an efficient method for simulation of noisy coupled multi dimensional oscillators we present an efficient computational method for the study of populations of noisy coupled oscillators by taking a population density approach in which the probability density of observing an oscillator at a point of state space is the primary variable instead of the states of all of the oscillators we are able to seamlessly account for intrinsic noise within the oscillators and global coupling within the population the population is assumed to consist of a large number of oscillators so that the noise process is well sampled over the population our numerical method is able to solve the governing equation even in the challenging case of limit cycle oscillators with a large number of state variables instead of simulating a prohibitive number of oscillators our particle method simulates relatively few particles allowing for the efficient solution of the governing equation

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

domain decomposition schemes with high order accuracy and unconditional stability parallel finite difference schemes with high order accuracy and unconditional stability for solving parabolic equations are presented the schemes are based on domain decomposition method i e interface values between subdomains are computed by the explicit scheme interior values are computed by the implicit scheme the numerical stability and error are derived in the h 1 norm in one dimensional case numerical results of both one and two dimensions examining the stability accuracy and parallelism of the procedure are also presented

unconditional stability of corrected explicit implicit domain decomposition algorithms for parallel approximation of heat equations a class of corrected explicit implicit domain decomposition ceidd methods is investigated for the parallel approximation of linear heat equations explicit implicit domain decomposition eidd methods are computationally and communicationally efficient for each time step but always suffer from small time step size restrictions by adding an interface correction step to kuznetsov s eidd the one dimensional ceidd procedure achieves unconditional stability without discarding the time stepwise efficiency of the eidd method in order to maintain the virtues of the ceidd method and improve the flexibility in domain partitioning for solving multidimensional problems special zigzag shaped interfaces are suggested in the ceidd method based on noncrossover and crossover types of zigzag interfaces the resulting ceidd zi algorithms are studied for two strategies of subdomain partition by the energy method it shows that the proposed algorithms including their degenerate cases the corrected explicit hopscotch schemes are convergent in the discrete h1 semi norm and l2 norm numerical experiments confirm the results in our analysis

unconditional stability of parallel difference schemes with second order accuracy for parabolic equation in this paper we investigate the parallel difference schemes of parabolic equation in particular two kinds of difference schemes with intrinsic parallelism are constructed firstly we combine the values of previous two time levels at the interface points to get the dirichlet boundary condition for the sub domain problems then the values in the sub domains are calculated by fully implicit scheme and then finally the values at the interface points are computed by fully implicit scheme the unconditional stability of these schemes is proved and the convergence rate of second order is also obtained numerical results are presented to examine the accuracy stability and parallelism of the parallel schemes

p restriction of partitions and homomorphisms between specht modules abstract let be a partition of n we show that the space of f s n homomorphisms between the specht modules s and s r is one dimensional where f is a field of characteristic p and r is the p restriction of equivalently our result proves the corresponding theorem for the homomorphism space hom f s n s reg s where reg is the p regularisation of as defined by james

second order in time and space corrected explicit implicit domain decomposition scheme for convection diffusion equations abstract we develop a second order corrected explicit implicit domain decomposition scheme sceidd for the parallel approximation of convection diffusion equations over multi block subdomains the stability and convergence properties of the sceidd scheme are analyzed it is proved that the scheme is unconditionally stable and it is second order accurate in time as well as space furthermore three different numerical experiments are performed to verify the theoretical results in all the experiments the sceidd scheme is compared with the eipcmu2d scheme which is first order in time

overlapping schwarz waveform relaxation method for the solution of the reaction diffusion equation abstract we are interested in solving time dependent problems using domain decomposition methods in the classical approach one discretizes first the time dimension and then one solves a sequence of steady problems by a domain decomposition method in this article we treat directly the time dependent problem and we study a schwarz waveform relaxation algorithm for the convection diffusion equation we study the convergence of the overlapping schwarz waveform relaxation method for solving the reaction diffusion equation over multi overlapped subdomains also we will show that the method converges linearly and superlinearly over long and short time intervals and the convergence depends on the size of overlap numerical results are presented from solutions of a specific model problems to demonstrate the convergence linear and superlinear and the role of the overlap size

a generalization of peaceman rachford fractional step method in this paper we develop a set of time integrators of type fractional step runge kutta methods which generalize the time integrator involved in the classical peaceman rachford scheme combining a time semidiscretization of this type with a standard spatial discretization we obtain a totally discrete algorithm capable of discretizing efficiently a general parabolic problem if suitable splittings of the elliptic operator are considered we prove that our proposal is second order consistent and stable even for an operator splitting in m terms which do not necessarily commute finally we illustrate the theoretical results with various applications such as alternating directions or evolutionary domain decomposition

