nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

chaotic blowup in the 3d incompressible euler equations on a logarithmic lattice the dispute on whether the three dimensional 3d incompressible euler equations develop an infinitely large vorticity in a finite time blowup keeps increasing due to ambiguous results from state of the art direct numerical simulations dns while the available simplified models fail to explain the intrinsic complexity and variety of observed structures here we propose a new model formally identical to the euler equations by imitating the calculus on a 3d logarithmic lattice this model clarifies the present controversy at the scales of existing dns and provides the unambiguous evidence of the following transition to the blowup explained as a chaotic attractor in a renormalized system the chaotic attractor spans over the anomalously large six decade interval of spatial scales for the original euler system our results suggest that the existing dns strategies at the resolution accessible now and presumably rather long into the future are unsuitable by far for the blowup analysis and establish new fundamental requirements for the approach to this long standing problem

cloaking via change of variables for the helmholtz equation the transformation optics approach to cloaking uses a singular change of coordinates which blows up a point to the region being cloaked this paper examines a natural regularization obtained by 1 blowing up a ball of radius rather than a point and 2 including a well chosen lossy layer at the inner edge of the cloak we assess the performance of the resulting near cloak as the regularization parameter tends to 0 in the context of dirichlet and neumann boundary measurements for the time harmonic helmholtz equation since the goal is to achieve cloaking regardless of the content of the cloaked region we focus on estimates that are uniform with respect to the physical properties of this region in three space dimensions our regularized construction performs relatively well the deviation from perfect cloaking is of order in two space dimensions it does much worse the deviation is of order 1 log in addition to proving these estimates we give numerical examples demonstrating their sharpness some authors have argued that perfect cloaking can be achieved without losses by using the singular change of variable based construction in our regularized setting the analogous statement is false without the lossy layer there are certain resonant inclusions depending in general on that have a huge effect on the boundary measurements 2010 wiley periodicals inc

bracket formulations and energy and helicity preserving numerical methods for the three dimensional vorticity equation abstract the vorticity equation for three dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the poisson or nambu bracket together with a dissipative bracket the budgets of kinetic energy helicity and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method in particular energy and helicity are conserved precisely in inviscid flow computations the energy and enstrophy dissipate properly owing to viscosity in viscous flow computations and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations the relationships between the stream function velocity and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens

a fully discrete bem fem for the exterior stokes problem in the plane we reformulate the johnson nedelec approach for the exterior two dimensional stokes problem taking advantage of the parameterization of the artificial boundary the main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem this one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals we give a complete error analysis of both the galerkin and fully discrete galerkin methods

remarks on the blow up of the euler equations and the related equations we consider the euler system for inviscid incompressible fluid flows and its perturbations in n n 2 we prove global well posedness of this perturbed euler system in the triebel lizorkin spaces for initial vorticity which is small in the critical besov norms comparison type theorems about the blow up of solutions are proved between the euler system and its perturbations we also study the possiblity of the self similar type of blow up of solutions to the equations

optimal control for electromagnetic cloaking metamaterial parameters design abstract in this paper we develop an optimal control problem to optimize the permittivity and permeability of the metamaterial which can have the invisible cloaking effect the inverse problem is calibrated as an optimization problem constrained by the time harmonic maxwell s equations proper objective functional is introduced with coefficients of the maxwell s equations as control variables we adopt the discontinuous galerkin method to solve the state equation several two dimensional cloaking benchmark problems are used to test our optimal control method numerical results demonstrate that the optimized permittivity and permeability of the metamaterial can achieve the cloaking effectively

gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes in this paper the gradient recovery type a posteriori error estimators for finite element approximations are proposed for irregular meshes both the global and the local a posteriori error estimates are derived moreover it is shown that the a posteriori error estimates is asymptotically exact on where the mesh is regular enough and the exact solution is smooth

the role of bkm type theorems in 3d euler navier stokes and cahn hilliard navier stokes analysis abstract the beale kato majda theorem contains a single criterion that controls the behaviour of solutions of the 3 d incompressible euler equations versions of this theorem are discussed in terms of the regularity issues surrounding the 3 d incompressible euler and navier stokes equations together with a phase field model for the statistical mechanics of binary mixtures called the 3 d cahn hilliard navier stokes chns equations a theorem of bkm type is established for the chns equations for the full parameter range moreover for this latter set it is shown that there exists a reynolds number and a bound on the energy dissipation rate that remarkably reproduces the r e 3 4 upper bound on the inverse kolmogorov length normally associated with the navier stokes equations alone an alternative length scale is introduced and discussed together with a set of pseudo spectral computations on a 12 8 3 grid

