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Plane wave approximation in linear elasticity
No full textWe consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes
A space-time DG method for the Schr\"odinger equation with variable potential
Full text linkWe 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
A space-time Trefftz discontinuous Galerkin method for the linear Schr\"odinger equation
Full text linkA space-time Trefftz discontinuous Galerkin method for the Schr\"odinger
equation with piecewise-constant potential is proposed and analyzed. Following
the spirit of Trefftz methods, trial and test spaces are spanned by
non-polynomial complex wave functions that satisfy the Schro\"odinger equation
locally on each element of the space-time mesh. This allows for a significant
reduction in the number of degrees of freedom in comparison with full
polynomial spaces. We prove well-posedness and stability of the method, and,
for the one- and two- dimensional cases, optimal, high-order, h-convergence
error estimates in a skeleton norm. Some numerical experiments validate the
theoretical results presented
Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations
No full textStability results for the time-harmonic Maxwell equations with impedance boundary conditions
No full textPlane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version
No full textA space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation
Full text linkWe 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. (2016, IMA J. Numer.
Anal., 36, 1599-1635). 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 -convergence bounds.Comment: 34 page
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Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes
No full textWe 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
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Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version
No full textWe consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns
Space-time virtual elements for the heat equation
Full text linkWe propose and analyze a space-time virtual element method for the
discretization of the heat equation in a space-time cylinder, based on a
standard Petrov-Galerkin formulation. Local discrete functions are solutions to
a heat equation problem with polynomial data. Global virtual element spaces are
nonconforming in space, so that the analysis and the design of the method are
independent of the spatial dimension. The information between time slabs is
transmitted by means of upwind terms involving polynomial projections of the
discrete functions. We prove well posedness and optimal error estimates for the
scheme, and validate them with several numerical tests
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