a multiscale flux basis for mortar mixed discretizations of stokes darcy flows abstract a multiscale flux basis algorithm is developed for the stokes darcy flow problem the method is based on a non overlapping domain decomposition algorithm where the global problem is reduced to a coarse scale mortar interface problem that is solved by an iterative solver subdomain solves are required at each interface iteration so the cost for the method without a multiscale basis can be high when the number of subdomains or the condition number of the interface problem is large the proposed algorithm involves precomputing a multiscale flux basis which consists of the flux or velocity trace response from each mortar degree of freedom it is computed by each subdomain independently before the interface iteration begins the subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis this may lead to a significant reduction in computational cost since the number of subdomain solves is fixed depending only on the number of mortar degrees of freedom associated with a subdomain several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation for large scale stokes darcy problems

approach to an irregular time series on the basis of the fractal theory we present a technique to measure the fractal dimension of the set of points t f t forming the graph of a function f defined on the unit interval first we apply it to a fractional brownian function 1 which has a property of self similarity for all scales and we can get the stable and precise fractal dimension this technique is also applied to the observational data of natural phenomena it does not show self similarity all over the scale but has a different self similarity across the characteristic time scale the present method gives us a stable characteristic time scale as well as the fractal dimension

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

subgrid upscaling and mixed multiscale finite elements second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale epsilon can be approximated on coarse meshes of spacing h gg epsilon only if one uses special techniques the mixed variational multiscale method also called subgrid upscaling can be used and this method is extended to allow oversampling of the local subgrid problems the method is shown to be equivalent to the multiscale finite element method when one uses the lowest order raviart thomas spaces and provided that there are no fine scale components in the source function f in the periodic setting a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is o epsilon hm sqrt epsilon h where m 1 moreover m 2 if one uses the second order brezzi douglas marini or brezzi douglas duran fortin spaces and no oversampling the error bounding constant depends only on the hm 1 norm of f and so is independent of small scales when m 1 when oversampling is not used a superconvergence result for the pressure approximation is shown

a global jacobian method for mortar discretizations of nonlinear porous media flows we describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method there are two main ideas 1 linearize the global system in both subdomain and interface variables simultaneously to yield a single newton iteration and 2 algebraically eliminate subdomain velocities and optionally subdomain pressures to solve linear systems for the 1st or the 2nd schur complements solving the 1st schur complement system gives the multiscale solution without the need to solve an interface iteration solving the 2nd schur complement system gives a linear interface problem for a nonlinear model the methods are less complex than a previously developed nonlinear mortar algorithm which requires two nested newton iterations and a forward difference approximation furthermore efficient linear preconditioners can be applied to speed up the iteration the methods are implemented in parallel and a numerical study is performed

stochastic multiscale flux basis for stokes darcy flows abstract three algorithms are developed for uncertainty quantification in modeling coupled stokes and darcy flows the porous media may consist of multiple regions with different properties the permeability is modeled as a non stationary stochastic variable with its log represented as a sum of local karhunen loeve kl expansions the problem is approximated by stochastic collocation on either tensor product or sparse grids coupled with a multiscale mortar mixed finite element method for the spatial discretization a non overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem which is solved by an iterative solver for each stochastic realization in the traditional domain decomposition implementation each subdomain solves a local dirichlet or neumann problem in every interface iteration to reduce this cost two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced the basis consists of the local flux or velocity trace responses from each mortar degree of freedom it is computed for each subdomain independently before the interface iteration begins the use of the multiscale flux basis avoids the need for subdomain solves on each iteration the deterministic basis is computed at each stochastic collocation and used only at this realization the stochastic basis is formed by further looping over all local realizations of a subdomain s kl region before the stochastic collocation begins it is reused over multiple realizations numerical tests are presented to illustrate the performance of the three algorithms with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms

adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics in this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method the error estimates are based on the solution of an appropriately defined adjoint problem we present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain the primal subdomains are discretized using either an interior penalty discontinuous galerkin method or a continuous galerkin method with weakly imposed dirichlet conditions the mixed subdomains are discretized using raviart thomas mixed finite elements the a posteriori error estimate also accounts for the errors due to adjoint inconsistent subdomain discretizations the coupling between the subdomain discretizations is achieved via a mortar space we show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive

a new coupled approach high accuracy numerical method for the solution of 3d non linear biharmonic equations in this paper we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three dimensional non linear biharmonic partial differential equations on a 19 point compact stencil using coupled approach the numerical solutions of unknown variable u x y z and its laplacian 2u are obtained at each internal grid point the resulting stencil algorithm is presented which can be used to solve many physical problems the proposed method allows us to use the dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary we also show that special treatment is required to handle the boundary conditions the new method is tested on three problems and the results are compared with the corresponding second order approximation which we also discuss using coupled approach

critical storage capacity of the j 1 neural network for neural networks in which the coupling jij are allowed to take on the values jij 1 or jij 1 the authors determine numerically the critical storage capacity for random unbiased patterns as a function of the stability they use an exact enumeration scheme based on the gray code and a continuous distribution for the patterns to control finite size effects results are presented for n 25 they indicate an optimal storage capacity of c 0 82 n

implementation of a mortar mixed finite element method using a multiscale flux basis this paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems the algorithm uses non overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem which is then solved using an iterative method the original implementation by arbogast pencheva wheeler and yotov multiscale model simul 2007 required solving one local dirichlet problem on each subdomain per interface iteration we alter this implementation by forming a multiscale flux basis this basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom on each subdomain independently the computation of these basis functions requires solving a fixed number of dirichlet subdomain problems taking linear combinations of the multiscale flux basis functions replaces the need to solve any dirichlet subdomain problems during the interface iteration this new implementation yields the same solution as the original implementation and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain the gain in computational efficiency increases with the number of subdomains

robust a posteriori error control and adaptivity for multiscale multinumerics and mortar coupling we consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains termed multinumerics coupled using the mortar technique the grids need not match along the interfaces we are also interested in the multiscale setting where the subdomains are partitioned by a mesh of size h whereas the interfaces are partitioned by a mesh of much coarser size h and where lower order polynomials are used in the subdomains and higher order polynomials are used on the mortar interface mesh we derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm our estimates are also locally efficient and one of them is robust with respect to the ratio h h under an assumption of sufficient regularity of the weak solution the present approach allows bounding separately and comparing mutually the subdomain and interface errors a subdomain interface adaptive refinement s

a new method to deduce high order compact difference schemes for two dimensional poisson equation abstract this paper introduces a novel method for deducing high order compact difference schemes for the two dimensional 2d poisson equation like finite volume method a dual partition is introduced combining simpson integral formula and parabolic interpolation a family of fourth order and sixth order compact difference schemes are obtained based on three different types of dual partitions moreover several new fourth order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth order schemes which have been presented in others work the outline for the nonlinear poisson equation is also given numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth order and sixth order compact difference schemes

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

the multiscale robin coupled method for flows in porous media abstract a multiscale mixed method aiming at the accurate approximation of velocity and pressure fields in heterogeneous porous media is proposed the procedure is based on a new domain decomposition method in which the local problems are subject to robin boundary conditions the domain decomposition procedure is defined in terms of two independent spaces on the skeleton of the decomposition corresponding to interface pressures and fluxes that can be chosen with great flexibility to accommodate local features of the underlying permeability fields the well posedness of the new domain decomposition procedure is established and its connection with the method of douglas et al 1993 12 is identified also allowing us to reinterpret the known procedure as an optimized schwarz or two lagrange multiplier method the multiscale property of the new domain decomposition method is indicated and its relation with the multiscale mortar mixed finite element method mmmfem and the multiscale hybrid mixed mhm finite element method is discussed numerical simulations are presented aiming at illustrating several features of the new method initially we illustrate the possibility of switching from mmmfem to mhm by suitably varying the robin condition parameter in the new multiscale method then we turn our attention to realistic flows in high contrast channelized porous formations we show that for a range of values of the robin condition parameter our method provides better approximations for pressure and velocity than those computed with either the mmmfem and the mhm this is an indication that our method has the potential to produce more accurate velocity fields in the presence of rough realistic permeability fields of petroleum reservoirs

