Justification of the fast multipole method for the stokes system. Part II. Exterior domain problems

We consider the exterior domain problems of Dirichlet and Neumann type of the two-dimensional Stokes equations. For the solution of this boundary value problem we choose a potential ansatz and show that for the reduction of the computational costs, the fast multipole method of Greengard and Rokhlin can be used. Therefore, we find a complex representation of the hydrodynamical potentials and provide statements about the corresponding multipole and Taylor expansions, as well as the appropriate translation, rotation and conversion operators. The theoretical statements are illustrated by numerical experiments. Bibliography: 15 titles.     

Splitting of resonant and scattering frequencies under shape deformation

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues, and the splitting may only become apparent at high orders in their Taylor expansion. In this paper, we address the splitting problem in the evaluation of resonant and scattering frequencies of the two-dimensional Laplacian operator under boundary variations of the domain. By using surface potentials we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions that are of Fredholm type with index 0. We then proceed from the generalized Rouché's theorem to investigate the splitting problem.     

Hele-Shaw flows with anisotropic surface tension

In this note, we study Hele-Shaw flows in the presence of anisotropic surface tension when the fluid domain is bounded. The flows are driven by a sink, by a multipole, or solely by anisotropic surface tension. For a sink flow, we show that if the center of mass of the initial domain is not located at a certain point which is determined by the anisotropic surface tension and intensity of the sink, then either the solution will break down before all the fluid is sucked out or the fluid domain will eventually become unbounded in diameter. For a multipole driven flow, we prove that if the anisotropic surface tension, the order, and intensity of the multipole do not satisfy a certain equality, either the flow will develop finite-time singularities or the fluid domain will become unbounded in diameter as time goes to infinity. For a flow driven purely by anisotropic surface tension, we show that the center of mass of the fluid domain moves in a constant velocity, which is determined explicitly.     

A Multipole Method for the Rapid Solution of the Stationary Linearized Stokes System

Here we will show how the multipole method can be used for the questions connected with the hydrodynamical potentials. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Average values of L-functions in characteristic two

Gauss made two conjectures about average values of class numbers of orders in quadratic number fields, later on proven by Lipschitz and Siegel. A version for function fields of odd characteristic was established by Hoffstein and Rosen. In this paper, we extend their results to the case of even characteristic. More precisely, we obtain formulas of average values of L-functions associated to orders in quadratic function fields over a constant field of characteristic two, and then derive formulas of average class numbers of these orders.     

Effective multipoles in random media

Abstract

In a homogeneous medium, the far field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole – but not the octupole – can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest. Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero- and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green’s formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials.     

On a Problem of the Constructive Theory of Harmonic Mappings

The problem of irremovable error appears in finite difference realization of the Winslow approach in the constructive theory of harmonic mappings. As an example, we consider the well-known Roache–Steinberg problem and demonstrate a new approach, which allows us to construct harmonic mappings of complicated domains effectively and with high precision. This possibility is given by the analytic-numerical method of multipoles with exponential convergence rate. It guarantees effective construction of a harmonic mapping with precision controlled by an a posteriori estimate in a uniform norm with respect to the domain.     

Instanton in the Field of a Pointlike Source of a Euclidean Non-Abelian Field

We consider the behavior of an (anti)instanton in the field of a pointlike source of a Euclidean non-Abelian field and investigate (anti)instanton deformations described by variations of its characteristic parameters. We formulate the variational problem of seeking the corresponding “crumpled” topological configurations and solve it algebraically using the Ritz method (of multipole decomposition of deformation fields). We investigate the domain of parameters specific for the instanton liquid model. We propose a simple method of taking contributions of such configurations in the functional integral into account approximately. In the framework of superposition analysis, we obtain an estimate for the mean energy of the source in the instanton liquid, and the estimate increases linearly with distance. In the case of a dipole in the color-singlet state, the energy is a linear function of the distance between the sources with the “strength” coefficient agreeing well with the known model and lattice estimates. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 267–298, February, 2006.     