a new bem fem coupling strategy for two dimensional fluid solid interaction problems we present a numerical method to solve a fluid solid interaction problem posed in the plane in this scheme we use a finite element method to approximate the solid vibrations and the near wave field the far field effects are taken into account by means of boundary integral equations posed on an artificial interface that contains the obstacle the boundary unknown involved in our formulation is approximated by a spectral method we obtain a fully discrete galerkin procedure whose main advantage is the simplicity of the quadratures used to approximate the weakly singular boundary integrals we provide numerical results that illustrate the accuracy of our method and the stability of the algorithm used to solve the linear systems of equations that arise from this discretization technique

curl recovery for the lowest order rectangular edge element abstract in this paper we present two recovered curl results of the lowest order rectangular edge element for 2 d time harmonic maxwell s equations the proposed methods are about local discrete least square fittings it is proved that the two methods are superconvergent numerical examples show that the recovery methods can obtain superconvergent curl approximations for time harmonic maxwell s equations

on absence and existence of the anomalous localized resonance without the quasi static approximation this paper considers transmission problems for the helmholtz equation with bodies that have negative material parameters such material parameters are used to model metals on optical frequencies and so called metamaterials as the absorption of the materials in the model tends to zero the fields may blow up when the speed of the blow up is suitable this is called the anomalous localized resonance alr in this paper we study this phenomenon and formulate a new condition the weak anomalous resonance w ar where the speed of the blow up of fields may be slower using this concept we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi static approximation we give simple geometric conditions under which w ar or alr may or may not appear in particular we show that in a case of a curved layer of negative material with a strictly convex boundary neither alr nor w ar appears with nonzero frequencies i e in the dynamic range in dimensions

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

mathematical analysis and finite element time domain simulation of arbitrary star shaped electromagnetic cloaks in this paper we establish the explicit expression for two dimensional 2d electromagnetic cloaks of arbitrary star shapes without explicit contour expressions of the objects furthermore 2d arbi

nearly cloaking the electromagnetic fields the approximate cloaking is investigated for time harmonic maxwell s equations via the approach of transformation optics the problem is reduced to certain boundary effect estimates due to an inhomogeneous electromagnetic inclusion with an asymptotically small support but an arbitrary content enclosed by a thin high conducting layer sharp estimates are established in terms of the asymptotic parameter which are independent of the material tensors of the small electromagnetic inclusion the result implies that the blow up a small region construction via the transformation optics approach yields a near cloak for the electromagnetic waves a novelty lies in the fact that the geometry of the cloaking construction of this work can be very general moreover by incorporating the conducting layer developed in the present paper right between the cloaked region and the cloaking region arbitrary electromagnetic contents can be nearly cloaked our mathematical technique extends the general one developed in h y

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

on approximate electromagnetic cloaking by transformation media we give a comprehensive study on regularized approximate electromagnetic cloaking in the spherical geometry via the transformation optics approach the following aspects are investigated i near invisibility cloaking of passive media as well as active radiating sources ii the existence of cloak busting inclusions without lossy medium lining iii overcoming the cloaking busts by employing a lossy layer outside the cloaked region and iv the frequency dependence of the cloaking performances we address these issues and connect the obtained asymptotic results to singular ideal cloaking numerical verifications and demonstrations are provided to show the sharpness of our analytical study

blowup or no blowup the interplay between theory and numerics the question of whether the 3d incompressible euler equations can develop a finite time singularity from smooth initial data has been an outstanding open problem in fluid dynamics and mathematics recent studies indicate that the local geometric regularity of vortex lines can lead to dynamic depletion of vortex stretching guided by the local non blowup theory we have performed large scale computations of the 3d euler equations on some of the most promising blowup candidates our results show that there is tremendous dynamic depletion of vortex stretching the local geometric regularity of vortex lines and the anisotropic solution structure play an important role in depleting the nonlinearity dynamically and thus prevents a finite time blowup

