The article is organized as follows. In Sec. II , we briefly study the Weertman equation, and discuss both the uniqueness of its solution and its interpretation as the long-time limit ofthe dynamical system ( 6 ). We formally derive asymptotes of solutions to ( 1 ) and state identities about the velocity c η in general cases. Also, we explain how to choose c(t) in ( 6 ) to solve this equation in a comoving frame, namely, one which follows the translational motion ofthe core. An analytical solution to ( 1 ) that exists in a particular case is recalled. In Sec. III , we introduce our numerical representation for η and discuss corresponding implementations ofthe diffusive operator −|∂ x | and the advective operator ∂ x , as well as methods to evaluate c(t). Also, we make use ofthe asymptotic behavior when |x| → +∞ ofthe solution to ( 1 ) to circumvent the issue ofthe infinite domain of integration in ( 2 ). Once these fundamental elements have been introduced, we build in Sec. IV a Preconditioned Collocation Scheme (PCS) that applies to our problem, and justify this denomination. In Sec. V , we use this numerical approach on two test cases: one with a simple potential F σ , for which the exact solution is known, and one with a more physically relevant potential F σ . We also illustrate the robustness of our approach, concluding that the algorithm presented is unconditionally stable with respect to the time step ∆t. We empirically derive error scalings with respect to the parameters involved in the discretization. A concluding discussion closes the article, underlining some limitations of our approach, and proposing a few possible extensions. An Appendix is devoted to examining further one such extension.
beyond this range the behavior is of polynomial type in x. Such tricky estimates are of course very specific to the case ofthe wave equation discretized using the finite-difference method (the discrete eigenvalues depend on the numerical scheme).
An interesting (and maybe difficult) open question that remains unclear after this work is the following: for a given range of filtration given by n 6 f (N ), can we determine precisely the minimal time that ensures the uniform controllability? If we compare with the results given in [5, Section 2.4], it is likely that one cannot do better than T > 2/ cos(Γ(f )π/2), which is sharp in the context where we filter the whole solution, but we have no intuition if it would be possible to recover this minimal time in our context.
Figure 14: Dispersion curve for κh taking values in 0.1 : 0.01 : 3. The slopes of each curve are approximately twice the discretization order, as predicted by the asymptotics (4).
In this project, we have used a method different from that in  and  to characterize and quantize the numerical dispersion associated with continuous Galerkin finite element. Without starting from an ansatz, we construct directly the numerical solution (on the whole real line) via the limiting absorption principle. We also obtain an asymptotic expansion for the difference between the analytic and numerical wavenumber, which agrees with the results of  and . With technique in complex analysis like contour deformation, we arrive at an explicit formula for the numerical solution, which allows the identification ofthe numerical wavenumber in terms of analytic poles ofan algebraic equation. This identification is useful since we can evaluate concretely the numerical wave number from analytic poles, and the latter are calculated via Guillaume’s algorithm.
u the least square approximationof u thresholded such that |˜ u| ≤ M .
This suggests that the slowest K(m) increases, the largest m can be, allowing a better reconstruction. The choice ofthe density ν is here important, as K is dependent on it. This means that choosing a adequate density allows to use a lower number of samples that, e.g. the uniform density. Note however that the choice ofthe density ν also affects the norm used in theorem 2 to measure the error, which can be different than the norm we are interested in. We are here interested in the stability for the standard L 2 norm. We thus choose a density ofthe form ν = (1 − α)λ + αν 0 where λ is the uniform density, and ν 0 an arbitrary fixed density. The choice ofthe density ν 0 and the parameter α is discussed in the next section for some particular cases.
Abstract. In this paper, we develop in a general framework a non overlapping Domain Decomposi- tion Method that is proven to be well-posed and converges exponentially fast, provided that specific transmission operators are used. These operators are necessarily non local and we provide a class of such operators in the form of integral operators. To reduce the numerical cost of these integral oper- ators, we show that a truncation process can be applied that preserves all the properties leading to an exponentially fast convergent method. A modal analysis is performed on a separable geometry to illustrate the theoretical properties ofthe method and we exhibit an optimization process to further reduce the convergence rate ofthe algorithm.
3. Formation of singularities. In the case of large data, we prove sufficient conditions for the breakdown of regular solutions in finite time. For this purpose, the 2 × 2 system (1.1) is first transformed into a scalar equation with a source term (Sec. 3.1). Then one considers successively the dissipative case ε > 0 (Sec. 3.2) and the conservative case ε = 0 (Sec. 3.3). In the case of small data, global smooth solutions are expected: the Sugimoto’s model has been defined to prevent from the occurence of shocks and to propagate acoustic solitons . The Shizuta-Kawashima is invocated to study the possible existence of global smooth solutions near equilibrium (Sec. 3.4). Numerical experiments are proposed to illustrate these properties (Sec. 3.5).