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

a survey of the theory of hypercube graphs abstract we present a comprehensive survey of the theory of hypercube graphs basic properties related to distance coloring domination and genus are reviewed the properties of the n cube defined by its subgraphs are considered next including thickness coarseness hamiltonian cycles and induced paths and cycles finally various embedding and packing problems are discussed including the determination of the cubical dimension of a given cubical graph

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

consistent treatment of boundaries with mortar contact formulations using dual lagrange multipliers abstract computational modelling of contact problems raises two basic questions which method should be used to enforce the contact conditions and how should this method be discretised the most popular enforcement methods are the lagrange multiplier method the penalty method and combinations of these two a frequently used discretisation method is the so called node to segment approach however this approach might lead to problems like jumps in contact forces loss of convergence or failure to pass the patch test thus in the last few years several segment to segment contact algorithms based on the mortar method were proposed combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations on the other hand a lagrange multiplier mortar method requires additional unknowns hence condensation of the lagrange multipliers is desirable to preserve the initial size of the system of equations this can be achieved by interpolating the lagrange multipliers with so called dual shape functions discretising two contacting bodies leads to opposed contact surface representations of finite element edges called slave and master elements respectively in current versions of dual lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element instead of the entire element belongs to the contact area we present a modified definition of the dual shape functions in such slave elements the basic idea is to construct dual shape functions that fulfill the so called biorthogonality condition within the contact area this leads to consistent mortar matrices also in the boundary region to avoid ill conditioning of the stiffness matrix the modified mortar matrices are weighted with appropriate weighting factors in doing so the corresponding modified lagrange multiplier nodal values are of the same order as the unmodified ones various examples demonstrate the performance of the modified mortar contact algorithm

first order corrections to the homogenised eigenvalues of a periodic composite medium a convergence proof let e be a dirichlet eigenvalue of the periodically rapidly oscillating elliptic operator a x e and let be a corresponding simple eigenvalue of the homogenised operator a we characterise the possible limit points of the ratio e e as e 0 our characterisation is quite explicit when the underlying domain is a planar convex classical polygon with sides of rational or infinite slopes in particular in this case it implies that there is often a continuum of such limit points

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

mortar multiscale finite element methods for stokes darcy flows we investigate mortar multiscale numerical methods for coupled stokes and darcy flows with the beavers joseph saffman interface condition the domain is decomposed into a series of subdomains coarse grid of either stokes or darcy type the subdomains are discretized by appropriate stokes or darcy finite elements the solution is resolved locally in each coarse element on a fine scale allowing for non matching grids across subdomain interfaces coarse scale mortar finite elements are introduced on the interfaces to approximate the normal stress and impose weakly continuity of the normal velocity stability and a priori error estimates in terms of the fine subdomain scale h h and the coarse mortar scale h h are established for fairly general grid configurations assuming that the mortar space satisfies a certain inf sup condition several examples of such spaces in two and three dimensions are given numerical experiments are presented in confirmation of the theory

