bddc and feti dp preconditioners for spectral element discretizations abstract two of the most recent and important nonoverlapping domain decomposition methods the bddc method balancing domain decomposition by constraints and the feti dp method dual primal finite element tearing and interconnecting are here extended to spectral element discretizations of second order elliptic problems in spite of the more severe ill conditioning of the spectral element discrete systems compared with low order finite elements and finite differences these methods retain their good properties of scalability quasi optimality and independence on the discontinuities of the elliptic operator coefficients across subdomain interfaces

numerical analysis of conservative and entropy schemes for the fokker planck landau equation conservatives and entropy schemes for the fokker planck landau fpl equation are studied we prove the existence of a unique positive and global in time solution for the homogeneous linear and nonlinear discretized either in the velocity space or both in the velocity space and in time fpl equation the stability analysis of these schemes leads to sufficient conditions on the time step that guarantee positivity and entropy decay

an asymptotic preserving monte carlo method for the boltzmann equation abstract in this work we propose an asymptotic preserving monte carlo method for the boltzmann equation that is more efficient than the currently available monte carlo methods in the fluid dynamic regime this method is based on the successive penalty method 39 which is an improved bgk penalization method originally proposed by filbet and jin 16 here we introduce the monte carlo implementation of the method which despite its lower order accuracy is very efficient in higher dimensions or simulating some complicated chemical processes this method allows the time step independent of the mean free time which is prohibitively small in the fluid dynamic regime we study some basic properties of this method and compare it with some other asymptotic preserving monte carlo methods in terms of numerical performance in different regimes from rarefied to fluid dynamic regimes and their computational efficiency

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

dual primal feti methods for three dimensional elliptic problems with heterogeneous coefficients in this paper certain iterative substructuring methods with lagrange multipliers are considered for elliptic problems in three dimensions the algorithms belong to the family of dual primal finite element tearing and interconnecting feti methods which recently have been introduced and analyzed successfully for elliptic problems in the plane the family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual primal preconditioner it is shown that the condition numbers of several of the dual primal feti methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains the mesh size and jumps in the coefficients these results closely parallel those of other successful iterative substructuring methods of primal as well as dual type

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

conservative finite difference schemes for the fokker planck equations not violating the law of an increasing entropy abstract two finite difference schemes for the fokker planck equation in the case of an isotropic function of the charge particles are constructed these schemes are first order accurate with respect to time and second order accurate with respect to a e 2 the schemes conserve the number of particles and energy and the entropy change is nonnegative a relation between the incresing entropy law and the stability of the finite difference schemes is established

low energy basis preconditioning for elliptic substructured solvers based on unstructured spectral hp element discretization abstract the development and application of three dimensional unstructured hierarchical spectral hp element algorithms has highlighted the need for efficient preconditioning for elliptic solvers building on the work of bica ph d thesis courant institute new york university 1997 we have developed an efficient preconditioning strategy for substructured solvers based on a transformation of the expansion basis to a low energy basis in this numerically derived basis the strong coupling between expansion modes in the original basis is reduced thus making it amenable to block diagonal preconditioning the efficiency of the algorithm is maintained by developing the new basis on a symmetric reference element and ignoring in the preconditioning step the role of the jacobian of the mapping from the reference to the global element by applying an additive schwarz block preconditioner to the low energy basis combined with a coarse space linear vertex solver we have observed reductions in execution time of up to three times for tetrahedral elements and 10 times for prismatic elements when compared to a standard diagonal preconditioner full details of the implementation and validation of the tetrahedral and prismatic element preconditioning strategy are set out below

bddc and feti dp without matrices or vectors abstract the connection between the bddc balancing domain decomposition by constraints algorithm and the feti dp finite element tearing and interconnecting dual primal algorithm is discussed in the language of function spaces operators and linear functionals

asymptotic preserving schemes for kinetic fluid modeling of disperse two phase flows abstract we consider a system coupling the incompressible navier stokes equations to the vlasov fokker planck equation such a problem arises in the description of particulate flows we design a numerical scheme to simulate the behavior of the system this scheme is asymptotic preserving thus efficient in both the kinetic and hydrodynamic regimes it has a numerical stability condition controlled by the non stiff convection operator with an implicit treatment of the stiff drag term and the fokker planck operator yet consistent to a standard asymptotic preserving fokker planck solver or an incompressible navier stokes solver only the conjugate gradient method and fast poisson and helmholtz solvers are needed numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme with several interesting applications