a geometric interpretation of coherent structures in navier stokes flows the pressure in the incompressible three dimensional navier stokes and euler equations is governed by poisson s equation this equation is studied using the geometry of three forms in six dimensions by studying the linear algebra of the vector space of three forms l 3 wwhere w is a six dimensional real vector space we relate the characterization of non degenerate elements of l 3 wto the sign of the laplacian of the pressure and hence to the balance between the vorticity and the rate of strain when the laplacian of the pressure dp satisfies dpo0 the three form associated with poisson s equation is the real part of a decomposable complex form and an almost complex structure can be identified when dp 0 a real decomposable structure is identified these results are discussed in the context of coherent structures in turbulence

developing a time domain finite element method for modeling of electromagnetic cylindrical cloaks in this paper we propose a time domain finite element method for modeling of electromagnetic cloaks the permittivity and permeability of the cloak model are described by the drude dispersion model the model to be solved is quite challenging in that we have to solve a coupled problem with different partial differential equations given in different regions our method is based on a mixed finite element method using edge elements with different types of meshes in different regions numerical results demonstrate that our algorithm is quite effective for simulating cloaks in time domain to our knowledge this is the first cloak simulation carried out by the time domain finite element method

on the blow up of solutions of the 3 d euler equations in a bounded domain it is shown that if 0 hat t is the maximal interval of existence of a smooth solutionu of the incompressible euler equations in a bounded simply connected domain subseteq r3 then int 0 hat t left omega cdot t right l infty omega dt infty where u is the vorticity crucial to this result is a special estimate proven in of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity

on the finite time singularities of the 3d incompressible euler equations we prove the finite time vorticity blowup in the pointwise sense for solutions of the 3d incompressible euler equations assuming some conditions on the initial data and its corresponding solutions near initial time these conditions are represented by the relation between the deformation tensor and the hessian of pressure both coupled with the vorticity directions associated with the initial data and solutions near initial time we also study the possibility of the enstrophy blowup for the 3d euler and the 3d navier stokes equations and prove the finite time enstrophy blowup for initial data satisfying suitable conditions finally we obtain a new blowup criterion that controls the blowup by a quantity containing the hessian of the pressure 2006 wiley periodicals inc

blowup in stagnation point form solutions of the inviscid 2d boussinesq equations abstract the 2d boussinesq equations model large scale atmospheric and oceanic flows whether its solutions develop a singularity in finite time remains a classical open problem in mathematical fluid dynamics in this work blowup from smooth nontrivial initial velocities in stagnation point form solutions of this system is established on an infinite strip x y 0 1 r we consider velocities of the form u f t x y f x t x with scalar temperature y t x assuming f x 0 x attains its global maximum only at points x i located on the boundary of 0 1 general criteria for finite time blowup of the vorticity y f x x t x i and the time integral of f x t x i are presented briefly for blowup to occur it is sufficient that 0 x 0 and f t x i 0 x i 0 while y f x x 0 x i 0 to illustrate how vorticity may suppress blowup we also construct a family of global exact solutions a local existence result and additional regularity criteria in terms of the time integral of f x t l 0 1 are also provided

finite time blow up for a 1d model of 2d boussinesq system the 2d conservative boussinesq system describes inviscid incompressible buoyant fluid flow in a gravity field the possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics we consider a 1d model of the 2d boussinesq system motivated by a particular finite time blow up scenario we prove that finite time blow up is possible for the solutions to the model system

efficient and stable model reduction scheme for the numerical simulation of broadband acoustic metamaterials this study proposes an efficient and stable model reduction scheme for the numerical simulation of broadband inhomogeneous and anisotropic acoustic systems unlike a conventional model reduction scheme the proposed model reduction scheme uses the adaptive quasi static ritz vector aqsrv as a basis vector the proposed aqsrv based model reduction scheme has the following two representative features 1 multiple frequency subintervals and 2 adaptive selection of the subinterval information i e the proper number and location of the center frequencies and basis vector at each subinterval using the error indicator multiple frequency subintervals means to divide the frequency band of interest into several frequency bands from the computational time viewpoint adaptive selection of the subinterval information and basis vector means to select a different number of subintervals and basis vectors for use according to the target system the proposed model reduction scheme is applied to the numerical simulation of the simple mass damping spring system and the acoustic metamaterial systems i e acoustic lens and acoustic cloaking device for the first time through these numerical examples the proposed model reduction scheme was verified from the efficiency and stability point of view the aqsrv scheme is developed for the efficient analysis of the acoustic systems the aqsrv scheme has always a guaranteed convergence of solution the aqsrv scheme automatically constructs the reduced model the heuristic algorithm automatically chooses the subintervals and basis vectors the aqsrv scheme is applied to the broadband acoustic metamaterial systems