a finite volume eulerian lagrangian localized adjoint method for solution of the contaminant transport equations in two dimensional multiphase flow systems an eulerian lagrangian localized adjoint method ellam numerical solution is developed for the multiphase contaminant transport equations in two dimensions the ellam uses finite volume test functions in the space time domain defined by the characteristics of the hyperbolic part of the governing equation the use of the characteristics results in an approximation that allows large time steps while still maintaining accurate solutions this greatly reduces the computational effort required to find solutions to the advection dispersion equation combination of the finite volume test functions with the conservative form of the governing equation results in a local conservation of mass property for problems with a constant saturation such as that of contaminant transport in the saturated zone this property is highly advantageous as the resulting numerical approximation conserves mass both globally and locally for problems with a variable saturation the requirement of local conservation of mass is too stringent particularly around first type boundaries where oscillations can occur because of errors inherent in the numerical determination of fluid velocities and in the backtracking routine a combined conservative nonconservative ellam is developed with an ellam formulation based on the nonconservative form of the governing equation being applied to subdomains intersecting first type boundaries and a conservative ellam being used for all other subdomains the combined conservative nonconservative ellam is compared to a galerkin finite element scheme and is found to have greatly superior performance requiring far fewer time steps to obtain a solution of equivalent accuracy

an ellam scheme for advection diffusion equations in two dimensions we develop an eulerian lagrangian localized adjoint method ellam to solve two dimensional advection diffusion equations with all combinations of inflow and outflow dirichlet neumann and flux boundary conditions the ellam formalism provides a systematic framework for implementation of general boundary conditions leading to mass conservative numerical schemes the computational advantages of the ellam approximation have been demonstrated for a number of one dimensional transport systems practical implementations of ellam schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper extensive numerical results are presented to compare the ellam scheme with many widely used numerical methods and to demonstrate the strength of the ellam scheme

a two step two field hybrid method for the static and dynamic analysis of substructure problems with conforming and non conforming interfaces abstract the need for assembling independent finite element substructure solutions arises in several engineering and scientific problems including the design and analysis of complex structural systems component mode synthesis global local analysis adaptive refinement and parallel processing in this paper we discuss the solution of such problems by a two field hybrid method where the substructures are jointed with low order polynomial or piece wise polynomial lagrange multipliers and present a rayleigh ritz based smoothing procedure for improving the accuracy of the computed coupled solution in the presence of various substructure heterogeneities we consider both conforming and nonconforming substructure meshes and demonstrate the benefits of the proposed two step solution method with several examples from structural mechanics

character formulas and frobenius subgroups of algebraic groups let g be a connected semisimple algebraic group over a field of non zero characteristic p with a maximal torus t on g we have the frobenius endomorphism f induced by the pth power map on the underlying field the scheme theoretic kernel of f is the first frobenius kernel g and the inverse image of t under f is the frobenius subgroup g t the purpose of this paper is to study certain questions in the rational representation theory of g t let b be a bore1 subgroup of g which contains t in 6 cline parshall and scott obtained a result which describes the functor induction from b to g as the composite of functors of the form induction from b to a minimal parabolic subgroup p over a sequence of minimal parabolics coming from a given reduced expression of the long word in the weyl group in section 5 we obtain the analogous result for g t in section 6 we develop a theory of intertwines for a certain class of g t modules this gives a direct proof of an extended strong linkage principle for g t independent of the strong linkage principle for g originally jantzen 17 proved that strong linkage for g t and more generally for g t n 1 follows from the strong linkage principle for g 1 recently we have applied this method to give a new proof of an extended strong linkage principle for g see 121 from the results of section 5 we can write down a formula for the image of one of the intertwines in our class of modules this also follows from results of jantzen 16 171 composing the intertwines in question relative to the class of modules corresponding to a given character of t gives a map whose image is the irreducible g t module of that highest weight in case the character is a restricted dominant character the image coincides with the irreducible g module of that highest weight by a result

the mass preserving and modified upwind splitting ddm scheme for time dependent convectiondiffusion equations in the paper a new mass preserving and modified upwind s ddm scheme over non overlapping multi block sub domains for solving time dependent convectiondiffusion equations in two dimensions is developed and analyzed on each sub domain the intermediate fluxes on the interfaces of sub domains are firstly computed by the modified semi implicit flux schemes then the solutions and fluxes in the interiors of sub domains are computed by the modified upwind splitting implicit solution and flux coupled schemes on the theoretical aspect analyzing the mass preserving and modified upwind s ddm scheme is difficult over non overlapping multi block sub domains due to the combination of the splitting technique the conservative modified upwind technique and the domain decomposition we prove the proposed scheme to be mass conservative and unconditionally stable in discrete l2 norm and further prove its convergence and obtain error estimates numerical experiments are presented to illustrate mass conservation convergence and parallelism we propose a new mass preserving and modified upwind splitting ddm scheme in 2d the solution and flux coupled scheme is to preserve mass over multi subdomains the developed scheme is mass conservative and unconditionally stable it is of first order convergence in time step and second order convergence in space