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

analysis of a fourth order compact scheme for convection diffusion in 1984 gupta et al introduced a compact fourth order finite difference convection diffusion operator with some very favorable properties in particular this scheme does not seem to suffer excessively from spurious oscillatory behavior and it converges with standard methods such as gauss seidel or sor hence multigrid regardless of the diffusion this scheme has been rederived developed including some variations and applied in both convection diffusion and navier stokes equations by several authors accurate solutions to high reynolds number flow problems at relatively coarse resolutions have been reported these solutions were often compared to those obtained by lower order discretizations such as second order central differences and first order upstream discretizations the latter it was stated achieved far less accurate results due to the artificial viscosity which the compact scheme did not include we show here that while the compact scheme indeed does not suffer from a cross stream artificial viscosity as does the first order upstream scheme when the characteristic direction is not aligned with the grid it does include a streamwise artificial viscosity that is inversely proportional to the natural viscosity this term is not always benign 7 refs 1 fig 1 tab

a posteriori error estimates for the mortar mixed finite element method several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived a residual based estimator provides optimal upper and lower bounds for the pressure error an efficient and reliable estimator for the velocity and mortar pressure error is also derived which is based on solving local element problems in a higher order space the interface flux jump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only

domain decomposition for linear elasticity with dg jumps and mortars in this article we present a domain decomposition method for solving a linear elasticity system data are transmitted by jumps as in a discontinuous galerkin method and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains a decoupling algorithm condenses the unknowns on the interface the matrix of the system is constructed by parallel computation numerical experiments confirming the theory are presented

particle method approximation of oscillatory solutions to hyperbolic differential equations particle methods approximating hyperbolic partial diferential equations with oscillatory solutions are studied convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid highly oscillatory solutions to the broadwell and variable coefficient carleman models are considered homogenization results are given and the approximations of more general systems are discussed numerical examples are presented

numerical simulation of singularly perturbed non linear elliptic boundary value problems using finite element method the present paper presents the finite element solution of two dimensional non linear singularly perturbed elliptic partial differential equation subject to appropriate dirichlet boundary conditions a new fifth order convergent newton type iterative method has been described and used to linearize the non linear problem the inclusion of this newton s method of fifth order convergence in finite element method for solving non linear system of equations reduces the number of iterations and hence the cost of computation to demonstrate the usefulness of the proposed scheme a non convex variational ginzburg landau equation is considered

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

an error analysis for the finite element method applied to convection diffusion problems abstract this paper analyzes the finite element method applied to a convection diffusion model problem linear elements are used for the trial space the error is measured in a norm closely related to the l sup p norm when the test space is composed of linear elements with parabolic upwinding the method is shown to be optimal when the input data is piecewise smooth a condition which is usually observed in practice without these smoothness assumptions the method is shown to be non optimal even if the class of test spaces is extended to include any elements which have a shape independent of the mesh size author

random walk generated by random permutations of 1 2 3 n 1 we study properties of a non markovian random walk x n l l 0 1 2 n evolving in discrete time l on a one dimensional lattice of integers whose moves to the right or to the left are prescribed by the rise and descent sequences characterizing random permutations of n 1 1 2 3 n 1 we determine exactly the probability of finding the end point xn x n n of the trajectory of such a permutation generated random walk pgrw at site x and show that in the limit n it converges to a normal distribution with a smaller compared to the conventional polya random walk diffusion coefficient we formulate as well an auxiliary stochastic process whose distribution is identical to the distribution of the intermediate points x n l l n which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of turns of the pgrw trajectories

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

a fractal set from the binary reflected gray code the permutation associated with the decimal expression of the binary reflected gray code with n bits is considered its cycle structure is studied considered as a set of points its self similarity is pointed out as a fractal it is shown to be the attractor of an ifs for large values of n the set is examined from the point of view of time series analysis

a class of efficient quadrature based predictor corrector methods for solving nonlinear systems we extend a class of quadrature based predictor corrector techniques for root finding to multivariate systems they are found to have a rate of convergence of 1 2 regardless of the degree of precision for the quadrature technique from which they are derived provided it is at least one by reusing the linear system from the previous iterate this class incorporates a significant improvement in computational time relative to the standard class through the inclusion of an lu decomposition during the iteration complexity is equivalent to newton s method as they only require knowledge of f x and f x