adaptive finite element method for the sound wave problems in two kinds of media abstract in this paper we consider the adaptive finite element method for sound wave propagation problems in two kinds of media which are the linear anisotropic acoustic materials the cloak metamaterials and penetrable media and the nonlinear acoustic materials a posteriori error estimator based on the new flux recovery technique and a residual type posteriori error estimator are proposed based on our a posteriori error estimators the adaptive finite element algorithm is given for numerical simulations of the helmholtz equations in different media extensive numerical results demonstrate the effectiveness of the adaptive algorithm

on a fem bem formulation for an exterior quasilinear problem in the plane we use a version of the fem bem method introduced by costabel boundary elements ix vol 1 c a brebbia et al eds springer verlag 1987 and han j comput math 8 1990 pp 223 232 to discretize an exterior quasilinear problem we provide error estimates for the galerkin method and propose a fully discrete scheme based on simple quadrature formulas furthermore we show that these numerical integration schemes preserve the optimal rates of convergence finally we present results of numerical experiments involving our discretization method

on the eddy resolving capability of high order discontinuous galerkin approaches to implicit les under resolved dns of euler turbulence we present estimates of spectral resolution power for under resolved turbulent euler flows obtained with high order discontinuous galerkin dg methods the 1 rule based on linear dispersion diffusion analysis introduced by moura et al 2015 10 is here adapted for 3d energy spectra and validated through the inviscid taylor green vortex problem the 1 rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra as the original rule relies on standard upwinding different riemann solvers are tested very good agreement is found for solvers which treat the different physical waves in a consistent manner relatively good agreement is still found for simpler solvers the latter however displayed spurious features attributed to the inconsistent treatment of different physical waves it is argued that in the limit of vanishing viscosity such features might have a significant impact on robustness and solution quality the estimates proposed are regarded as useful guidelines for no model dg based simulations of free turbulence at very high reynolds numbers

optimal order multigrid methods for solving exterior boundary value problems the coupling of boundary elements and finite elements combines the advantage of boundary elements for treating domains extended to infinity and that of finite elements in treating the nonhomogeneity of equations and the complexity of domains in the case of the laplacian by taking a circle or a sphere as the artificial coupling boundary it is shown that the corresponding boundary integral equation can be solved without any cost and the coupled system is reduced to a simple finite element system two multigrid methods are proposed to solve this finite element linear system both methods are of optimal order and can be used to solve such finite element equations as efficiently as to solve those arising from interior boundary value problems numerical experiments are included to show the efficiency and advantages of the methods an apparent significance of the methods is that the boundary elements appear neither in the discretization nor in the coding

an adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures an edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures the unbounded computational domain is truncated to a bounded one by a perfectly matched layer pml technique the pml parameters such as the thickness of the layer and the medium properties are determined through sharp a posteriori error estimates numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method

interplay between the beale kato majda theorem and the analyticity strip method to investigate numerically the incompressible euler singularity problem decrease to zero in order to sustain a finite time singularity consistent with the bkm theorem a simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the bkm theorem with the analyticity strip method it is shown that a finite time blowup can exist only if t vanishes sufficiently fast at the singularity time in particular if a power law is assumed for t then its exponent must be greater than some critical value thus providing a new test that is applied to our 4096 3 taylor green numerical simulation our main conclusion is that the numerical results are not inconsistent with a singularity but that higher resolution studies are needed to extend the time interval on which a well resolved power law behavior of t takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay

computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements in this paper we analyze a galerkin procedure based on a combination of finite and spectral elements for approximating a time harmonic acoustic wave scattered by a bounded inhomogeneity the finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition which consists in a linear integral equation this integral equation is discretized by a spectral galerkin approximation method we provide error estimates for the galerkin method propose fully discrete schemes based on elementary quadrature formulas and show that the perturbation due to this numerical integration gives rise to a quasi optimal rate of convergence we also suggest a method for implementing the algorithm using the preconditioned gmres method and provide some numerical results