a stabilized explicit lagrange multiplier based domain decomposition method for parabolic problems a fully explicit stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations pdes is presented writing the semi discretized equations as a differential algebraic equation dae system where the interface continuity constraints between subdomains are enforced by lagrange multipliers the method uses the runge kutta chebyshev projection scheme to integrate the dae explicitly and to enforce the constraints by a projection with mass lumping techniques and node to node matching grids the method is fully explicit without solving any linear system a stability analysis is presented to show the extended stability property of the method the method is straightforward to implement and to parallelize numerical results demonstrate that it has excellent performance

loewy series of certain indecomposable modules for frobenius subgroups we imitate some approaches in infinite dimensional representation theory of complex semisimple lie algebras by using the truncated category method in the categories of modules for certain frobenius subgroups of a semisimple algebraic group over an algebraically closed field of characteristic p 0 by studying the translation functors from p singular weights to p regular weights we obtain some results on l wy series of certain indecomposable modules

numerical analyses of operator splitting algorithms for the two dimensional advection diffusion equation abstract holly and preissmann s hp scheme has been the basis of a large number of operator splitting algorithms for the solution of the advection diffusion equation however these algorithms including hp are first order accurate in time due to splitting errors error analyses of these algorithms incorporating splitting error and errors resulting from numerical solutions of the split advection and diffusion equations are lacking in this paper error analysis of a second order accurate adaptation of the hp scheme ahp is presented for the two dimensional advection diffusion equation a modified ahp scheme mahp is suggested to remove the ad hoc nature of boundary conditions during the diffusion step of computations in both hp and ahp as boundary conditions specified for the advection diffusion equation are not applicable to the split equations second order accurate boundary conditions for the split advection and diffusion equations are derived an analysis of numerical dispersion and dissipation associated with the numerical procedure for the advection equation is presented the analysis establishes a criterion so that computational errors are small in two dimensional advection dominated transport problems several numerical examples are presented to verify the numerical analyses presented in the paper in addition a review of the current status of operator splitting algorithms for the advection diffusion equation is presented the objective of the review is to identify the issues that have not been addressed in the previous studies

stabilized explicit implicit domain decomposition methods for the numerical solution of parabolic equations we report a class of stabilized explicit implicit domain decomposition seidd methods for the numerical solution of parabolic equations explicit implicit domain decomposition eidd methods are globally noniterative nonoverlapping domain decomposition methods which when compared with schwarz algorithm based parabolic solvers are computationally and communicationally efficient for each simulation time step but suffer from small time step size restrictions by adding a stabilization step to eidd the seidd methods retain the time stepwise efficiency in computation and communication of the eidd methods but exhibit much better numerical stability three seidd algorithms are presented in this paper which are experimentally tested to show excellent stability computation and communication efficiencies and high parallel speedup and scalability

the new mass conserving s ddm scheme for two dimensional parabolic equations with variable coefficients abstract in the article a new and efficient mass conserving operator splitting domain decomposition method s ddm is proposed and analyzed for solving two dimensional variable coefficient parabolic equations with reaction term the domain is divided into multiple non overlapping block divided subdomains on each block divided subdomain the interface fluxes are first computed explicitly by local multi point weighted schemes and the solutions in the interior of subdomain are computed by the one directional operator splitting implicit schemes at each time step the scheme is proved to satisfy mass conservation over the whole domain of domain decomposition by combining with some auxiliary lemmas and applying the energy method we analyze theoretically the stability of our scheme and prove it to have second order accuracy in space step in the l2 norm numerical experiments are performed to illustrate its accuracy conservation stability efficiency and parallelism our scheme not only keeps the excellent advantages of the non overlapping domain decomposition and the operator splitting technique but also preserves the global mass

on the structure of the cohomology of line bundles on gb one of the most intriguing problems in characteristic p representation theory for semisimple algebraic groups is the problem of finding the formal characters of the irreducible modules an equivalent problem is to find the composition factors of weyl modules it is a non trivial fact that the weyl modules occur as the top cohomology groups of certain line bundles on homogeneous spaces one could therfore more generally ask for the composition factors of all such cohomology groups references 2 and 3 deal with aspects of this question in the present paper we use a method primarily due to j c jantzen 151 to extend the results in the abovementioned works our main results are i a translation principle theorem 2 5 below generalizing jantzen s theorem in 15 1 ii a proof of the carter lusztig conjecture see 6 p 2391 compare also 3 theorem 4 11 and 71 on the existence of intertwining homomorphisms between weyl modules theorem 3 3 below together with some related results on higher ext groups and iii a rather general vanishing theorem for cohomology groups of line bundles on homogeneous spaces theorem 4 1 below as will be apparent from a comparison with 151 many of jantzen s arguments in the proof of his translation principle can be directly carried over to our situation however the fact that we deal with all cohomology groups not just the highest allows us sometimes to shorten the proofs in particular this is the case for the proof of proposition 2 3 in which we give the effect of the translation functor on irreducible modules in an earlier preprint 4 we showed that the vanishing theorem for dominant line bundles on homogeneous spaces can be proved for p big using translation arguments shortly after that preprint was written the author 151 and independently w haboush 111 found an even simpler way of proving that result in order to describe the vanishing behaviour of the cohomology of non ample line bundles the method is however still useful this is demonstrated in the last section of this paper where we 245 0021 8693 81 070245 14 02 00 o

