analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

a hybrid method for hydrodynamic kinetic flow part ii coupling of hydrodynamic and kinetic models in this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic kinetic problems in which the knudsen number may span from equilibrium to highly rarefied regimes our approach is characterized by using the full boltzmann equation for the kinetic regime the compressible euler equations for equilibrium with a buffer zone in which the bgk es equation is used to represent the transition between fully kinetic to equilibrium flows in this fashion the boltzmann solver is used only when the collision integral is non stiff and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities thus in principle the same mesh size and time steps can be used in the whole computation moreover the time step is limited only by convective terms since the boltzmann solver is applied only in wholly kinetic regimes we use the reduced noise dsmc scheme we have proposed in part i of the present work this ensures a smooth exchange of information across the different domains with a natural way to construct interface numerical fluxes several tests comparing our hybrid scheme with full boltzmann dsmc computations show the good agreement between the two solutions on a wide range of knudsen numbers

on finite groups in which coprime commutators are covered by few cyclic subgroups the coprime commutators j and j were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders every element of a finite group g is both a 1 commutator and a 0 commutator now let j 2 and let x be the set of all elements of g that are powers of j 1 commutators an element g is a j commutator if there exist a x and b g such that g a b and a b 1 for j 1 let y be the set of all elements of g that are powers of j 1 commutators an element g is a j commutator if there exist a b y such that g a b and a b 1 the subgroups of g generated by all j commutators and all j commutators are denoted by j g and j g respectively for every j 2 the subgroup j g is precisely the last term g of the lower central series of g while for every j 1 the subgroup j g is precisely the last term of the lower central series of j 1 g that is j g j 1 g in the present paper we prove that if g possesses m cyclic subgroups whose union contains all j commutators of g then j g contains a subgroup of m bounded order which is normal in g and has the property that j g is cyclic if j 2 and g possesses m cyclic subgroups whose union contains all j commutators of g then the order of j g is m bounded

v cycle multigrid for cell centered finite differences we introduce and analyze a v cycle multigrid algorithm for cell centered finite difference methods applied to second order elliptic boundary value problems unlike conventional cell centered multigrid algorithms that use the natural injection operator for prolongation we use a new prolongation operator whose energy norm we prove is bounded by 1 in the constant coefficient case and 1 ch in the nonconstant case we are thus able to use general finite element multigrid theory to conclude that the v cycle either converges well or serves as a reasonably good preconditioner respectively while our theory does not establish optimal performance our numerical experiments do show that the resulting algorithm converges much faster than the conventional schemes in fact these results show that the energy norm convergence factor is small and remains bounded uniformly in the finest mesh size while that of the conventional algorithm grows

a note on convergence of the multigrid v cycle several recent papers have discussed the convergence of the multigrid v cycle in particular there are several results for the symmetric case where the numbers of smoothings before the fine to coarse transfer and after the coarse to fine transfer are the same in most instances the smoother h i e 1a has been limited to the case where e is positive definite and the eigenvalues h of h satisfy 0

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

a p cycle analysis of multigrid method we suggest a new strategy for multigrid iterations based on our analysis for p cycle multigrid method first we improve a recent result on multigrid v cycle analysis given by bramble and pasciak to p cycle multigrid method we show that the convergence rate improves slowly with p eventually bounded by a constant which is in agreement with many previous experiments based on the this observation we use p cycle only in the first few multigrid iterations which is more effective than the standard p cycle

commutators and commutator subgroups in profinite groups abstract let g be a profinite group we prove that the commutator subgroup g is finite by procyclic if and only if the set of all commutators of g is contained in a union of countably many procyclic subgroups

ellipsoidal bgk model for polyatomic molecules near maxwellians a dichotomy in the dissipation estimate abstract we consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal bgk model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic maxwellian we observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter reaches zero accordingly we employ two different sets of micro macro system to derive the full coercivity and close the energy estimate