on the coupling of boundary integral and mixed finite element methods we present a numerical method for solving an exterior dirichlet problem in the plane the technique consists in coupling boundary integral and mixed finite element methods an artificial boundary is introduced separating an interior region from an exterior one from an integral representation of the solution in the exterior domain we deduce two integral equations which relate the solution and its normal derivative over the artificial boundary these integral equations are incorporated into the so called mixed formulation of the problem in the interior region and a finite element method is used to approximate the resulting variational problem

on the collapse of tubes carried by 3d incompressible flows we define the notion of a regular tube and prove that a regular tube convected by a 3d incompressible flow cannot collapse to zero thickness in finite time

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

asymptotically exact a posteriori error estimators part i grids with superconvergence in part i of this work we develop superconvergence estimates for piecewise linear finite element approximations on quasi uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in particular we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant ui we then analyze a postprocessing gradient recovery scheme showing that q h nabla u h is a superconvergent approximation to nabla u here qh is the global l2 projection in part ii we analyze a superconvergent gradient recovery scheme for general unstructured shape regular triangulations this is the foundation for an a posteriori error estimate and local error indicators

note on loss of regularity for solutions of the 3 d incompressible euler and related equations one of the central problems in the mathematical theory of turbulence is that of breakdown of smooth indefinitely differentiable solutions to the equations of motion in 1934 j leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3 d incompressible navier stokes equations the problem is still open we show in this report that breakdown of smooth solutions to the 3 d incompressible slightly viscous i e corresponding to high reynolds numbers or highly turbulent navier stokes equations cannot occur without breakdown in the corresponding solution of the incompressible euler ideal fluid equation we prove then that solutions of distorted euler equations which are equations closely related to the euler equations for short term intervals do breakdown

on the coupling of mixed fem and bem for an exterior helmholtz problem in the plane this paper deals with an elliptic boundary value problem posed in the plane with variable coefficients but whose restriction to the exterior of a bounded domain reduces to a helmholtz equation we consider a mixed variational formulation in a bounded domain that contains the heterogeneous medium coupled with a boundary integral method applied to the helmholtz equation in we utilize suitable auxiliary problems duality arguments and fredholm alternative to show that the resulting formulation of the problem is well posed then we define a corresponding galerkin scheme by using rotated raviart thomas subspaces and spectral elements on the interface we show that the discrete problem is uniquely solvable and convergent and prove optimal error estimates finally we illustrate our analysis with some results from computational experiments

nonsymmetric coupling of bem and mixed fem on polyhedral interfaces in this paper we propose and analyze some new methods for coupling mixed finite element and boundary element methods for the model problem of the laplace equation in free space or in the exterior of a bounded domain as opposed to the existing methods which use the complete matrix of operators of the calderon projector to obtain a symmetric coupled system we propose methods with only one integral equation the system can be considered as a further generalization of the johnson nedelec coupling of bem fem to the case of mixed formulations in the bounded domain using some recent analytical tools we are able to prove stability and convergence of galerkin methods with very general conditions on the discrete spaces and no restriction relating the finite and boundary element spaces this can be done for general lipschitz interfaces and in particular the coupling boundary can be taken to be a lipschitz polyhedron both the indirect and the direct approaches for the boundary integral formulation are explored

centroidal voronoi tessellation based finite element superconvergence in this article a finding on finite element superconvergence is reported the laplacian operator with dirichlet boundary condition is considered the linear finite element solutions have an o h2 0 5 superconvergence in l2 norm at nodes on an almost equilateral triangular mesh generated based on centroidal voronoi tessellation for an arbitrary 2d bounded domain extensive numerical examples are presented to demonstrate the superconvergence property copyright 2008 john wiley sons ltd