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

cumulant approach of arbitrary truncated levy flight abstract the problem of an arbitrary truncated levy flight description using the method of cumulant approach has been solved the set of cumulants of the truncated levy distribution given the assumption of arbitrary truncation has been found the influence of truncation shape on the truncated levy flight properties in the gaussian and the levy regimes has been investigated

exponential decay for soft potentials near maxwellian we consider both soft potentials with angular cutoff and landau collision kernels in the boltzmann theory inside a periodic box we prove that any smooth perturbation near a given maxwellian approaches zero at the rate of e lambda t p for some 0 and 0 p 1 our method is based on an unified energy estimate with appropriate exponential velocity weight our results extend the classical result caflisch of 2 to the case of very soft potential and coulomb interactions and also improve the recent almost exponential decay results by 5 14

a numerical method for the accurate solution of the fokker planck landau equation in the nonhomogeneous case abstract a new approach for the accurate numerical solution of the fokker planck landau fpl equation in the nonhomogeneous case is presented the method couples through a time splitting algorithm a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator recently introduced the scheme allows the use of different grids in the velocity space for the transport and the collision phases the use of a suitable explicit runge kutta solver for the time integration of the collision phase permits avoid once of excessive small time steps induced by the stiffness of the diffusive collision operator numerical results for both space homogeneous and space nonhomogeneous situations show the efficiency and the accuracy of the present method

implicit schemes for the fokker planck landau equation we propose time implicit schemes to solve the homogeneous fokker planck landau equation in both the isotropic and 3 dimensional geometries these schemes have properties of conservation and entropy moreover they allow for large time steps making them faster than the usual explicit schemes to solve the involved linear systems we prove that the use of krylov like solvers preserves the conservation properties we show in particular that the conjugate gradient method can be used numerical tests are performed for the isotropic case and demonstrate an important gain in terms of cpu time with the same accuracy as explicit schemes this work is a first step to the development of fast implicit schemes to solve a class of inhomogeneous kinetic equations

discretization methods for one dimensional fokker planck operators abstract several new numerical methods for solving a general class of linear and nonlinear 1 dimensional time dependent fokker planck equations are suggested these methods are positive simple to implement and equilibrium preserving in the sense that every equilibrium solution of the analytic equations is also an equilibrium solution of the discretized equations the new methods are all applied to the nonlinear problem of compton and inverse compton scattering and numerical results are compared our numerical comparison includes results from a standard technique now in use the chang cooper method and we discuss the relationship between this technique and the new methods in addition we point out certain shortcomings of the chang cooper technique which these new methods remedy

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

a class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources in this paper we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the boltzmann kinetic and related equations numerically solving these equations are challenging due to the nonlinear stiff collision source terms induced by small mean free or relaxation time we propose to penalize the nonlinear collision term by a bgk type relaxation term which can be solved explicitly even if discretized implicitly in time moreover the bgk type relaxation operator helps to drive the density distribution toward the local maxwellian thus naturally imposes an asymptotic preserving scheme in the euler limit the scheme so designed does not need any nonlinear iterative solver or the use of wild sum it is uniformly stable in terms of the possibly small knudsen number and can capture the macroscopic fluid dynamic euler limit even if the small scale determined by the knudsen number is not numerically resolved it is also consistent to the compressible navier stokes equations if the viscosity and heat conductivity are numerically resolved the method is applicable to many other related problems such as hyperbolic systems with stiff relaxation and high order parabolic equations