Figure 6: Best reconstruction error for the proposed method, using GCV and optimal value of m.
6.2 More general shapes
While the results obtained here inform us on the importance of sampling on the border ofthe considered domain, a numerical test on another simple shape shows that the density on the border is critical. The theoretical analysis for the disk and the ball was based on the fact that the Fourier-Bessel functions were already an orthogonal basis ofthe space V m b . We focus here on the square [−1, 1] 2 . As neither the Fourier-Bessel functions, nor the plane waves, form an orthogonal basis, we construct one by orthogonalizing the Fourier-Bessel functions.
particular the application to other wave equations) ofthe Engquist–Majda conditions. It is not possible to give here an exhaustive bibliography, and we will refer the reader to recent review papers on the subject by Hagstr¨ om ,  and Givoli . In the last decade, alternative solutions have been progressively developed and, especially, researchers have tried to promote again the use of exact nonlocal boundary either by using speciﬁc geometries for the absorbing boundaries, as in the works by Grote and Keller , , or by exploiting the recent progress in rapid algorithms (multipoles) and rational approximation, as in the work of Alpert, Greengard, and Hagstr¨ om , . Approximately during the same period, the introduction by B´ erenger ofthe perfectly matched layers (PMLs) technique ,  partly revolutionized the subject. The philosophy here is to replace the absorbing boundary with an absorbing layer (or sponge layer) which is such that any wave propagating in the computational domain is transmitted to the absorbing layer without being reﬂected. This method quickly attracted many researchers in diﬀerent ﬁelds of application, in particular because of its good practical performances and its easy implementation.
[CSV] E. J. Cand` es, T. Strohmer and V. Voroninski PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Comm. Pure and Appl. Math. 66 (2013), 1241–1274.
[FD] F. Feng & D. Dai, Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in math, 2013, Springer, New York. [FBGJ] A. Fernandez-Bertolin, K. Gr¨ ochenig & Ph. Jaming On unique
POSTERIORI ERROR ESTIMATES FOR THEHELMHOLTZEQUATION ?
T. CHAUMONT-FRELET 1,2 , A. ERN 3,4 , AND M. VOHRAL´IK 4,3
Abstract. We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed finite element problems. We show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase. We also derive a fully computable upper bound on the prefactor for several common settings of domains and boundary conditions. This leads to a guaranteed estimate without any assumption on the mesh size or the polynomial degree, though the obtained guaranteed bound may lead to large error overestimation. We next demonstrate that the estimator is locally efficient, robust in all regimes with respect to the polynomial degree, and asymptotically robust with respect to the wavenumber. Finally we present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp.
The method proposed in this study owns other additional interesting properties. As a DG method, it is formulated as a symmetri
DG method, that is, as a symmetri
variational formu- lation ofthe
orresponding boundary-value problem. Its derivation follows the path devised in [4℄ (see also [37, p. 122℄) for designing Symmetri
Interior Penalty (SIP) methods but in a bit dierent way, more straightforward in our opinion. Additionally, when the penalty terms enfor
- ing the
ontinuity ofthe normal tra
es (really the dual variables) are dis
arded, this symmetry here yields an important gain. The storage ofthe boundary integral operators involved in the formulation is then avoided: the
ontribution ofthe BIEs then being element-wise only. It is also worth noting that all the degrees of freedom ofthe dis
rete problem to be solved are lo
ated on the skeleton ofthe mesh, that is, the boundaries ofthe elements. Su
h a feature is
to the redu
tion of unknowns yielded by HDG methods even if here there still remains unknowns on both sides ofthe interfa
es. Last but not least perhaps the most important advantage ofthe proposed approa
h lies on the
e ofthe lo
al shape fun
ount for all kinds of waves: ev anes
ent, propagative, et
. This is in
ontrast with usual Tretz methods whi
ally use plane,
al waves, multipoles, et
f., for example, [3, 23, 34, 20, 10℄ to
ite a few). It should be noted also that, even if the method, whi
onsidered here, is of Tretz type, the lo
al approximations are done by means of a Boundary Element Method (BEM) (
f., for example, [40, 6℄). As a result, these approximations are ultimately performed in terms of pie
ewise polynomial fun
tions on a BEM mesh. In
ontrast then to usual Tretz methods, h or
Werner integral equation based on the use ofthe Dirichlet-Neumann (DN) and Neumann- Dirichlet (ND) maps. This construction starts from the ideas developed in . In section 4, we build an efficient and simple approximationof both the DN and ND operators by applying some similar techniques as in the Beam Propagation Methods [40, 32] working in a generalized coordinates system. Section 5 is devoted to the well-posedness ofthe new integral equations at any frequency. In section 6, we calculate the eigenvalues ofthe new operators and numerically show that the new integral equations have an excellent eigenvalue clustering even for high- frequencies for the scattering problem of a plane wave by a sphere. In section 7, we extend the ideas to a generalization ofthe usual CFIE of Harrington and Mautz . We develop several aspects linked to the implementation ofthe new integral equations in a Krylov iterative solver in the 8th section. An essential aspect is that the approximations ofthe DN and ND operators are computed by a paraxialization technique . In section 9, we perform some numerical experiments to show that the generalized integral formulations have some interesting convergence rates.