adaptive finite element methods for elliptic pdes based on conforming centroidal voronoi delaunay triangulations a new triangular mesh adaptivity algorithm for elliptic pdes that combines a posteriori error estimation with centroidal voronoi delaunay tessellations of domains in two dimensions is proposed and tested the ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to generate superior unstructured grids result in a mesh adaptivity algorithm that has several very desirable features including the following errors are very well equidistributed over the triangles at all levels of refinement the triangles remain very well shaped even if the grid size at any particular refinement level when viewed globally varies by several orders of magnitude and the convergence rates achieved are the best obtainable using piecewise linear finite elements this methodology can be easily extended to higher order finite element approximations or mixed finite element formulations although only the linear approximation is considered in this paper

scale dependence of the coarse grained velocity derivative tensor structure in turbulence velocity fluctuations in hydrodynamic turbulence have a nontrivial structure characterized by correlations of the velocity gradient tensor in this paper we consider a phenomenological model incorporating the main features of hydrodynamic fluid turbulence aimed at predicting the structure of the velocity gradient tensor m coarse grained at a spatial scale r this model m chertkov a pumir and b i shraiman phys fluids 11 2394 1999 is formulated as a set of stochastic ordinary differential equations with three dimensionless parameters characterizing the reduction of the nonlinearity induced by the pressure term the reisotropization effect of the small scale velocity field and the influence of the small scales on the coarse grained velocity derivative tensor semiclassical solutions of this model are obtained and compared with direct numerical simulations dns data the dns data show that the joint probability distribution function of the second and third invariants of m becomes increasingly skewed as the scale r decreases in the inertial range the model results correctly reproduce this behavior provided the parameter that controls nonlinearity reduction is finely tuned the influence of the other parameters in the model is much weaker

a dual dual mixed formulation for nonlinear exterior transmission problems we combine a dual mixed finite element method with a dirichletto neumann mapping derived by the boundary integral equation method to study the solvability and galerkin approximations of a class of exterior nonlinear transmission problems in the plane as a model problem we consider a nonlinear elliptic equation in divergence form coupled with the laplace equation in an unbounded region of the plane our combined approach leads to what we call a dual dual mixed variational formulation since the main operator involved has itself a dual type structure we establish existence and uniqueness of solution for the continuous and discrete formulations and provide the corresponding error analysis by using raviart thomas elements the main tool of our analysis is given by a generalization of the usual babuska brezzi theory to a class of nonlinear variational problems with constraints

on the nonuniqueness of weak solution of the euler equation weak solution of the euler equations is defined as an l2 vector field satisfying the integral relations expressing the mass and momentum balance their general nature has been quite unclear in this work an example of a weak solution on a 2 dimensional torus is constructed that is identically zero outside a finite time interval this example is simpler and more transparent than the previous example of v scheffer j geom anal 3 4 1993 pp 343 401 1997 john wiley sons inc

convergence of an aefem for time harmonic maxwell equations with variable coefficients abstract an adaptive edge finite element method aefem for three dimensional indefinite time harmonic maxwell equations with variable coefficients is proved to be convergent and arbitrary order nedelec edge elements are considered a posteriori upper bound quasi orthogonality and the contraction of the error estimator are provided to prove the contraction of the sum of the energy error and the scaled error estimator between two consecutive adaptive loops numerical experiments are presented to support the theoretical results

numerical simulations of possible finite time singularities in the incompressible euler equations comparison of numerical methods abstract the numerical simulation of the 3d incompressible euler equations is analyzed with respect to different integration methods the numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods based on this comparison a kida pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given this estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective

an optimal iterative process for the johnson nedelec method of coupling boundary and finite elements we reformulate the discretization of the johnson nedelec method 11 of coupling boundary elements and finite elements for an exterior bidimensional laplacian this new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix we show that if the parameter of discretization is sufficiently small the fully discrete scheme is well posed and the error estimates remain unaltered the rest of the paper is devoted to the study of an efficient algorithm for solving the resulting discrete linear systems

