We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

The classical Koksma Hlawka inequality does not apply to functions with simple discontinuities. Here we state a Koksma Hlawka type inequality which applies to piecewise smooth functions f χ Ω , with f smooth and Ω a Borel subset of [0, 1] d :

The algorithm devised by Feige and Schechtman for partitioning higher dimensional spheres into regions of equal measure and small diameter is combined with David and Christ's construction of dyadic cubes to yield a partition algorithm suitable to any connected Ahlfors regular metric measure space of finite measure.

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In this work we consider the Optimized Schwarz Method for the three-dimensional (3D) diffusion-reaction problem. In particular, we treat the case of cylindrical interfaces between the subdomains, and we provide a convergence analysis of the Schwarz method, both in the case of Dirichlet interface conditions and in that of general transmission conditions. This allows us to recover, for the latter case, optimal symbols for the interface conditions, which are supposed to work well for geometries which feature cylindrical interfaces. Moreover, starting from these optimal symbols, we propose effective and easily computable constant interface parameters, derived both from Taylor expansions and from an optimization procedure. We finally present several 3D numerical results aiming at validating the theoretical findings.1. Introduction. The classical Schwarz method for the numerical solution of a partial differential equation consists of splitting the computational domain in two (or even more) subdomains, with or without overlap, and in the solution of the equation at hand in such subdomains in an iterative framework, through the exchange at the interface of the trace of the solution. It is known that this method features a slow convergence in general and does not converge without overlap [23, 26, 5] unless a nonconforming discretization is employed [8,3,2]. For this reason, Lions introduced different transmission conditions, of Robin type, which improved the convergence properties and also reach converge without overlap [17]. Successively, this method has been generalized by considering more general performing interface conditions involving nonlocal transmission operators (generalized Schwarz method; see, e.g., [6,20]).The choice of suitable parameters in such operators is crucial to guarantee good convergence properties. This is usually driven by the minimization of the reduction factor related to the iterations (Optimized Schwarz Method). To obtain the reduction factor the Fourier transform is usually applied to some variables leading to ordinary differential equations [14]. This optimization strategy has been applied to a great variety of problems. We cite, for example, the advection-reaction-diffusion problem [9,15], the Helmholtz equation [12,19], the coupling of heterogeneous media [11,18,10], the shallow-water equations [22], the Maxwell equations [7], the fluid-structure interaction problem [13], and the scattering problem [25].

Revisiting and extending a recent result of M. Huxley, we estimate the Lp($\mathbb{T}$d) and Weak–Lp($\mathbb{T}$d) norms of the discrepancy between the volume and the number of integer points in translated domains.

Abstract. We consider Fourier transforms µ of densities supported on curves in R d . We obtain sharp lower and close to sharp upper bounds for the decay rates of µ(R·) L q (S d−1 ) , as R → ∞.

We propose a unified convergence analysis of the generalized Schwarz method applied to a linear elliptic problem for a general interface (flat, cylindrical or spherical) in any dimension. In particular, we provide the exact convergence set of the interface symbols related to the operators involved in the transmission conditions. We also provide a general procedure to obtain estimates of the optimized interface symbols within the constants. Finally, we apply such general results to a fluid-structure interaction model problem, and we assess the effectiveness of the theoretical findings through three-dimensional numerical experiments in the haemodynamic context.