93 research outputs found
A Plane Wave Virtual Element Method for the Helmholtz Problem
We introduce and analyze a virtual element method (VEM) for the Helmholtz
problem with approximating spaces made of products of low order VEM functions
and plane waves. We restrict ourselves to the 2D Helmholtz equation with
impedance boundary conditions on the whole domain boundary. The main
ingredients of the plane wave VEM scheme are: i) a low frequency space made of
VEM functions, whose basis functions are not explicitly computed in the element
interiors; ii) a proper local projection operator onto the high-frequency
space, made of plane waves; iii) an approximate stabilization term. A
convergence result for the h-version of the method is proved, and numerical
results testing its performance on general polygonal meshes are presented
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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Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes
We extend the a priori error analysis of Trefftz-discontinuous Galerkin methods for time-harmonic wave propagation problems developed in previous papers to acoustic scattering problems and locally refined meshes. To this aim, we prove refined regularity and stability results with explicit dependence of the stability constant on the wave number for non convex domains with non connected boundaries. Moreover, we devise a new choice of numerical flux parameters for which we can prove L2-error estimates in the case of locally refined meshes near the scatterer. This is the setting needed to develop a complete hp-convergence analysis
An entropy structure preserving space-time Galerkin method for cross-diffusion systems
Cross-diffusion systems are systems of nonlinear parabolic partial
differential equations that are used to describe dynamical processes in several
application, including chemical concentrations and cell biology. We present a
space-time approach to the proof of existence of bounded weak solutions of
cross-diffusion systems, making use of the system entropy to examine long-term
behavior and to show that the solution is nonnegative, even when a maximum
principle is not available. This approach naturally gives rise to a novel
space-time Galerkin method for the numerical approximation of cross-diffusion
systems that conserves their entropy structure. We prove existence and
convergence of the discrete solutions, and present numerical results for the
porous medium, the Fisher-KPP, and the Maxwell-Stefan problem.Comment: 33 page
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Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations
In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived
Design and performance of a space-time virtual element method for the heat equation on prismatic meshes
We present a space-time virtual element method for the discretization of the
heat equation, which is defined on general prismatic meshes and variable
degrees of accuracy. Strategies to handle efficiently the space-time mesh
structure are discussed. We perform convergence tests for the - and
-versions of the method in case of smooth and singular solutions, and test
space-time adaptive mesh refinements driven by a residual-type error indicator
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