a successive penalty based asymptotic preserving scheme for kinetic equations we propose an asymptotic preserving ap scheme for kinetic equations that is efficient also in the hydrodynamic regimes this scheme is based on the bhantnagar gross krook bgk penalty method introduced by filbet and jin j comput phys 229 2010 pp 7625 7648 but uses the penalization successively to achieve the desired asymptotic property this method possesses a stronger ap property than the original method of filbet and jin with the additional feature of being positivity preserving when applied on the boltzmann equation it is also general enough to be applicable to several important classes of kinetic equations including the boltzmann equation and the landau equation numerical experiments verify these properties

an entropy scheme for the fokker planck collision operator of plasma kinetic theory we propose a finite difference scheme to approximate the fokker planck collision operator in 3 velocity dimensions the principal feature of this scheme is to provide a decay of the numerical entropy as a consequence it preserves the collisional invariants and its stationary solutions are the discrete maxwellians we consider both the whole velocity space problem and the bounded velocity problem in the latter case we provide artificial boundary conditions which preserve the decay of the entropy

quasi optimal schwarz methods for the conforming spectral element discretization the spectral element method is used to discretize self adjoint elliptic equations in three dimensional domains the domain is decomposed into hexahedral elements and in each of the elements the discretization space is the set of polynomials of degree n in each variable a conforming galerkin formulation is used the corresponding integrals are computed approximately with gauss lobatto legendre gll quadrature rules of order n and a lagrange interpolation basis associated with the gll nodes is used fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method the conforming finite element space on the gll mesh consisting of piecewise q1 or p1 functions produces a stiffness matrix kh that is known to be spectrally equivalent to the spectral element stiffness matrix kn kh is replaced by a preconditioner tilde k h which is well adapted to parallel computer architectures the preconditioned operator is then tilde k h 1 k n techniques for nonregular meshes are developed which make it possible to estimate the condition number of tilde k h 1 k n where tilde k h is a standard finite element preconditioner of kh based on the gll mesh two finite element based preconditioners the wirebasket method of smith and the overlapping schwarz algorithm for the spectral element method are given as examples of the use of these tools numerical experiments performed by pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions

a stochastic asymptotic preserving scheme for a kinetic fluid model for disperse two phase flows with uncertainty in this paper we consider a kinetic fluid model for disperse two phase flows with uncertainty we propose a stochastic asymptotic preserving s ap scheme in the generalized polynomial chaos stochastic galerkin gpc sg framework which allows the efficient computation of the problem in both kinetic and hydrodynamic regimes the s ap property is proved by deriving the equilibrium of the gpc version of the fokker planck operator the coefficient matrices that arise in a helmholtz equation and a poisson equation essential ingredients of the algorithms are proved to be positive definite under reasonable and mild assumptions the computation of the gpc version of a translation operator that arises in the inversion of the fokker planck operator is accelerated by a spectrally accurate splitting method numerical examples illustrate the s ap property and the efficiency of the gpc sg method in various asymptotic regimes

on the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds we prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails in particular boltzmann type equations with smoothed soft potentials we compensate the lack of uniform in time estimates by the use of precise logarithmic sobolev type inequalities and the assumption that the initial datum decays rapidly at large velocities our method not only gives explicit results on the times of convergence but is also able to cover situations in which compactness arguments apparently do not apply even mere convergence to equilibrium was an open problem for soft potentials

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

the mosaic of high performance domain decomposition methods for structural mechanics formulation interrelation and numerical efficiency of primal and dual methods a multitude of domain decomposition methods ddm for structural mechanics is available in the literature today a unified framework for formulating primal and dual ddm is thus presented in this paper aiming at providing a mathematical platform for a uniform treatment of high performance ddm in structural mechanics a novel approach for developing new ddm from existing methods is also proposed and is applied to dual and primal methods in the field of the feti methods this approach leads to a new category of methods derived from existing feti variants furthermore two alternative formulations of the balancing domain decomposition method are described while interrelations between the introduced and existing methods are established finally comparative numerical tests demonstrate the differences in the computational performance of the methods in question