explicit implicit conservative domain decomposition procedures for parabolic problems based on block centered finite differences domain decomposition procedures for solving parabolic equations are considered the underlying discretization consists of block centered finite differences in these procedures fluxes at subdomain interfaces are calculated from the solution at the previous time level these fluxes serve as neumann boundary data for implicit block centered discretizations in the subdomains a priori error estimates are presented and numerical results examining the stability accuracy and parallelism of the scheme are presented

intermediate boundary conditions for time split methods applied to hyperbolic partial differential equations determination des conditions aux limites intermediaires appropriees pour les methodes de decomposition appliquees aux equations aux derivees partielles hyperboliques

an efficient s ddm iterative approach for compressible contamination fluid flows in porous media in this paper we develop an efficient splitting domain decomposition method s ddm for compressible contamination fluid flows in porous media over multiple block divided sub domains by combining the non overlapping domain decomposition splitting linearization and extrapolation techniques the proposed s ddm iterative approach divides the large domain into multiple block sub domains in each time interval the s ddm scheme is applied to solve the water head equation in which an efficient local multilevel scheme is used for computing the values of water head on the interfaces of sub domains and the splitting implicit scheme is used for computing the interior values of water head in sub domains and the s ddm scheme is then proposed to solve the concentration equation by combining the upstream volume technique numerical experiments are performed and analyzed to illustrate the efficiency of the s ddm iterative approach for simulating compressible contamination fluid flows in porous media the developed method takes the excellent attractive advantages of both the non overlapping domain decomposition and the splitting technique and reduces computational complexities large memory requirements and long computational durations

structure of cohomology of line bundles on gb for semisimple groups let g be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p 0 t a maximal torus of g and b a borel subgroup containing t each weight in x t determines a line bundle on the flag varietyg b it turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of b modules to the category of g modules for each 1 dimensional b module defined by a weight in x t further indgb h g b turns out to be the dual of a weyl module one of the main problems is to calculate the characters of the irreducible g modules for p 0 the character of the irreducible g module of highest weight is given by weyl s character formula and the g module structures of the cohomology of line bundles are well understood however for p 0 the story is quite different and many of the results remain conjectural since characters of weyl modules are given by weyl s character formula understanding the structure ofh g b turns out to be the main problem it is also interesting to understand the structure of the higher cohomology which might help us to understand h h andersen has a series of papers toward the understanding of the higher cohomology such as the simple socle of h and filtrations of h i 4 in 5 using the representations of infinitesimal subgroup schemes of g he proved generically that h g b w has simple socle and simple head and their highest weights can be calculated in 6 he proved that the socle series of h comes generically from the

a new class of second order linearly implicit fractional step methods a new family of linearly implicit fractional step methods is proposed and analysed in this paper the combination of one of these time integrators with a suitable spatial discretization permits a very efficient numerical solution of semilinear parabolic problems the main quality of this new family of methods compared to other existing time integrators of this type is that they are stable when the spatial differential operator is decomposed in a number m of simpler operators which do not necessarily commute we prove that these methods satisfy this general stability result as well as they are second order consistent both consistency and stability are proven for an operator splitting in an arbitrary number m of terms m 2 finally a numerical experiment illustrates these theoretical results in the last section of the paper

domain decomposition for an advection diffusion problem with parabolic layers this paper deals with multidomain domain decomposition algorithms applied to the solution of a singularly perturbed advection diffusion problem with parabolic layers our purpose is to study the algorithms applied to finite difference approximations of the problem the finite difference approximations make use of special piecewise uniform meshes we consider two algorithms the first one is based on decomposition of the computational domain into nonoverlapping vertical subdomains and second uses decomposition into nonoverlapping horizontal subdomains convergence properties of the algorithms are established numerical results for a test problem are presented

an efficient parallel algorithm with application to computational fluid dynamics abstract when solving time dependent partial differential equations on parallel computers using the nonoverlapping domain decomposition method one often needs numerical boundary conditions on the boundaries between subdomains these numerical boundary conditions can significantly affect the stability and accuracy of the final algorithm in this paper a stability and accuracy analysis of the existing methods for generating numerical boundary conditions will be presented and a new approach based on explicit predictors and implicit correctors will be used to solve convection diffusion equations on parallel computers with application to aerospace engineering for the solution of euler equations in computational fluid dynamics simulations both theoretical analyses and numerical results demonstrate significant improvement in stability and accuracy by using the new approach

a two scale approximation of the schur complement and its use for non intrusive coupling this paper presents a two scale approximation of the schur complement of a subdomain s stiffness matrix obtained by combining local ie element strips and global ie homogenized contributions this approximation is used in the context of a coupling strategy that is designed to embed local plasticity and geometric details into a small region of a large linear elastic structure the strategy consists in creating a local model which contains the desired features of the concerned region and then substituting it into the global problem by the means of a non intrusive solver coupling technique adapted from domain decomposition methods using the two scale approximation of the schur complement as a robin condition on the local model enables to reach high efficiency examples include a large 3d problem provided by our industrial partner snecma

