outputs found
Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems
We consider interior penalty discontinuous Galerkin discretizations of
time-harmonic wave propagation problems modeled by the Helmholtz equation, and
derive novel a priori and a posteriori estimates. Our analysis classically
relies on duality arguments of Aubin-Nitsche type, and its originality is that
it applies under minimal regularity assumptions. The estimates we obtain
directly generalize known results for conforming discretizations, namely that
the discrete solution is optimal in a suitable energy norm and that the error
can be explicitly controlled by a posteriori estimators, provided the mesh is
sufficiently fine
Frequency-explicit a posteriori error estimates for discontinuous Galerkin discretizations of Maxwell's equations
We propose a new residual-based a posteriori error estimator for
discontinuous Galerkin discretizations of time-harmonic Maxwell's equations in
first-order form. We establish that the estimator is reliable and efficient,
and the dependency of the reliability and efficiency constants on the frequency
is analyzed and discussed. The proposed estimates generalize similar results
previously obtained for the Helmholtz equation and conforming finite element
discretization of Maxwell's equations. In addition, for the discontinuous
Galerkin scheme considered here, we also show that the proposed estimator is
asymptotically constant-free for smooth solutions. We also present
two-dimensional numerical examples that highlight our key theoretical findings
and suggest that the proposed estimator is suited to drive - and
-adaptive iterative refinements.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0920
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
Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems
We consider interior penalty discontinuous Galerkin discretizations of timeharmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin-Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine
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Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
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
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