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

parallel hybrid algebraic multilevel iterative methods abstract for the large scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self adjoint elliptic boundary value problem by making use of the blocked multilevel iteration idea we construct preconditioning matrices for the coefficient matrix and set up a class of parallel hybrid algebraic multilevel iterative methods for solving this kind of system of linear equations theoretical analysis shows that not only do these new methods lend themselves to parallel computation but also their convergence rates are independent of both the sizes and the level numbers of the grids and their computational work loads are also bounded by linear functions of the step sizes of the finest grids

the analysis of multigrid algorithms for pseudodifferential operators of order minus one multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one classical multigrid theory deals with the case of differential operators of positive order the pseudodifferential operator gives rise to a coercive form on h sup minus 1 2 omega effective multigrid algorithms are developed for this problem these algorithms are novel in that they use the inner product on h sup minus 1 omega as a base inner product for the multigrid development the authors show that the resulting rate of iterative convergence can at worst depend linearly on the number of levels in these novel multigrid algorithms in addition it is shown that the convergence rate is independent of the number of levels and unknowns in the case of a pseudodifferential operator defined by a single layer potential finally the results of numerical experiments illustrating the theory are presented 19 refs 1 fig 2 tabs

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

numerical comparison between the boltzmann and es bgk models for rarefied gases abstract rarefied gas flows obey the boltzmann equation but numerical simulations of this equation are not always possible so that simpler models have been introduced the es bgk equation is one of these models it gives the correct transport coefficients for the navier stokes approximation so that boltzmann or es bgk simulations are expected to give the same results for dense gases but in the case of a rarefied flow complete numerical comparisons are needed in this paper we present numerical comparisons between the two models in transitional regimes where the es bgk model is expected to be useful for reentry flows around a compression ramp and a plate we also emphasize that the es bgk model gives flow predictions closer to the boltzmann result than the simpler bgk model

two level hierarchical fem method for modeling passive microwave devices in recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh version of the finite element method in this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep version of the finite element analysis applied to indefinite problems a two levelv cycle algorithm has been implemented and studied with a gauss seidel iterative scheme used as a smoother the convergence of the method has been investigated and numerical results for a number of numerical examples are presented

a cal v cycle multigrid approach for mortar finite elements mortar methods based on dual lagrange multipliers provide a flexible tool for the numerical approximation of partial differential equations the associated finite element spaces are in general nonconforming and nonnested optimal multigrid results have previously been established for cal w cycle and the variable cal v cycle multigrid methods in this paper we introduce a new multigrid method based on a nested sequence of modified mortar spaces for which we can establish that the cal v cycle with one smoothing step has contraction numbers uniformly bounded away from one to obtain nested mortar spaces we apply a product form of certain corrections at the interfaces numerical results demonstrate the efficiency of the resulting multigrid solver

a multi grid method with a priori and a posteriori level choice for the regularization of nonlinear ill posed problems in this paper we study a multi grid method for the numerical solution of nonlinear systems of equations arising from the discretization of ill posed problems where the special eigensystem structure of the underlying operator equation makes it necessary to use special smoothers we provide uniform contraction factor estimates and show that a nested multigrid iteration together with an a priori or a posteriori chosen stopping index defines a regularization method for the ill posed problem i e a stable solution method that converges to an exact solution of the underlying infinite dimensional problem as the data noise level goes to zero with optimal rates under additional regularity conditions

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

multigrid in a weighted space arising from axisymmetric electromagnetics consider the space of two dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable this space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates we prove that if the original three dimensional domain is convex then the multigrid v cycle applied to the inner product in this space converges provided certain modern smoothers are used for the convergence analysis we first prove several intermediate results e g the approximation properties of a commuting projector in weighted norms and a superconvergence estimate for a dual mixed method in weighted spaces the uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments

microscopically implicit macroscopically explicit schemes for the bgk equation in this work a new class of numerical methods for the bgk model of kinetic equations is introduced the schemes proposed are implicit with respect to the distribution function while the macroscopic moments are evolved explicitly in this fashion the stability condition on the time step coincides with a macroscopic cfl evaluated using estimated values for the macroscopic velocity and sound speed thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either with the technique proposed here the updating of the distribution function requires the solution of a linear system of equations even though the bgk model is highly non linear thus the proposed schemes are particularly effective for high or moderate mach numbers where the macroscopic cfl condition is comparable to accuracy requirements we show results for schemes of order 1 and 2 and the generalization to higher order is sketched

on groups in which commutators are covered by finitely many cyclic subgroups abstract we show that if g is a group in which all commutators are contained in finitely many cyclic subgroups then g is either finite or cyclic