resonance phenomenon for the galerkin truncated burgers and euler equations it is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial fourier modes having wave numbers in excess of a threshold k mathrm g exhibit unexpected features the study is carried out for both the one dimensional burgers equation and the two dimensional incompressible euler equation for large k mathrm g and smooth initial conditions the first symptom of truncation a localized short wavelength oscillation which we call a tyger is caused by a resonant interaction between fluid particle motion and truncation waves generated by small scale features shocks layers with strong vorticity gradients etc these tygers appear when complex space singularities come within one galerkin wavelength ensuremath lambda mathrm g 2 ensuremath pi k mathrm g from the real domain and typically arise far away from preexisting small scale structures at locations whose velocities match that of such structures tygers are weak and strongly localized at first in the burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to k mathrm g ensuremath 2 3 and k mathrm g ensuremath 1 3 respectively but grow and eventually invade the whole flow they are thus the first manifestations of the thermalization predicted by t d lee q j appl math 10 69 1952 the sudden dissipative anomaly the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t ensuremath star which is well known for the burgers equation and sometimes conjectured for the three dimensional euler equation has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit k mathrm g ensuremath rightarrow ensuremath infty this leads to reynolds stresses acting on scales larger than the galerkin wavelength and thus prevents the flow from converging to the inviscid limit solution there are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid limit behavior

singular weak and absent solutions of the euler equations we will describe necessary and sufficient conditions for blowup and discuss weak solutions for the incompressible euler equations we will also describe a result concerning anomalous dissipation of energy

dissipative anomalies in singular euler flows abstract we discuss onsager s conjecture that non vanishing energy dissipation in high reynolds number turbulence is associated to singular distributional solutions of the incompressible euler equations we carefully explain the physical and mathematical meaning of the conjecture and also review relevant theoretical experimental and numerical work emphasizing some of the dramatic successes of onsager s point of view finally we present several new ideas and results on lagrangian dynamics of circulations and vortex lines that we believe will be important for future progress

on the lagrangian dynamics of the axisymmetric 3d euler equations abstract we study the dynamics along the particle trajectories for the 3d axisymmetric euler equations in particular by rewriting the system of equations we find that there exists a complex riccati type of structure in the system on the whole of r 3 which generalizes substantially the previous results in 5 d chae on the blow up problem for the axisymmetric 3d euler equations nonlinearity 21 2008 2053 2060 using this structure of equations we deduce the new blow up criterion that the radial increment of pressure is not consistent with the global regularity of classical solution we also derive a much more refined version of the lagrangian dynamics than that of 6 d chae on the lagrangian dynamics for the 3d incompressible euler equations comm math phys 269 2 2007 557 569 in the case of axisymmetry

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

a quaternionic structure in the three dimensional euler and ideal magneto hydrodynamics equations by considering the three dimensional incompressible euler equations a 4 vector is constructed out of a combination of scalar and vector products of the vorticity and the vortex stretching vector u s the evolution equation for can then be cast naturally into a quaternionic riccati equation this is easily transformed into a quaternionic zero eigenvalue schrodinger equation whose potential depends on the hessian matrix of the pressure with minor modifications this system can alternatively be written in complex notation an infinite set of solutions of scalar zero eigenvalue schrodinger equations has been found by adler and moser which are discussed in the context of the present problem similarly when the equations for ideal magneto hydrodynamics mhd are written in elsasser variables a pair of 4 vectors are governed by coupled

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

convergence and optimality of adaptive edge finite element methods for time harmonic maxwell equations we consider a standard adaptive edge finite element method aefem based on arbitrary order nedelec edge elements for three dimensional indefinite time harmonic maxwell equations we prove that the aefem gives a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops provided the initial mesh is fine enough using the geometric decay we show that the aefem yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom the main technical contribution of the paper is in the establishment of a quasi orthogonality and a localized a posteriori error estimator

a fully discrete bem fem method for an exterior elasticity system in the plane abstract we present a modified version of the usual bem fem coupling for the exterior elasticity problem in the plane cf costabel and stephan siam j numer anal 27 1990 1212 1226 this new formulation allows us to take advantage of techniques from hsiao et al computing 25 1980 557 566 to compute the boundary integral terms using simple quadrature formulas we provide error estimates for the galerkin method and prove that the corresponding fully discrete scheme preserves the optimal rates of convergence

analysis of recovery type a posteriori error estimators for mildly structured grids some recovery type error estimators for linear finite elements are analyzed under o h 1 0 regular grids superconvergence of order o h 1 0 p is established for recovered gradients by three different methods as a consequence a posteriori error estimators based on those recovery methods are asymptotically exact