multiscale mortar expanded mixed discretization of nonlinear elliptic problems abstract we consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method this is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non overlapping subdomains an unknown lagrange multiplier is introduced on the interfaces which serve as pressure dirichlet boundary condition for local subdomain problems a finite element space is defined on interface to approximate the pressure boundary such that the normal fluxes match weakly on interface we demonstrated solvability of discretization and established a priori l2 error estimates for both vector and scalar approximations we proved the optimal order convergence rates by an appropriate choice of mortar space and polynomial degree of approximation the uniqueness of discrete problem is shown for sufficiently small values of mesh size an error estimate for the mortar pressure is derived via linear interface formulation with pressure dependent coefficient we also present the analysis of the linear second order elliptic problem and prove the similar results the computational experiments are presented to validate theory

high order variable mesh off step discretization for the solution of 1 d non linear hyperbolic equation abstract in this paper we propose a new high order three level implicit method based on off step discretization on a non uniform mesh for the solution of 1 d non linear hyperbolic partial differential equation of the form u tt u xx g x t u u x u t subject to appropriate initial and dirichlet boundary conditions we use only three evaluations of the function g and three grid points at each time level in a compact cell our method is directly applicable to the wave equation in polar coordinates and we do not require any special technique to handle singular coefficients of the differential equation the method is convergent for uniform mesh numerical results are provided to justify the usefulness of the proposed method

high order implicit finite difference schemes for the two dimensional poisson equation abstract in this paper a new family of high order finite difference schemes is proposed to solve the two dimensional poisson equation by implicit finite difference formulas of 2 m 1 operator points the implicit formulation is obtained from taylor series expansion and wave plane theory analysis and it is constructed from a few modifications to the standard finite difference schemes the approximations achieve 2 m 4 order accuracy for the inner grid points and up to eighth order accuracy for the boundary grid points using a successive over relaxation method the high order implicit schemes have faster convergence as m is increased compensating the additional computation of more operator points thus the proposed solver results in an attractive method easy to implement with higher order accuracy but nearly the same computation cost as those of explicit or compact formulation in addition particular case m 1 yields a new compact finite difference schemes of sixth order of accuracy numerical experiments are presented to verify the feasibility of the proposed method and the high accuracy of these difference schemes

random walks and random permutations a connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the gue random hermitian matrices the scaling of the distribution gives the maximum mean displacement after t time steps as 2t 1 2 with standard deviation proportional to 1 3 the exponent 1 3 is typical of a large class of two dimensional growth problems

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

a stochastic mortar mixed finite element method for flow in porous media with multiple rock types this paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types the governing equations are based on darcy s law with nonstationary stochastic permeability represented as a sum of local karhunen loeve expansions the approximation uses stochastic collocation on either a tensor product or a sparse grid coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method the latter method requires solving a coarse scale mortar interface problem via an iterative procedure the traditional implementation requires the solution of local fine scale linear systems on each iteration we employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration in the stochastic setting the basis is further reused over multiple realizations leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude error analysis and numerical experiments are presented

first order corrections to the homogenized eigenvalues of a periodic composite medium this paper studies the eigenvalue problem associated with the vibration of a com posite medium assumed to have a periodic microstructure the investigation concentrates on the first order correction to the homogenized eigenvalues a major difficulty arises due to the interaction of the periodic microstructure with the boundary of the medium the way in which the boundary intersects the underlying periodic structure strongly influences the value of the correction a rigorous derivation is given of the eigenvalue correction formula the properties of the eigenvalue correction are studied in one and two dimensional numerical calculations

a survey of combinatorial gray codes the term combinatorial gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified small way this notion generalizes the classical binary reflected gray code scheme for listing n bit binary numbers so that successive numbers differ in exactly one bit position as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families including permutations and combinations the area of combinatorial gray codes was popularized by herbert wilf in his invited address at the siam conference on discrete mathematics in 1988 and his subsequent siam monograph combinatorial algorithms an update 1989 in which he posed some open problems and variations on the theme this resulted in much recent activity in the area and most of the problems posed by wilf are now solved in this paper we survey the area of combinatorial gray codes describe recent results variations and trends and highlight some open problems