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

an application of the abstract multilevel theory to nonconforming finite element methods a multilevel product algorithm and its additive version or a multilevel preconditioner are proposed for the crouzeix raviart nonconforming finite element methods for second order elliptic equations the authors show that the preconditioned matrix has a condition number bounded uniformly from above if the domain can be partitioned into overlapping convex subdomains a uniform convergence for the multilevel product algorithm is also established the results are regularity free for the continuous problem and can be applied to problems with rough solutions and local grid refinement numerical results that illustrate the theory are presented

cauchy problem for the ellipsoidal bgk model of the boltzmann equation ellipsoidal bgk model es bgk is a generalized version of the original bgk model designed to reproduce the physically correct prandtl number in the navier stokes limit in this paper we study the cauchy problem for the es bgk model under the condition of finite initial mass energy and entropy equivalence type estimates for the temperature tensor are crucially used

multigrid algorithms for a vertex centered covolume method for elliptic problems we analyze v cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem as an application we study the convergence of multigrid algorithms for a covolume method or a vertex centered finite volume element method for variable coefficient elliptic problems on polygonal domains as in standard finite element methods the v cycle algorithm with one pre smoothing converges with a rate independent of the number of levels various types of smoothers including point or line jacobi and gauss seidel relaxation are considered

on profinite groups in which commutators are covered by finitely many subgroups for a family of group words w we show that if g is a profinite group in which all w values are contained in a union of finitely many subgroups with a prescribed property then the verbal subgroup w g has the same property as well in particular we show this in the case where the subgroups are periodic or of finite rank if g contains finitely many subgroups g1 g2 gs of finite exponent e whose union contains all k values in g it is shown that k g has finite e k s bounded exponent if g contains finitely many subgroups g1 g2 gs of finite rank r whose union contains all k values it is shown that k g has finite k r s bounded rank

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

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

coupling boltzmann and navier stokes equations by friction the aim of this paper is to introduce and validate a coupled navier stokes boltzman approach for the calculation of hypersonic rarefied flows around manoeuvering vehicles the proposed strategy uses locally a kinetic model in the boundary layer coupled through wall friction forces to a global navier stokes solver dif ferent numerical experiments illustrate the potentialities of the method

remarks on multigrid convergence theorems multigrid has become an important iterative method for the solution of discrete elliptic equations however there is much to be done in the theory of convergence proofs at the present time there are two general two level methods for general convergence proofs an algebraic approach and a duality approach while these theories do not give sharp estimates they provide good general rigorous convergence theorems in this note we study the relationship between these theories while the approach and thought process leading to these theories are different the results are essentially the same indeed the basic estimates required by these theories are the same

a two level algorithm for the weak galerkin discretization of diffusion problems this paper analyzes a two level algorithm for the weak galerkin wg finite element methods based on local raviart thomas rt and brezzi douglas marini bdm mixed elements for two and three dimensional diffusion problems with dirichlet condition we first show the condition numbers of the stiffness matrices arising from the wg methods are of o h 2 we use an extended version of the xu zikatanov xz identity to derive the convergence of the algorithm without any regularity assumption finally we provide some numerical results

classical solutions for the ellipsoidal bgk model with fixed collision frequency abstract we establish the existence of global in time smooth solutions for the ellipsoidal bgk model which is a variant of the bgk model for the boltzmann equation designed to yield the correct prandtl number in the hydrodynamic approximation at the navier stokes level for this we carefully design a function space which captures the growth of the solution in a weighted sobolev norm and show that the ellipsoidal relaxation operator is lipschitz continuous in the induced metric this approach is restricted to the case when the collision frequency does not depend on the macroscopic field but no smallness on the initial data is required

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

multigrid analysis of finite element methods with numerical integration the author analyzes multigrid convergence rates when elliptic boundary value problems are discretized using finite element methods with numerical integration the resulting discrete problem does not fall into the standard variational framework for analyzing multigrid methods since the bilinear forms on different grid levels are not suitably related to each other he first discusses extensions of the standard variational multigrid theory and then apply these results to the case of numerical quadrature in particular it is shown that the v cycle algorithm has a convergence rate independent of grid size under suitable conditions

on the multigrid f cycle in a recent paper a bound was derived on the convergence of the multigrid v cycle for the case when the solution is in the sobolev space h 1 a but not in h 1 a a 62 a showing that the convergence factor approaches one only as 1 o k a 1 a for a large number of levels k we now extend the technique to obtain the asymptotically better bound 1 o k 1 a 2 a on the multigrid f cycle we also show that in many cases for practical values of k one gets the same bound for the f cycle as for the v cycle with a 1

