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

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 $\Omega\subset\mathbb R^n$, $\mathcal{D}(\Omega)$ is dense in $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue measure and $\Omega$ is "thick" (in the sense of Triebel); and (ii) for a $d$-set $\Gamma\subset\mathbb R^n$ ($0<d<n$), $\{u\in H^{s_1}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ is dense in $\{u\in H^{s_2}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ whenever $-\frac{n-d}{2}-m-1<s_{2}\leq s_{1}<-\frac{n-d}{2}-m$ for some $m\in\mathbb N_0$. For (ii), we provide concrete examples, for any $m\in\mathbb N_0$, where density fails when $s_1$ and $s_2$ are on opposite sides of $-\frac{n-d}{2}-m$. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}=\{0\}$ for a given closed set $\Gamma\subset\mathbb R^n$ and $s\in \mathbb R$. 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

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

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 $h$-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 $k\to\infty$, $h$ must decrease with $k$ at the same rate as for the standard formulation). We prove $k$-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 $k$, 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

Comparative study of advanced heat pumps

A numerical simulation study is reported on the thermodynamic performance of several non-CFC refrigeration devices. The study includes complex compound absorption, Brayton, Stirling, and thermoelectric devices. Comparisons are made to the more commonly applied vapor compression systems, including those using R-134a. The study examines the effect of thermal resistances between the device and the heat rejection or heat absorption space. A cool side temperature difference between 0 and 20{dollar}\sp\circ{dollar}C is investigated, and this temperature difference accounts for both thermal resistance and cooling load. An outside temperature ranging between 35{dollar}\sp\circ{dollar}C and 46{dollar}\sp\circ{dollar}C is considered in the calculations, with a cooled space temperature of 22{dollar}\sp\circ{dollar}C assumed throughout. Evaluations of the coefficients of performance for each of the units show the vapor compression machines demonstrate superior performance over the complete range of operating conditions examined. However, additional requirements, such as maintenance and environmental factors, indicate other desirable options

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) $H^1$ norm of the solution is bounded by the $L^2$ 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~$h$-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

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in $${\mathbb R^{2}}$$ to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907-1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N≥2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane wave