Tilted symmetric orders are symmetric orders

We prove in this note that a ring which is derived equivalent to a symmetric order is again a symmetric order. Group rings of finite groups over an integral domain of characteristic 0 are symmetric orders.     

Spaces of orders and their Turing degree spectra

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s (2004) [28] investigation of orders on groups in topology and Solomon’s (2002) [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group F_n of rank n>1 has a bi-order in every Turing degree.     

Layer potentials C*-algebras of domains with conical points

To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ?Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.     

Generalized boundary values of the solutions of semilinear elliptic equations from weighted functional spaces

In weighted C-spaces, we establish the solvability of a boundary-value problem for a semilinear elliptic equation of order 2m in a bounded domain with generalized functions given on its boundary, strong power singularities at some points of the boundary, and finite orders of singularities on the entire boundary. The behavior of the solution near the boundary of the domain is analyzed.     

On preconditioners for the Laplace double‐layer in 2D

The discretization of the double‐layer potential integral equation for the interior Dirichlet–Laplace problem in a domain with smooth boundary results in a linear system that has a bounded condition number. Thus, the number of iterations required for the convergence of a Krylov method is, asymptotically, independent of the discretization size N. Using the fast multipole method to accelerate the matrix–vector products, we obtain an optimal solver. In practice, however, when the geometry is complicated, the number of Krylov iterations can be quite large—to the extend that necessitates the use of preconditioning. We summarize the different methodologies that have appeared in the literature (single‐grid, multigrid, approximate sparse inverses), and we propose a new class of preconditioners based on a fast multipole method‐based spatial decomposition of the double‐layer operator. We present an experimental study in which we compare the different approaches, and we discuss the merits and shortcomings of our approach. Our method can be easily extended to other second‐kind integral equations with non‐oscillatory kernels in two and three dimensions. Copyright © 2014 John Wiley & Sons, Ltd.     

Diffraction by a rigid,mobile inclusion

The long-wavelength problem of scattering by a rigid inclusion in an elastic medium is studied. Taking into account the mobility of the inclusion leads to non-classical boundary conditions. At infinity the solution is sought in the form of a multipole ansatz. Near the scatterer a series of static problems is obtained. In the course of the solution the integral characteristic of the rigid mobile inclusion, whose scalar analog is the tensor e_ij studied by Polya and Szegö, arises in a natural manner.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 61–68, 1986.     

Inverse problem for Schrödinger equations with Yang–Mills potentials in a slab

In this work we consider the inverse boundary value problem for Schrödinger equations with Yang–Mills potentials in the domain of infinite slab type. We prove that the potentials can be determined uniquely up to a gauge equivalent class assuming that only partial measurements are known on the boundary hyperplanes.     

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, II

In this paper, we consider mixed problems with a timelike boundary derivative (or a Dirichlet) condition for semilinear wave equations with exponential nonlinearities in a quarter plane. The case when the boundary vector field is tangent to the characteristic which leaves the domain in the future is also considered. We show that solutions either are global or blow up on a C¹ curve which is spacelike except at the point where it meets the boundary; at that point, it is tangent to the characteristic which leaves the domain in the future.     

Potential comparison and asymptotics in scalar conservation laws without convexity

Two kinds of optimal convergence orders in L¹-norm to a self-similar solution are proved or conjectured for various evolutionary problems so far. The first convergence order is of the magnitude of the similarity solution itself and the second one is of order 1/t. Employing a potential comparison technique to scalar conservation laws we may easily see that these asymptotic convergence orders are related to space and time translation of potentials. We present the technique clearly in the simple setting of scalar conservation laws in one space dimension.     

The planar Dirichlet problem for the Stokes equations

The Dirichlet problem for the Stokes equations is studied in a planar domain. We construct a solution of this problem in form of appropriate potentials and determine the unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence we determine a solution of the Dirichlet problem for a compressible Stokes system and a solution of a boundary value problem on a domain with cracks. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in ?². By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.