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beyond this range **the** behavior is **of** polynomial type in x. Such tricky estimates are **of** course very specific to **the** case **of** **the** 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.

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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).
7 Conclusion
In this project, we have used a method different from that in [1] and [8] 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** [1] and [8]. With technique in complex **analysis** like contour deformation, we arrive at **an** explicit formula for **the** numerical solution, which allows **the** identification **of** **the** numerical wavenumber in terms **of** analytic poles **of** **an** 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.

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u **the** least square **approximation** **of** 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 **of** **the** 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 **of** **the** 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 **of** **the** form ν = (1 − α)λ + αν 0 where λ is **the** uniform density, and ν 0 **an** arbitrary fixed density. **The** choice **of** **the** density ν 0 and **the** parameter α is discussed in **the** next section for some particular cases.

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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 **of** **the** method and we exhibit **an** optimization process to further reduce **the** convergence rate **of** **the** algorithm.

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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 [37]. **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).

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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 **of** **the** 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 **of** **the** 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.

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particular **the** application to other wave equations) **of** **the** 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 [18], [19] and Givoli [13]. 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 [14], [15], or by exploiting **the** recent progress in rapid algorithms (multipoles) and rational **approximation**, as in **the** work **of** Alpert, Greengard, and Hagstr¨ om [2], [1]. Approximately during **the** same period, **the** introduction by **B**´ erenger **of** **the** perfectly matched layers (PMLs) technique [6], [5] 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.

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[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 **THE** **HELMHOLTZ** **EQUATION** ?
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.

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Werner integral **equation** based on **the** use **of** **the** Dirichlet-Neumann (DN) and Neumann- Dirichlet (ND) maps. This construction starts from **the** ideas developed in [7]. In section 4, we build **an** efficient and simple **approximation** **of** 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 **of** **the** new integral equations at any frequency. In section 6, we calculate **the** eigenvalues **of** **the** 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 **of** **the** usual CFIE **of** Harrington and Mautz [31]. We develop several aspects linked to **the** implementation **of** **the** new integral equations in a Krylov iterative solver in **the** 8th section. **An** essential aspect is that **the** approximations **of** **the** DN and ND operators are computed by a paraxialization technique [42]. In section 9, we perform some numerical experiments to show that **the** generalized integral formulations have some interesting convergence rates.

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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 **approximation** **of** **the** propagator to prove that, microlocally, **the** solution **of** **the** **Helmholtz** **equation** is a finite sum **of** lagrangian distributions. In particular, abstract estimates **of** **the** solution are only used for **the** large times control, and this part **of** **the** 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.

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Since **the** branching fractions **of** multi-body decays are large [6], 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[11], **the** invariant mass spectrum **of** **the** proton-antiproton and **the** invariant mass spectra **of** **the** 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**.

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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. [50] or Kibler et al. [51].

and ε > 0 is a measure **of** **the** 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 **of** **the** environmental and developmental constraints) and that **the** displacement **of** **the** 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.

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1 INTRODUCTION
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 **of** **the** two-body baryonic decay rates compared to three-body decay rates and **the** possible connection to production mechanisms for baryons in **B** decays. **Analysis** **of** **the** charmed three-body baryonic **B** decay reveals that **the** invariant mass **of** **the** baryon-antibaryon system is peaked near threshold [1]. Charmless two-body baryonic **B** decays (which have not yet been observed [2, 3]) may be used to measure direct CP violation in **the** **B** 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 **of** **B** mesons to baryons. **The** Feynman diagrams for these decays are shown in Figure 1, in which **the** **B** meson decays weakly via internal W emission to Λ + c **p**(π).

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i =1
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 [9] for more details). Due to **the** advection–diffusion character **of** Eq. (23) **an** appropriate stabilization **of** **the** 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