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

the specified boundary stiffness force sbsf method for finite element subregion analysis it is difficult to analyse a large complex structure in sufficient detail to obtain accurate results everywhere one approach is simply to refine the whole structure model in the regions of interest another approach is to identify a subregion of the structure and develop a separate refined model of the subregion it is difficult to assure accuracy in this subregion model because of uncertainties in specifying boundary conditions and loading from the whole structure model solution in the current literature three methods other than whole model refinement have been described to deal with this problem these are called the specified boundary displacement method the linear constraint method and the zooming method this paper describes a new approach to this problem of modelling subregions this approach uses the stiffnesses and forces from the whole model solution at the nodes on the boundary of the sub region model accurate displacement and stress solutions are obtained with this method as it takes into account the interaction between the new stiffness of the subregion and the rest of the structure this approach is similar to substructuring however the equations outside the subregion are discarded rather than condensed out which results in much less computation effort examples of the application of this method to the problem of a plate with a centre hole in tensile loading are presented the results compared favourably with the results of the same problem solved using other methods with significant improvement in accuracy over the specified boundary displacement method also presented are some results from design modification of the subregion which illustrate the potential of this method for redesign application

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

conservative domain decomposition procedure with unconditional stability and second order accuracy a new derivation of the conservative domain decomposition procedure for solving the parabolic equation is presented in this procedure fluxes at subdomain interfaces are calculated from the solution at the previous time level then these fluxes serve as neumann boundary data for implicit block centered discretization in the subdomain the unconditional stability and the second order accuracy of solution values as well as fluxes are proved numerical results examining the stability accuracy and parallelism of the procedure are also presented

on the treatment of time dependent boundary conditions in splitting methods for parabolic differential equations splitting methods for time dependent partial differential equations usually exhibit a drop in accuracy if boundary conditions become time dependent this phenomenon is investigated for a class of splitting methods for two space dimensional parabolic partial differential equations a boundary value correction discussed in a paper by fairweather and mitchell for the laplace equation with dirichlet conditions is generalized for a wide class of initial boundary value problems a numerical comparison is made for the adi method of peaceman rachford and the lod method of yanenko applied to problems with dirichlet boundary conditions and non dirichlet boundary conditions

parallel domain decomposition procedures of improved d d type for parabolic problems two parallel domain decomposition procedures for solving initial boundary value problems of parabolic partial differential equations are proposed one is the extended d d type algorithm which extends the explicit implicit conservative galerkin domain decomposition procedures given in 5 from a rectangle domain and its decomposition that consisted of a stripe of sub rectangles into a general domain and its general decomposition with a net like structure an almost optimal error estimate without the factor h 1 2 given in dawson dupont s error estimate is proved another is the parallel domain decomposition algorithm of improved d d type in which an additional term is introduced to produce an approximation of an optimal error accuracy in l 2 norm

domain decomposition operator splittings for the solution of parabolic equations we study domain decomposition counterparts of the classical alternating direction implicit adi and fractional step fs methods for solving the large linear systems arising from the implicit time stepping of parabolic equations in the classical adi and fs methods for parabolic equations the elliptic operator is split along coordinate axes they yield tridiagonal linear systems whenever a uniform grid is used and when mixed derivative terms are not present in the differential equation unlike coordinate axes based splittings we employ domain decomposition splittings based on a partition of unity such splittings are applicable to problems on nonuniform meshes and even when mixed derivative terms are present in the differential equation and they require the solution of one problem on each subdomain per time step without iteration however the truncation error in our proposed method deteriorates with smaller overlap amongst the subdomains unless a smaller time step is chosen estimates are presented for the asymptotic truncation error along with computational results comparing the standard crank nicolson method with the proposed method

locally linearized fractional step methods for nonlinear parabolic problems this work deals with the efficient numerical solution of a class of nonlinear time dependent reaction diffusion equations via the method of lines approach we first perform the spatial discretization of the original problem by applying a mimetic finite difference scheme the system of ordinary differential equations arising from that process is then integrated in time with a linearly implicit fractional step method for that purpose we locally decompose the discrete nonlinear diffusion operator using suitable taylor expansions and a domain decomposition splitting technique the totally discrete scheme considers implicit time integrations for the linear terms while explicitly handling the nonlinear ones as a result the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelizable linear subsystems the convergence of the proposed methods is illustrated by numerical experiments

on a domain decomposition algorithm for a singularly perturbed reaction diffusion problem this paper deals with an iterative algorithm for domain decomposition applied to the solution of a singularly perturbed reaction diffusion problem this algorithm is based on finite difference domain decomposition approach and suitable for parallel computing convergence properties of the algorithm are established numerical results for a test singularly perturbed problem are presented

