59 research outputs found
Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation
We describe and analyse a space-time Trefftz discontinuous Galerkin method
for the wave equation. The method is defined for unstructured meshes whose
internal faces need not be aligned to the space-time axes. We show that the
scheme is well-posed and dissipative, and we prove a priori error bounds for
general Trefftz discrete spaces. A concrete discretisation can be obtained
using piecewise polynomials that satisfy the wave equation elementwise.Comment: 8 pages, submitted to the XXIV CEDYA / XIV CMA conference, Cadiz 8-12
June 201
Plane wave approximation in linear elasticity
We 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
Density results for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets
We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin
spaces on rough sets. Our main results, stated in the simplest Sobolev space
setting, are that: (i) for an open set ,
is dense in whenever has zero Lebesgue
measure and is "thick" (in the sense of Triebel); and (ii) for a
-set (), is dense in whenever for some . For (ii), we provide
concrete examples, for any , where density fails when
and are on opposite sides of . The results (i) and (ii)
are related in a number of ways, including via their connection to the question
of whether for a
given closed set and . They also
both arise naturally in the study of boundary integral equation formulations of
acoustic wave scattering by fractal screens. We additionally provide analogous
results in the more general setting of Besov and Triebel--Lizorkin spaces.Comment: 38 pages, 6 figure
Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
A new, coercive formulation of the Helmholtz equation was introduced in
[Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate -version
Galerkin discretisations of this formulation, and the iterative solution of the
resulting linear systems. We find that the coercive formulation behaves
similarly to the standard formulation in terms of the pollution effect (i.e. to
maintain accuracy as , must decrease with at the same rate
as for the standard formulation). We prove -explicit bounds on the number of
GMRES iterations required to solve the linear system of the new formulation
when it is preconditioned with a prescribed symmetric positive-definite matrix.
Even though the number of iterations grows with , these are the first such
rigorous bounds on the number of GMRES iterations for a preconditioned
formulation of the Helmholtz equation, where the preconditioner is a symmetric
positive-definite matrix.Comment: 27 pages, 7 figure
On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space
This paper concerns the following question: given a subset E of Rn with empty interior and an integrability parameter 1<p<infinity, what is the maximal regularity s in R for which there exists a non-zero distribution in the Bessel potential Sobolev space Hs,p(Rn) that is supported in E?
For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of E, sharpening previous results.
Furthermore, we provide a full classification of all possible maximal regularities, as functions of p, together with the sets of values of p for which the maximal regularity is attained, and construct concrete examples for each case.
Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions.
We collect several results characterising the regularity that can be achieved on certain special classes of sets, such as d-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations
Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions
We consider the Helmholtz transmission problem with one penetrable
star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of
the wavenumbers, we prove bounds on the solution in terms of the data, with
these bounds explicit in all parameters. In particular, the (weighted)
norm of the solution is bounded by the norm of the source term,
independently of the wavenumber. These bounds then imply the existence of a
resonance-free strip beneath the real axis. The main novelty is that the only
comparable results currently in the literature are for smooth, convex obstacles
with strictly positive curvature, while here we assume only Lipschitz
regularity and star-shapedness with respect to a point. Furthermore, our bounds
are obtained using identities first introduced by Morawetz (essentially
integration by parts), whereas the existing bounds use the much-more
sophisticated technology of microlocal analysis and propagation of
singularities. We also recap existing results that show that if the assumption
on the wavenumbers is lifted, then no bound with polynomial dependence on the
wavenumber is possible.Comment: 26 pages, 2 figure
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
A note on properties of the restriction operator on Sobolev spaces
In our companion paper [3] we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset Ω of Rn, defined as closed subspaces of the classical Bessel potential spaces Hs(Rn) for s∈R. These spaces are mapped by the restriction operator to certain spaces of distributions on Ω. In this note we make some observations about the relation between these spaces of global and local distributions. In particular, we study conditions under which the restriction operator is or is not injective, surjective and isometric between given pairs of spaces. We also provide an explicit formula for minimal norm extension (an inverse of the restriction operator in appropriate spaces) in a special case
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
Plane wave approximation of homogeneous Helmholtz solutions
In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω 2 u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations
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