homogenization and convergence of the vortex method for 2 d euler equations with oscillatory vorticity fields the 2 d incompressible euler equations with oscillatory vorticity fields are studied a homogenization result for 2 d euler equations in velocity vorticity formulation is obtained and weak continuity of the equations is proved convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles numerical results are given comparisons are made with the corresponding finite difference approximation

a new two level implicit discretization of o k 2 kh 2 h 4 for the solution of singularly perturbed two space dimensional non linear parabolic equations we propose a new two level implicit difference method of o k2 kh2 h4 for the solution of singularly perturbed non linear parabolic differential equation e uxx uyy f x y t u ux u y ut 0 0 subject to appropriate initial and dirichlet boundary conditions where k 0 and h 0 are grid sizes in time and space directions respectively and e 0 is a small parameter we also develop new methods of o kh2 h4 for the estimates of u x and u y in all cases we use 9 spatial grid points and a single computational cell the proposed methods are directly applicable to singular problems we do not require any special scheme to solve singular problems we also discuss alternating direction implicit adi method for solving diffusion equation in polar cylindrical coordinates this method permits multiple use of the one dimensional tri diagonal algorithm with a considerable saving in computing time and produces a very efficient solver it is shown that the adi method is unconditionally stable numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

analysis of multiscale mortar mixed approximation of nonlinear elliptic equations abstract a multiscale mortar mixed finite element method is established to approximate non linear second order elliptic equations the method is based on non overlapping domain decomposition and mortar finite element methods the existence and uniqueness of the approximation are demonstrated and a priori l 2 error estimates for the velocity and pressure are derived an error bound for mortar pressure is proved convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete pressure dependent coefficient optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation the quadratic convergence of the newton raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem numerical experiments are performed to support theoretic results

dynamical analysis and perturbation solution of an seir epidemic model abstract in this paper we present a kind of technique to obtain the asymptotical solution of an seir where s is the susceptible population e is the exposed population i is the infectious population and r is the recovered population epidemic model by employing the method of perturbation at first we investigate the epidemic model by analysing its dynamical behavior then we use the method of perturbation to obtain the analytical solution of the model we assign values for parameters and draw figures to observe the magnitude of error of the perturbation method in a macro view finally we analyse macroscopically the two comparison chart on analytical solution and the exact solution and know that it is feasible to analyse the solution of the epidemic model by using the perturbation method

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a new fast algorithm based on half step discretization for one space dimensional quasilinear hyperbolic equations abstract in this article we describe a new compact three level implicit method of order four in time and space based on half step discretization for one space dimensional quasilinear hyperbolic equation u tt a x t u u xx f x t u u x u t defined in the semi infinite solution region where a 0 we require only nine grid points for the unknown variable u x t and two extra half step points each for x and t variables the proposed method is directly applicable to wave equation with singular coefficients which is main attraction of our work we do not require extra grid points for computation we describe the derivation of the method in detail the proposed method when applied to damped wave equation is shown to be unconditionally stable many benchmark problems are solved to confirm the fourth order convergence of the proposed method

a lagrange multiplier method for a stokes biot fluid poroelastic structure interaction model we study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium the free fluid is governed by the stokes equations while the flow in the poroelastic medium is modeled using the biot poroelasticity system equilibrium and kinematic conditions are imposed on the interface a mixed darcy formulation is employed resulting in continuity of flux condition of essential type a lagrange multiplier method is employed to impose weakly this condition a stability and error analysis is performed for the semi discrete continuous in time and the fully discrete formulations a series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters

order h 4 difference methods for a class of singular two space elliptic boundary value problems abstract in this article we derive difference methods of o h4 for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points we obtain two sets of fouth order difference methods one in the absence of mixed derivatives second when the coefficients of uxy are not equal to zero and the coefficients of uxx and uyy are equal there do not exist fourth order schemes involving nine grid points for the general case the method having two variables has been tested on two dimensional viscous incompressible steady state navier stokes model equations in polar coordinates the proposed difference method for scalar equation is also applied to the poisson s equation in polar coordinates some numerical examples are provided to illustrate the fourth order convergence of the proposed methods