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

averaging for superconvergence verification and application of 2d edge elements to maxwell s equations in metamaterials abstract in this paper a simple averaging technique for edge elements is justified to improve the standard convergence rate hence achieving superconvergence when they are used to solve the maxwell s equations for simplicity here we focus on the lowest order triangular edge element which is widely used in practice though there exists no natural superconvergence points for the numerical solution obtained on such an edge element superconvergence can be obtained after a simple average of solutions over the neighboring elements a comprehensive analysis for the lowest order triangular edge element is carried out and one order higher convergence rate than the standard interpolation error estimate is proved for the averaged solution at midpoints of those interior edges of parallelograms formed by two triangles i e superconvergence happens at the parallelogram centers we also provide detailed analysis to explain why several cases do not have superconvergence extensive numerical results consistent with our analysis are presented

regularized transformation optics cloaking for the helmholtz equation from partial cloak to full cloak we develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the helmholtz equation in any space dimensions n geq 2 via the approach of transformation optics there are four major ingredients in our proposed theory 1 the non singular cloaking medium is obtained by the push forwarding construction through a transformation that blows up a subset k varepsilon in the virtual space where varepsilon ll 1 is an asymptotic regularization parameter k varepsilon will degenerate to k0 as varepsilon rightarrow 0 and in our theory k0 could be any convex compact set in mathbb r n or any set whose boundary consists of lipschitz hypersurfaces or a finite combination of those sets 2 a general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions 3 the contents being cloaked could also be extremely general possibly including at the same time generic mediums and sound soft sound hard and impedance type obstacles as well as some sources or sinks 4 in order to achieve a cloaking device of compact size particularly for the case when k varepsilon is not uniformly small an assembly by components the abc geometry is developed for both the virtual and physical spaces and the blow up construction is based on concatenating different components

new implementation techniques for the exterior stokes problem in the plane abstract we present a method to solve numerically two dimensional stokes problems on exterior domains our scheme is based on the fully discrete bem fem formulation proposed in 21 whose main advantage is that only elemental quadratures are used to approximate the weakly singular boundary integrals we show in this article that it is possible to maintain this important property without using curved triangles in the discretization process this modification makes the method easier to implement and the numerical experiments reveal that it still keeps the optimal order of convergence of the original scheme we also introduce in this paper a new iterative method to solve the complicated linear systems of equations that arises from this type of bem fem discretizations

an adaptive edge finite element method for electromagnetic cloaking simulation in this paper we develop an adaptive edge finite element method based on a reliable and efficient recovery type a posteriori error estimator for time harmonic maxwell equations the asymptotically exact a posteriori error estimator is based on the superconvergence result proved for the lowest order edge element on triangular grids where most pairs of triangles sharing a common edge form approximate parallelograms the efficiency and robustness of the proposed method is demonstrated by extensive numerical experiments for electromagnetic cloaking problems with highly anisotropic permittivity and permeability

cloaking via change of variables for the helmholtz equation in the whole space this paper is devoted to the study of a cloaking device that is composed of a standard near cloak based on a regularization of the transformation optics i e a change of variables that blows up a small ball to the cloaked region and a fixed lossy layer for the helmholtz equation in the whole space of dimension 2 or 3 with the outgoing condition at infinity we establish a degree of near invisibility which is independent of the content inside the cloaked region for this device we also show that the lossy layer is necessary to ensure the validity of the degree of near invisibility when no constraint on physical properties inside the cloaked region is imposed 2010 wiley periodicals inc

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence

the cauchy lagrangian method for numerical analysis of euler flow a novel semi lagrangian method is introduced to solve numerically the euler equation for ideal incompressible flow in arbitrary space dimension it exploits the time analyticity of fluid particle trajectories and requires in principle only limited spatial smoothness of the initial data efficient generation of high order time taylor coefficients is made possible by a recurrence relation that follows from the cauchy invariants formulation of the euler equation zheligovsky and frisch 2014 44 truncated time taylor series of very high order allow the use of time steps vastly exceeding the courant friedrichs lewy limit without compromising the accuracy of the solution tests performed on the two dimensional euler equation indicate that the cauchy lagrangian method is more and occasionally much more efficient and less prone to instability than eulerian runge kutta methods and less prone to rapid growth of rounding errors than the high order eulerian time taylor algorithm we also develop tools of analysis adapted to the cauchy lagrangian method such as the monitoring of the radius of convergence of the time taylor series certain other fluid equations can be handled similarly