a domain decomposition discretization of parabolic problems in recent years domain decomposition methods have attracted much attention due to their successful application to many elliptic and parabolic problems domain decomposition methods treat problems based on a domain substructuring which is attractive for parallel computation due to the independence among the subdomains in principle domain decomposition methods may be applied to the system resulting from a standard discretization of the parabolic problems or directly be carried out through a discretization of parabolic problems in this paper a direct domain decomposition method is introduced to discretize the parabolic problems the stability and convergence of this algorithm are analyzed

on the intertwinings between higher cohomology modules of line bundles on gb let g be a simple simply connected affine algebraic group over an algebraically closed field of characteristic p let b 7 be a bore1 subgroup and maximal torus and let x t be the character group let r be the set of roots of g t r the set of roots of bopp t s the set of simple roots in r and x t the set of dominant characters let 1 be a character on b which is an element of x t and let mn be the irreducible g module of highest weight 1 w is the weyl group of g t wo is the longest word in w and yl is the coroot of a root y in 4 we constructed the family of operators jw mje w on the set x t oz w see the preliminary section for 1 e x t in generic dominant position we showed that mjw l w e is a set of multiplicity one composition factors of h j here we call those factors the scaffolding factors of h l and that h w 1 has socle m and unique top factor m wow n at the same time that serves to determine the top and socle factors of the image of an intertwining between any two higher cohomology modules in this paper we augment that information about the image of the non zero intertwining cp h l w a h z w2 a by determining which of the scaffolding factors appear in the image section 2 in 1 1 we show the following facts about extensions among the scaffolding pieces if ext jm 1 mjw2 j is non zero then w2 sow for

an efficient characteristic finite difference s ddm scheme for convection diffusion equations abstract in this paper an efficient splitting domain decomposition method scheme for solving time dependent convection diffusion reaction equations is analyzed a three step method along each direction is used to solve the solution over each block divided sub domain at every time interval the new solutions are firstly solved by the quadratic interpolation then the intermediate fluxes on the interfaces of sub domains are computed by local multi point weighted average from the new solutions lastly the solutions and fluxes in the interiors of sub domains are computed by the splitting implicit characteristic difference method by some auxiliary lemmas and the defined intermediate exact solution the stability and error estimate are given in discrete l 2 norm we further prove that our scheme is of second order convergence in space and of first order convergence in time numerical experiments are presented to validate theoretical result

a fractional step ellam approach to high dimensional convection diffusion problems with forward particle tracking in this paper a fractional step method combined with an eulerian lagrangian localized adjoint method ellam is proposed to solve high dimensional convection diffusion problems the method reduces high dimensional problems to a series of uncoupled one dimensional problems in each time step interval in which one dimensional ellam is used to solve the one dimensional splitting equations the approach takes the attractive advantages of the ellam method and the fractional step technique it reduces computational complexities large memory requirements and long computation durations due to the application of the splitting technique it reduces temporal errors and generates accurate numerical solutions even if large time and coarse spatial step sizes are used in computation it effectively eliminates non physical oscillation or excessive numerical dispersion and treats boundary conditions well and in a natural way numerical experiments show the efficient performance of the approach

ipic domain decomposition algorithm for parabolic problems a domain decomposition algorithm which is based on a semi implicit prediction and then a fully implicit correction at the interface for solving parabolic partial differential equations is presented the numerical results show that the algorithm is unconditionally stable and very efficient in this paper high speed up is analyzed with respect to the spectral radius and operation counts

efficient parallel algorithms for parabolic problems domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation implicit schemes are used in order to march with large time steps parallelization is realized by approximating interface values using explicit computation various techniques are examined including a multistep second order explicit scheme and a one step high order scheme we show that the resulting schemes are of second order global accuracy in space and stable in the sense of osher or in l infty they are optimized with respect to the parallel efficiency

extensions between simple modules for frobenius kernels introduction let g be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p 0 t g is a maximal torus and r is the root system relative to t x t is the weight lattice let b t be a borel subgroup corresponding to the negative roots r r denote by gr the r th frobenius kernel of g the socle and radical structures of the cohomology groups of line bundles on the flag variety g b are determined by those structures of the grt modules zr ind grt brt cf 12 so the study of the grt structure of these modules turns out to be more interesting calculation of the extensions between simple modules plays an important rule in determining the socle structure in this paper we calculate the socle series of the weyl modules with p singular highest weight for the group of type g2 by studying the extensions between simple modules for frobenius kernel in the first section of this paper we study the properties of ext gr l h which turns out to be semisimple and to have a good filtration for large p and prestricted weights and some of the vanishing properties of these modules are also studied then we use these properties to calculate ext gr l l which will lead a calculation of the extensions between simple grt modules the results in this section will be used in section 3 to calculate the socle series of z1 with p singular weights for the group of type g2 the author 13 ch3 used the method of doty and sullivan 7 and calculated the socle series of h for the p singular weights in the bottom p alcove for the groups of type a2 and b2 however when the multiplicities of simple modules in