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

statistical estimates of the n bit gray codes by restricted random generation of permutations of 1 to 2 n the number of n bit gray codes is the number in a well defined subset of the permutations of the integers 1 to 2 n generating random permutations with associated estimates under suitably restrictive selection rules produces a discrete distribution whose expectation value is the number of such codes the number of hamiltonian circuits on the n cube cyclic gray codes is a further subset which can readily be estimated also reliable statistical estimates up to n 6 were produced with reasonable speed by computer implementation of this monte carlo process excellent agreement with the exact values for n 4 and 5 was obtained proofs are given of the validity of the technique and of an upper bound for the total number of gray codes the technique could also be used to count permutation subsets other than gray codes

poisson equation solver with fourth order accuracy by using interpolated differential operator scheme abstract a poisson equation solver with fourth order spatial accuracy has been developed by using the interpolated differential operator ido scheme the number of grids required for obtaining the results of numerical computation with the same accuracy as that by the second order center finite difference method is drastically reduced the consistency is confirmed to the use of the multigrid method and the red black algorithm the decrease of the cpu time by the multigrid method and by the parallel computing indicates the applicability of the ido poisson solver to large scale problems

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

a new high accuracy two level implicit off step discretization for the system of two space dimensional quasi linear parabolic partial differential equations abstract in this article we propose a two level compact implicit off step discretization for the solution of the system of 2d quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions new methods for the estimates of first order space derivatives of the solution are also derived these methods are fourth order accurate in space and second order accurate in time and involve only nine spatial grid points of a single compact cell we further develop the alternating direction implicit adi scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates the proposed methods are directly applicable to singular problems without the need of any special technique which is the main advantage of this work the method is effectively applied to the time dependent navier stokes model equations in polar coordinates numerical examples are provided to illustrate the accuracy of the methods

approximate static condensation algorithm for solving multi material diffusion problems on meshes non aligned with material interfaces abstract in this paper we describe a new algorithm for solving multi material diffusion problem when material interfaces are not aligned with the mesh in this case interface reconstruction methods are used to construct approximate representation of interfaces between materials they produce so called multi material cells in which materials are represented by material polygons that contain only one material the reconstructed interface is not continuous between cells we suggest the new method for solving multi material diffusion problems on such meshes and compare its performance with known homogenization methods

a new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems abstract in this article we derive a new difference method of o h4 so called arithmetic average discretization for the solution of two dimensional non linear singularly perturbed elliptic partial differential equation of the form e uxx uyy f x y u ux uy 0 0 is a small parameter we also derive new methods of o h4 for the estimates of u n which are quite often of interest in many physical problems in all cases we require only 9 grid points and a single computational cell the main advantage of the proposed methods is that the methods are directly applicable to singular problems we do not require any special technique or modification to solve singular problems numerical results are provided to demonstrate the usefulness of the methods discussed

the smart blage algorithm for singularly perturbed 2d elliptic partial differential equations we discuss a new smart block alternating group explicit smart blage algorithm for solving the symmetric block linear systems derived from nine point fourth order method based on arithmetic average discretizations to solve the singularly perturbed two dimensional elliptic equations the smart blage algorithm is predicted to save time as the idea involved is to eliminate evaluating two similar terms appearing in the algorithm the error analysis of the smart blage algorithm is discussed briefly numerical tests are performed to demonstrate the usefulness of the proposed smart blage method and compared with the corresponding block successive over relaxation bsor method by considering two test problems