multigrid convergence for nonsymmetric indefinite variational problems and one smoothing step we prove that the multigrid method for linear systems arising from nonsymmetric and indefinite elliptic variational problems converges for all sufficiently small h with one smoothing step only previous proofs required that the number of smoothing steps be within certain bounds which were quite difficult to compute our theory is based on sharp energy norm convergence estimates for the symmetric definite case and lower order terms are treated as perturbations the resulting estimates are so sharp that we obtain even the convergence of the v cycle

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

a hybrid particle continuum method applied to shock waves a hybrid numerical scheme designed for hypersonic non equilibrium flows is presented which solves the navier stokes equations in regions of near equilibrium and uses the direct simulation monte carlo method where the flow is in non equilibrium detailed analysis of each stage of the hybrid cycle illustrates the difficulty in defining physically correct dsmc boundary conditions in regards to both macroscopic state and velocity distribution however results also show that dsmc boundary conditions have little effect on a previously initialized interior particle domain a sub relaxation technique capable of determining macroscopic hydrodynamic properties in a dsmc simulation is used to determine low scatter boundary conditions for the ns domain particle and continuum domains adapt during the hybrid simulation through application of a continuum breakdown parameter based on the gradient length knudsen number the hybrid code reproduces experimental results and full dsmc simulations in half the time for a large range of 1d shock waves in argon and diatomic nitrogen gas

nonconforming multigrid method for nonsymmetric and indefinite problems abstract the convergence of w cycle multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems using p 1 nonconforming finite elements is proved

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

finsler geometric local indicator of chaos for single orbits in the henon heiles hamiltonian translating the dynamics of the h enon heiles hamiltonian as a geodesic flow on a finsler manifold we obtain a local and synthetic geometric indicator of chaos gic which represents a link between local quantities and asymptotic behavior of orbits and gives a strikingly evident one to one correspondence between geometry and instability going beyond the results attained using the customary dynamical approach and improving also on the global criteria established within the riemannian framework the gic is able to discriminate between regular and stochastic orbits on a given energy surface simply on the basis of the value it assumes along a relatively small piece of the trajectory without long integrations of the dynamics and without any reference to a perturbation

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

regularity and multigrid analysis for laplace type axisymmetric equations consider axisymmetric equations associated with laplace type operators we establish full regularity estimates in high order kondrat vetype spaces for possible singular solutions due to the non smoothness of the domain and to the singular coefficients in the operator then we show suitable graded meshes can be used in high order finite element methods to achieve the optimal convergence rate even when the solution is singular using these results we further propose multigrid v cycle algorithms solving the system from linear finite element discretizations on optimal graded meshes we prove the multigrid algorithm is a contraction with the contraction number independent of the mesh level numerical tests are provided to verify the theorem

multigrid convergence for a singular perturbation problem abstract so far there has been no analysis of multigrid methods applied to singularly perturbed dirichlet boundary value problems only for periodic boundary conditions does the fourier transformation mode analysis apply and it is not obvious that the convergence results carry over to the dirichlet case since the eigenfunctions are quite different in the two cases in this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the dirichlet case

the convergence rate of a multigrid method with gauss seidel relaxation for the poisson equation the numerical solution of the poisson equation is treated by a multigrid method for a uniform grid the convergence rate can be estimated even for the iteration with a v cycle independently of the shape of the domain as long as it is convex and polygonal the smoothing effect of the gaus seidel relaxation is described by a discrete seminorm which is weaker than the energy norm

convergence of a semi lagrangian scheme for the bgk model of the boltzmann equation recently a new class of semi lagrangian methods for the bgk model of the boltzmann equation has been introduced f filbet and g russo kinet relat models 2 2009 pp 231 250 g russo and p santagati a new class of large time step methods for the bgk models of the boltzmann equation arxiv 1103 5247 p santagati high order semi lagrangian methods for the bgk model of the boltzmann equation ph d thesis university of catania italy 2007 these methods work in a satisfactory way either in a rarefied or a fluid regime moreover because of the semi lagrangian feature the stability property is not restricted by the cfl condition these aspects make them very attractive for practical applications in this paper we prove that the discrete solution of the scheme converges in a weighted l 1 norm to the unique smooth solution by deriving an explicit error estimate