homomorphisms between weyl modules for sl3 k we classify all homomorphisms between weyl modules for sl3 k when k is an algebraically closed field of characteristic at least three and show that the hom spaces are all at most one dimensional as a corollary we obtain all homomorphisms between specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three we conclude by showing how a result of fayers and lyle on hom spaces for specht modules is related to earlier work of donkin for algebraic groups let g be a reductive algebraic group over an algebraically closed field of characteristic p 0 an important class of modules for such a group are the weyl modules labelled by dominant weights these can be constructed relatively explicitly and their heads provide a full set of simple modules for g equivalently one can study the duals of these modules denoted which have the advantage of being induced from one dimensional modules for a borel subgroup in determining the structure of such modules or indeed their cohomology the calculation of hom spaces between them is a useful tool relatively little is known in general about such hom spaces in type a when andare related by a suitable single reflection explicit non zero homomorphisms from to were constructed with some restrictions by carter and lusztig 3 and more generally by carter and payne 4 the corresponding cases in other types were considered by franklin 14 while it is clear that there should be a hierarchy of families of homomorphisms corresponding to different powers of p the only case where the above results provide a complete classification is when g is sl2 where it is relatively easy to determine all hom spaces exactly 6 the only other general results in this area by andersen 1 and koppinen 18 concern homo morphisms between modules labelled by weights which are close together typically such results show that certain hom spaces are non zero or in some cases one dimensional for weights which are far apart and not related by a single reflection almost nothing is known in this paper we will determine all homomorphisms between weyl modules for sl3 when p 3 and provide a recursive procedure for determining the composition factors arising in the image or kernel of such maps in most cases from these results we will also classify all homomorphisms between specht modules for the symmetric groups corresponding to three part partitions when p 3 after a section of preliminaries we review the sl3 data concerning p filtrations that we will need from 21 this describes certain filtrations of induced modules which will allow us to proceed by induction together with the set of p good homomorphisms which will be fundamental in our later constructions we also recall a theorem of carter and payne 4 on the existence of certain homomorphisms these will be the two key sets of data which we need to determine all possible homomorphisms with the notation developed up to that point in place in section 4 we can give the strategy to be followed in the remainder of the paper and in particular the translation functor arguments

the efficient s ddm scheme and its analysis for solving parabolic equations abstract in this paper we develop and analyze the efficient splitting domain decomposition method for solving parabolic equations the domain is divided into non overlapping multi block subdomains on interfaces of sub domains the local multilevel explicit scheme is proposed to solve the interface values of solution and then the splitting implicit scheme is proposed to solve the solution in interiors of subdomains on the theoretical aspect analyzing the s ddm scheme over non overlapping multi block sub domains is much challenging we strictly prove the stability and convergence of the s ddm scheme and obtain the optimal error estimates over non overlapping multi block subdomains numerical experiments are given to illustrate the accuracy and efficiency of the s ddm scheme the developed method takes the most attractive advantages of both the non overlapping domain decomposition and the splitting technique which overcomes the limitation of the stripe divided sub domains for parabolic equations in the previous methods the developed s ddm scheme reduces computational complexities large memory requirements and computational costs

a high order upwind method for the convection diffusion problem abstract in this paper a high order upwind method for the convection diffusion problems is proposed the scheme satisfies the maximum principle overcomes numerical oscillation and has astonishingly high order computing accuracy at the same time the error estimateso h3 c 4 0 in discretel2 h1 andlx norms in which the constants ino h3 c 4 0 do not depend on the diffusion coefficienta the velocity b the solutionc and the domain partition are obtained by using a new analytic technique several numerical examples are given in order to demonstrate the efficiency of the method

eulerian lagrangian localized adjoint methods for a nonlinear advection diffusion equation eulerian lagrangian localized adjoint methods ellam are developed for the non linear buckley leverett equation which is characterized by degenerate diffusion and sharpening near shock solutions the ellam methodology employs space time finite elements with edges oriented along flow paths and space time test functions that satisfy a local adjoint condition this combination extends eulerian lagrangian concepts in a systematic mass conservative fashion to problems with general boundary conditions various kinds of boundary conditions are considered and a local time stepping procedure is developed for a no flow outlet condition that leads to a boundary layer numerical experiments illustrate the potential of these methods

adi spectral collocation methods for parabolic problems we discuss the crank nicolson and laplace modified alternating direction implicit legendre and chebyshev spectral collocation methods for a linear variable coefficient parabolic initial boundary value problem on a rectangular domain with the solution subject to non zero dirichlet boundary conditions the discretization of the problems by the above methods yields matrices which possess banded structures this along with the use of fast fourier transforms makes the cost of one step of each of the chebyshev spectral collocation methods proportional except for a logarithmic term to the number of the unknowns we present the convergence analysis for the legendre spectral collocation methods in the special case of the heat equation using numerical tests we demonstrate the second order accuracy in time of the chebyshev spectral collocation methods for general linear variable coefficient parabolic problems