numerical approximation of the spatially homogeneous fokkerplancklandau equation a numerical method is developed for the spatially homogeneous fokkerplancklandau fpl equation for the case of coulomb collisions the equation is put into a form so that finite difference methods for parabolic type pdes can be applied through a transformation of variables the velocity space computation is reduced to a bounded domain and finite differencing is done with a well defined exact boundary condition corresponding to v at infinity for the discretization in time implicit differencing is used an analysis is carried out on the discrete approximation as applied to a linear fpl equation and for the linear equation it is proved that the numerical method is unconditionally stable and convergent computational work is then done that demonstrates the unconditional stability and accuracy of the numerical method for the nonlinear fpl equation

a nodal triangle based spectral element method for the shallow water equations on the sphere a nodal triangle based spectral element se method for the shallow water equations on the sphere is presented the original se method uses quadrilateral elements and high order nodal lagrange polynomials constructed from a tensor product of the legendre gauss lobatto points in this work we construct the high order lagrange polynomials directly on the triangle using nodal sets obtained from the electrostatics principle j s hesthaven from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex siam journal on numerical analysis 35 1998 655 676 and fekete points m a taylor b a wingate r e vincent an algorithm for computing fekete points in the triangle siam journal on numerical analysis 38 2000 1707 1720 these points have good approximation properties and far better lebesgue constants than any other nodal set derived for the triangle by employing triangular elements as the basic building blocks of the se method and the cartesian coordinate form of the equations we can use any grid imaginable including adaptive unstructured grids results for six test cases are presented to confirm the accuracy and stability of the method the results show that the triangle based se method yields the expected exponential convergence and that it can be more accurate than the quadrilateral based se method even while using 30 60 fewer grid points especially when adaptive grids are used to align the grid with the flow direction however at the moment the quadrilateral based se method is twice as fast as the triangle based se method because the latter does not yield a diagonal mass matrix

sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation we derive a new lower bound for the entropy dissipation associated with the spatially homogeneous boltzmann equation this bound is expressed in terms of the relative entropy with respect to the equilibrium and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy with an explicit rate our result gives a considerable refinement of the analogous estimate by carlen and carvalho 9 10 under very little additional assumptions our proof takes advantage of the structure of boltzmann s collision operator with respect to the tensor product and its links with fokker planck and landau equations several variants are discussed

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

positive definite balancing neumann neumann preconditioners for nearly incompressible elasticity in this paper a positive definite balancing neumann neumann bnn solver for the linear elasticity system is constructed and analyzed the solver implicitly eliminates the interior degrees of freedom in each subdomain and solves iteratively the resulting schur complement involving only interface displacements using a bnn preconditioner based on the solution of a coarse elasticity problem and local elasticity problems with natural and essential boundary conditions while the schur complement becomes increasingly ill conditioned as the materials becomes almost incompressible the bnn preconditioned operator remains well conditioned the main theoretical result of the paper shows that the proposed bnn method is scalable and quasi optimal in the constant coefficient case this bound holds for material parameters arbitrarily close to the incompressible limit while this result is due to an underlying mixed formulation of the problem both the interface problem and the preconditioner are positive definite numerical results in two and three dimensions confirm these good convergence properties and the robustness of the methods with respect to the almost incompressibility of the material

a class of asymptotic preserving schemes for the fokker planck landau equation we present a class of asymptotic preserving ap schemes for the nonhomogeneous fokker planck landau nfpl equation filbet and jin 16 designed a class of ap schemes for the classical boltzmann equation by penalization with the bgk operator so they become efficient in the fluid dynamic regime we generalize their idea to the nfpl equation with a different penalization operator the fokker planck operator that can be inverted by the conjugate gradient method we compare the effects of different penalization operators and conclude that the fokker planck fp operator is a good choice such schemes overcome the stiffness of the collision operator in the fluid regime and can capture the fluid dynamic limit without numerically resolving the small knudsen number numerical experiments demonstrate that the schemes possess the ap property for general initial data with numerical accuracy uniformly in the knudsen number