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

monotone enclosure for nonlinear pdes using multigrid techniques two possibilities to include numerical solutions of weakly nonlinear elliptic boundary value problems are considered if one single equation is given it is usually more efficient to solve it by nonlinear multigrid methods than to linearize it and to use a linear multigrid algorithm afterwards to get a monotone including nonlinear multigrid method we present a modified fas called mmg3 if a weakly coupled system of equations is given it is sometimes more convenient to linearize it with a conventional monotone iteration method and to use a linear multigrid algorithm as inner iteration this is demonstrated here by considering volterra lotka systems with diffusion

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

a parallel geometric multigrid method for finite elements on octree meshes in this article we present a parallel geometric multigrid algorithm for solving variable coefficient elliptic partial differential equations on the unit box with dirichlet or neumann boundary conditions using highly nonuniform octree based conforming finite element discretizations our octrees are 2 1 balanced that is we allow no more than one octree level difference between octants that share a face edge or vertex we describe a parallel algorithm whose input is an arbitrary 2 1 balanced fine grid octree and whose output is a set of coarser 2 1 balanced octrees that are used in the multigrid scheme also we derive matrix free schemes for the discretized finite element operators and the intergrid transfer operations the overall scheme is second order accurate for sufficiently smooth right hand sides and material properties its complexity for nearly uniform trees is mathcal o frac n n p log frac n n p mathcal o n p log n p where n is the number of octree nodes and n p is the number of processors our implementation uses the message passing interface standard we present numerical experiments for the laplace and navier linear elasticity operators that demonstrate the scalability of our method our largest run was a highly nonuniform 8 billion unknown elasticity calculation using 32 000 processors on the teragrid system ranger at the texas advanced computing center our implementation is publically available in the dendro library which is built on top of the petsc library from argonne national laboratory

a conjugate gradient method and a multigrid algorithm for morley s finite element approximation of the biharmonic equation the numerical solution of the linear equations arising from morley s nonconforming displacement method is studied a suitable preconditioning for the conjugate gradient method is described furthermore the nonconformity of the discretization necessitates some non standard ingredients of multigrid algorithms

convergence analysis of perturbed two grid and multigrid methods we consider multigrid methods for symmetric positive definite linear systems we present a new algebraic convergence analysis of two grid schemes with inexact solution of the coarse grid system this analysis allows us to bound the convergence factor of such perturbed two grid schemes assuming only a certain bound on the convergence factor for the unperturbed scheme with exact solution of the coarse grid system applied to multigrid methods with the standard w cycle this analysis shows that if the convergence factor of the unperturbed two grid method is uniformly bounded by sigma 1 2 then the convergence factor of the multigrid method is uniformly bounded by sigma 1 sigma the analysis is purely algebraic and requires only that pre and postsmoothing are applied in a symmetric way it covers both geometric and algebraic multigrid methods and the coarse grid matrix may be of any type not necessarily galerkin

an optimal order multigrid method for p1 nonconforming finite elements etude d une methode multigrille d ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l aide d elements finis non conformes

gas kinetic numerical studies of three dimensional complex flows on spacecraft re entry the gas kinetic numerical algorithm solving the boltzmann model equation is extended and developed to study the three dimensional hypersonic flows of spacecraft re entry into the atmosphere in perfect gas in this study the simplified velocity distribution function equation for various flow regimes is presented on the basis of the kinetic boltzmann shakhov model the discrete velocity ordinate technique and numerical quadrature methods such as the gauss quadrature formulas with the weight function 2 p 1 2exp v 2 and the gauss legendre numerical quadrature rule are studied to resolve the barrier in simulating complex flows from low mach numbers to hypersonic problems specially the gas kinetic finite difference scheme is constructed for the computation of three dimensional flow problems which directly captures the time evolution of the molecular velocity distribution function the gas kinetic boundary conditions and numerical procedures are studied and implemented by directly acting on the velocity distribution function the hpf high performance fortran parallel implementation technique for the gas kinetic numerical method is developed and applied to study the hypersonic flows around three dimensional complex bodies the main purpose of the current research is to provide a way to extend the gas kinetic numerical algorithm to the flow computation of three dimensional complex hypersonic problems with high mach numbers to verify the current method and simulate gas transport phenomena covering various flow regimes the three dimensional hypersonic flows around sphere and spacecraft shape with different knudsen numbers and mach numbers are studied by hpf parallel computing excellent results have been obtained for all examples computed