spectral fractional step runge kutta discretizations for initial boundary value problems with time dependent boundary conditions in this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting alternating direction or more generally fractional step methods such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive runge kutta method which is called the fractional step runge kutta method and a standard space discretization which can be of type finite differences finite elements or spectral methods among others spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps the numerical experiences performed also show the increase of accuracy that this technique provides

on the generic structure of cohomology modules for semisimple algebraic groups let c be a connected semisimple algebraic group over a field of positive characteristic denote by b a borel subgroup our main result says that generically the cohomology modules for line bundles on g b have simple socles and simple heads and wc identify the corresponding highest weights as one of the conse quences we discover a certain symmetry among extensions of simple modules for g in this paper we continue the study of the 1nodule structure of the cohomology groups h x h g b t x for g a sesimple algebraic group over a field of positive characteristic p b a borel subgroup of g and 2 x the line bundle on g b induced by the character x of b for earlier work on various aspects of this question see e g 1 4 7 9 11 13 here we will consider the case where x is generic i e lies far away from the walls of the weyl chamber containing it see 2 1 for the precise condition in this case h x is nonzero in exactly one degree our main result theorem 2 2 says that this nonzero cohomology module has simple socle and simple head for x dominant it is a classical result that h x has simple socle the first nondominant case was treated in 2 where we proved that h1 x also always has a simple socle it should likewise be mentioned that j c jantzen has proved that weyl modules which may be identified with hn k n dimg b in the generic case have simple socles 12 6 3 and that j e humphreys in 10 has formulated conjectures which suggest that for x generic h x has the same module structure as a weyl module recently doty and sullivan have announced results which include special cases of our result see their forthcoming paper 8 to formulate some of the consequences of our main theorem we need a little more notation we set w equal to the weyl group of g relative to a maximal torus t in b for w e w we denote by l w the length of w and if x is a generic dominant weight then we let l xw be the simple g module which by our theorem occurs as

energy conserved splitting fdtd methods for maxwell s equations in this paper two new energy conserved splitting methods ec s fdtdi and ec s fdtdii for maxwell s equations in two dimensions are proposed both algorithms are energy conserved unconditionally stable and can be computed efficiently the convergence results are analyzed based on the energy method which show that the ec s fdtdi scheme is of first order in time and of second order in space and the ec s fdtdii scheme is of second order both in time and space we also obtain two identities of the discrete divergence of electric fields for these two schemes for the ec s fdtdii scheme we prove that the discrete divergence is of first order to approximate the exact divergence condition numerical dispersion analysis shows that these two schemes are non dissipative numerical experiments confirm well the theoretical analysis results

multiplicative schwarz methods for parabolic problems the class of multiplicative schwarz methods originated from the classical schwarz alternating method it has been shown to be one of the most powerful methods for solving finite element or finite d

a finite difference domain decomposition algorithm for numerical solution of the heat equation a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented in this procedure interface values between subdomains are found by an explicit finite difference formula once these values are calculated interior values are determined by backward differencing in time a natural extension of this method allows for the use of different time steps in different subdomains maximum norm error estimates for these procedures are derived which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters

error analysis of multipoint flux domain decomposition methods for evolutionary diffusion problems abstract we study space and time discretizations for mixed formulations of parabolic problems the spatial approximation is based on the multipoint flux mixed finite element method which reduces to an efficient cell centered pressure system on general grids including triangles quadrilaterals tetrahedra and hexahedra the time integration is performed by using a domain decomposition time splitting technique combined with multiterm fractional step diagonally implicit runge kutta methods the resulting scheme is unconditionally stable and computationally efficient as it reduces the global system to a collection of uncoupled subdomain problems that can be solved in parallel without the need for schwarz type iteration convergence analysis for both the semidiscrete and fully discrete schemes is presented

an approximation to miscible fluid flows in porous media with point sources and sinks by an eulerian lagrangian localized adjoint method and mixed finite element methods we develop an eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solution technique for accurate numerical simulation of coupled systems of partial differential equations pdes which describe complex fluid flow processes in porous media an ellam which was shown previously to outperform many widely used methods in the context of linear convection diffusion pdes is presented to solve the transport equation for concentration since accurate fluid velocities are crucial in numerical simulations an mfem is used to solve the pressure equation for the pressure and darcy velocity this minimizes the numerical difficulties occurring in standard methods for approximating velocities caused by differentiation of the pressure and then multiplication by rough coefficients the ellam mfem solution technique significantly reduces temporal errors symmetrizes the governing transport equation eliminates nonphysical oscillation and or excessive numerical dispersion in many simulators conserves mass and treats boundary conditions accurately numerical experiments show that the ellam mfem solution technique simulates miscible displacements of incompressible fluid flows in porous media accurately with fairly coarse spatial grids and very large time steps which are one or two orders of magnitude larger than the time steps used in many methods moreover the ellam mfem solution technique can treat large mobility ratios discontinuous permeabilities and porosities anisotropic dispersion in tensor form and point sources and sinks