Here we are going to use the point of view of J.-F. Bony (see [Bon]). He considers the case of a source which concentrates on one or two points (with V1 6= 0) using a time-dependant method based on a BKW approximationofthe propagator to prove that, microlocally, the solution oftheHelmholtzequation is a finite sum of lagrangian distributions. In particular, abstract estimates ofthe solution are only used for the large times control, and this part ofthe solution has no contribution for the semiclassical measure, so the measure is actually constructed explicitely. Moreover, this method requires a geometrical assumption weaker than the Virial hypothesis used in the previous works.
Since the branching fractions of multi-body decays are large , it is natural to ask whether such final states are actually the products of intermediate two- body channels. If this is the case, then these initial two-body decays could involve proton-antiproton bound states (pp) [7, 8], or charmed pentaquarks [9, 10], or heavy charmed baryons. Motivated by these consider- ations, in particular the claim of a charmed pentaquark at 3.1 GeV/c 2 by the H1 collaboration, the invariant mass spectrum ofthe proton-antiproton and the invariant mass spectra ofthe charmed meson and proton are inves- tigated. Throughout this paper, we shall use the terms “exotic” and “non-exotic” to refer to the “Dp” pair with total quark content cquud and cquud respectively (where q is u or d). Specifically, the “exotic” combinations refer to D (∗)− p and D (∗)0 p while the “non-exotic” combina- tions are D (∗)− p and D (∗)0 p.
We have presented here patterns of mo- dulus of solutions to the NLS focusing equation in the (x, t) plane.
These study can be useful at the same time for hydrodynamics as well for non- linear optics; many applications in these fields have been realized, as it can be seen in recent works of Chabchoub et al.  or Kibler et al. .
and ε > 0 is a measure ofthe range of dispersal.
Before going any further, let us say a brief word about the meaning of ( 1.3 ). The key idea behind this model relies on the notion of dispersal budget (introduced in [ 25 ]). In a nutshell, it consists in assuming that the amount of energy per individual that the species can use to disperse is fixed (because ofthe environmental and developmental constraints) and that the displacement ofthe individuals has a cost (reflecting the amount of energy required to disperse) which, for simplicity, is assumed to be proportional to c(x) = |x| m with 0 6 m 6 2.
Charmed baryonic B decays are experimentally accessible and provide a way to check predictions given by various theoretical models for exclusive baryonic B decays. There is theoretical inter- est in the suppression ofthe two-body baryonic decay rates compared to three-body decay rates and the possible connection to production mechanisms for baryons in B decays. Analysisofthe charmed three-body baryonic B decay reveals that the invariant mass ofthe baryon-antibaryon system is peaked near threshold . Charmless two-body baryonic B decays (which have not yet been observed [2, 3]) may be used to measure direct CP violation in theB system. Their charmed counterparts, however, have branching fractions at least an order of magnitude higher than the charmless modes, and thus can help distinguish between theoretical models that predict the charmless decay rates ofB mesons to baryons. The Feynman diagrams for these decays are shown in Figure 1, in which theB meson decays weakly via internal W emission to Λ + c p(π).
N i ( x ) i (24)
where n is the total number of nodes, N i are compact support
shape functions for each node i . They take the value 1 for the node i and vanish for all other nodes (see for example  for more details). Due to the advection–diffusion character of Eq. (23) an appropriate stabilization ofthe Finite Element scheme is needed to avoid numerical instabilities induced by the convection term. An upwinding formulation is considered here, which modiﬁes the weighting function