a multilevel algorithm for the biharmonic problem a finite element discretization of the mixed variable formulation of the biharmonic problem is considered a multilevel algorithm for the numerical solution of the discrete equations is described convergence is proved under the assumption ofh 3 regularity

existence uniqueness and positivity of solutions for bgk models for mixtures abstract we consider kinetic models for a multi component gas mixture without chemical reactions in the literature one can find two types of bgk models in order to describe gas mixtures one type has a sum of bgk type interaction terms in the relaxation operator for example the model described by klingenberg pirner and puppo 20 which contains well known models of physicists and engineers for example hamel 16 and gross and krook 15 as special cases the other type contains only one collision term on the right hand side for example the well known model of andries aoki and perthame 1 for each of these two models 20 and 1 we prove existence uniqueness and positivity of solutions in the first part of the paper in the second part we use the first model 20 in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by dellacherie 11

split level iteration many techniques are used for solving large sparse linear systems some of the most popular methods for solving the more difficult problems combine direct and iterative algorithms to yield efficient multistage procedures although each stage is standard the manner in which stages are combined is often novel both multigrid and preconditioned conjugate gradients fall into this class of numerical methods in lo two promising algorithms were described with supportive numerical studies the first was cyclic application of model problem adi iteration followed by a conjugate gradient iteration on the deviation of the model problem from the actual problem some new developments relating to this procedure are described in a companion paper in this journal 121 the second was a two level iteration in which a variational correction with a coarse grid basis was applied cyclically to the result of conventional iteration over a fine grid this was the first published implementation of a multigrid method based on the variational techniques that provide the foundations for finite element computation in these initial two level variational studies certain advantages of additive over multiplicative coarse mesh correction were not recognized the coefficient matrix for additive correction does not depend on the fine mesh iterate and need thus be computed only once moreover it need only be factored once for direct solution the right hand side of the correction equations varies each cycle brandt 4 and others have exploited this in multigrid and related methods some numerical comparisons of additive and multiplicative correction for two level iteration are given in 111 also it was not recognized that under suitable conditions one may use the two level iteration as a preconditioner for conjugate gradients thus combining the two initially distinct methods as suggested subsequently in 1 7 9 some insight into the nature of such an iteration may be gained by a qualitative analysis of error behavior the fine mesh iteration acts primarily on high mcde error components and the coarse mesh correction reduces low mode components in some implementations the two iteration matrices commute but in most they do not in the commuting case careful choice of iteration parameters precludes benefits from the conjugate gradient third stage in the noncommuting case the conjugate gradient stage should act on error persisting because of component mixing between the other two stages the automatic extrapolation in the conjugate gradient method greatly reduces convergence sensitivity to choice of iteration parameters for the fine mesh iteration

a preconditioned gmres method in this paper we show the convergence of gmres method for nonsymmetric problems preconditioned with a multigrid method based on the original operator this is simpler and more natural to implement than using a preconditioner based on the principal part as suggested by cai and xu

multigrid algorithms for c 0 interior penalty methods multigrid algorithms for c 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper it is shown that v cycle f cycle and w cycle algorithms are contractions if the number of smoothing steps is sufficiently large the contraction numbers of these algorithms are bounded by cm alpha where m is the number of presmoothing and postsmoothing steps alpha is the index of elliptic regularity and the positive constant c is mesh independent these estimates are established for a smoothing scheme that uses a poisson solve as a preconditioner which can be easily implemented because the c 0 finite element spaces are standard spaces for second order problems furthermore the variable v cycle algorithm is also shown to be an optimal preconditioner

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

cauchy problem for the ellipsoidal bgk model for polyatomic particles abstract we establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal bgk model which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level

a coupled ordinates method for solution acceleration of rarefied gas dynamics simulations non equilibrium rarefied flows are frequently encountered in a wide range of applications including atmospheric re entry vehicles vacuum technology and microscale devices rarefied flows at the microscale can be effectively modeled using the ellipsoidal statistical bhatnagar gross krook esbgk form of the boltzmann kinetic equation numerical solutions of these equations are often based on the finite volume method fvm in physical space and the discrete ordinates method in velocity space however existing solvers use a sequential solution procedure wherein the velocity distribution functions are implicitly coupled in physical space but are solved sequentially in velocity space this leads to explicit coupling of the distribution function values in velocity space and slows down convergence in systems with low knudsen numbers furthermore this also makes it difficult to solve multiscale problems or problems in which there is a large range of knudsen numbers in this paper we extend the coupled ordinates method comet previously developed to study participating radiative heat transfer to solve the esbgk equations in this method at each cell in the physical domain distribution function values for all velocity ordinates are solved simultaneously this coupled solution is used as a relaxation sweep in a geometric multigrid method in the spatial domain enhancements to comet to account for the non linearity of the esbgk equations as well as the coupled implementation of boundary conditions are presented the methodology works well with arbitrary convex polyhedral meshes and is shown to give significantly faster solutions than the conventional sequential solution procedure acceleration factors of 5 9 are obtained for low to moderate knudsen numbers on single processor platforms

time metric equivalence and dimension change under time reparameterizations we study the behavior of dynamical systems under time reparameterizations which is important not only to characterize chaos in relativistic systems but also to probe the invariance of dynamical quantities we first show that time transformations are locally equivalent to metric transformations a result that leads to a transformation rule for all lyapunov exponents on arbitrary riemannian phase spaces we then show that time transformations preserve the spectrum of generalized dimensions d q except for the information dimension d1 which interestingly transforms in a nontrivial way despite previous assertions of invariance the discontinuous behavior at q 1 can be used to constrain and extend the formulation of the kaplan yorke conjecture

a generalized bpx multigrid framework covering nonnested v cycle methods more than a decade ago bramble pasciak and xu developed a framework in analyzing the multigrid methods with nonnested spaces or non inherited quadratic forms it was subsequently known as the bpx multigrid framework which was widely used in the analysis of multigrid and domain decomposition methods however the framework has an apparent limit in the analysis of nonnested v cycle methods and it produces a variable v cycle or nonuniform convergence rate v cycle methods or other nonoptimal results in analysis thus far this paper completes a long time effort in extending the bpx multigrid framework so that it truly covers the nonnested v cycle we will apply the extended bpx framework to the analysis of many v cycle nonnested multigrid methods some of them were proven previously only for two level and w cycle iterations some numerical results are presented to support the theoretical analysis of this paper

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

iterative solution methods for plate bending problems multigrid and preconditioned cg algorithm the numerical solution of the linear equations arising from a discretization of the plate bending problem based on the zienkiewicz triangle is studied the finite element scheme proceeds from the mindlin reissner formulation with modified shear energy however the kirchhoff condition is imposed on discrete points a multigrid algorithm is analyzed and numerical examples are provided furthermore a suitable preconditioning is established for the use of conjugate gradients

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

overcoming corner singularities using multigrid methods we consider the poisson equation delta u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain o we develop a finite element multigrid method on quasi uniform grids that obtains cal o h m 1 epsilon convergence in the h 1 o norm for any positive epsilon when f in h m o the cost of the method is proportional to the number of elements in the triangulation the results of this paper can be generalized to other equations and other boundary conditions

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

verbal sets and cyclic coverings abstract we consider groups g such that the set of all values of a fixed word w in g is covered by a finite set of cyclic subgroups fernandez alcober and shumyatsky studied such groups in the case when w is the word x 1 x 2 and proved that in this case the corresponding verbal subgroup g is either cyclic or finite answering a question asked by them we show that this is far from being the general rule however we prove a weaker form of their result in the case when w is either a lower commutator word or a non commutator word showing that in the given hypothesis the verbal subgroup w g must be finite by cyclic even this weaker conclusion is not universally valid it fails for verbose words

relativistic invariance of lyapunov exponents in bounded and unbounded systems the study of chaos in relativistic systems has been hampered by the observer dependence of lyapunov exponents les and of conditions such as orbit boundedness invoked in the interpretation of les as indicators of chaos here we establish a general framework that overcomes both difficulties and apply the resulting approach to address three fundamental questions how les transform under lorentz and rindler transformations and under transformations to uniformly rotating frames the answers to the first and third questions show that inertial and uniformly rotating observers agree on a characterization of chaos based on les the second question on the other hand is an ill posed problem due to the event horizons inherent to uniformly accelerated observers