finite volume element methods for non definite problems the error estimates for finite volume element method applied to 2 and 3 d non definite problems are derived a simple upwind scheme is proven to be unconditionally stable and first order accurate

an algorithm using the finite volume element method and its splitting extrapolation this paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation this method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation such as a high order of accuracy a high degree of parallelism less computational complexity and more flexibility than a richardson extrapolation because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid a large scale multidimensional problem is turned into a set of smaller discrete subproblems additionally this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition

two grid methods for finite volume element approximations of nonlinear parabolic equations two grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method the methods are based on one coarse grid space and one fine grid space the nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid the two grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 3 lnh as a result solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

generalized matrix equivalence theorem for polarization theory a generalized equivalence theorem for polarization theory is formulated and proven it is shown that anisotropic properties of homogeneous nondepolarizing media can be presented as a combination of four basic mechanisms linear and circular phase and linear and circular amplitude anisotropy expressions for the generalized effect operators of algebraic or operator optics are obtained and the inverse problem of crystal optics is solved in terms of physically realizable anisotropy parameters

a numerical framework for diffusion controlled bimolecular reactive systems to enforce maximum principles and the non negative constraint we present a novel computational framework for diffusive reactive systems that satisfies the non negative constraint and maximum principles on general computational grids the governing equations for the concentration of reactants and product are written in terms of tensorial diffusion reaction equations we restrict our studies to fast irreversible bimolecular reactions if one assumes that the reaction is diffusion limited and all chemical species have the same diffusion coefficient one can employ a linear transformation to rewrite the governing equations in terms of invariants which are unaffected by the reaction this results in two uncoupled tensorial diffusion equations in terms of these invariants which are solved using a novel non negative solver for tensorial diffusion type equations the concentrations of the reactants and the product are then calculated from invariants using algebraic manipulations the novel aspect of the proposed computational framework is that it will always produce physically meaningful non negative values for the concentrations of all chemical species several representative numerical examples are presented to illustrate the robustness convergence and the numerical performance of the proposed computational framework we will also compare the proposed framework with other popular formulations in particular we will show that the galerkin formulation which is the standard single field formulation does not produce reliable solutions and the reason can be attributed to the fact that the single field formulation does not guarantee non negative solutions we will also show that the clipping procedure which produces non negative solutions but is considered as a variational crime does not give accurate results when compared with the proposed computational framework

chirality of wavefunctions for three coalescing levels the analytic structure in the vicinity of three coalescing eigenvalues ep3 of a matrix problem is investigated it is argued that the three eigenfunctions also coalescing at the ep3 invoke a true chiral behaviour in the vicinity of the ep3 and that they can be related to a three dimensional helix the orientation of the helix depends on the distribution of the widths of the three levels in the vicinity of the ep3

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

a feti based domain decomposition technique for time dependent first order systems based on a dae approach we present a novel partitioned coupling algorithm to solve first order time dependent non linear problems e g transient heat conduction the spatial domain is partitioned into a set of totally disconnected subdomains the continuity conditions at the interface are modeled using a dual schur formulation where the lagrange multipliers represent the interface fluxes or the reaction forces that are required to maintain the continuity conditions the interface equations along with the subdomain equations lead to a system of differential algebraic equations daes for the resulting equations a numerical algorithm is developed which includes choosing appropriate constraint stabilization techniques the algorithm first solves for the interface lagrange multipliers which are subsequently used to advance the solution in the subdomains the proposed coupling algorithm enables arbitrary numeric schemes to be coupled with different time steps i e it allows subcycling in each subdomain this implies that existing software and numerical techniques can be used to solve each subdomain separately the coupling algorithm can also be applied to multiple subdomains and is suitable for parallel computers we present examples showing the feasibility of the proposed coupling algorithm copyright 2008 john wiley sons ltd

initial and boundary conditions for the lattice boltzmann method an alternative approach of implementing initial and boundary conditions for the lattice boltzmann method is presented the basic idea is to calculate the lattice boltzmann populations at a boundary node from the fluid variables that are specified at this node using the gradients of the fluid velocity the numerical performance of the lattice boltzmann method is tested on several problems with exact solutions and is also compared to an explicit finite difference projection method the discretization error of the lattice boltzmann method decreases quadratically with finer resolution both in space and in time the roundoff error of the lattice boltzmann method creates problems unless double precision arithmetic is used

the lattice boltzmann equation a new tool for computational fluid dynamics abstract we present a series of applications which demonstrate that the lattice boltzmann equation is an adequate computational tool to address problems spanning a wide spectrum of fluid regimes ranging from laminar to fully turbulent flows in two and three dimensions

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

error estimates in l 2 h 1 and l in covolume methods for elliptic and parabolic p roblems a unified approach in this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane we introduce a general approach for connecting these methods with finite element method analysis this unified approach is used to prove known convergence results in the h 1 l 2 norms and new results in the max norm for the elliptic problems we demonstrate that the error u u h between the exact solution u and the approximate solution u h in the maximum norm is o h 2 ln h in the linear element case furthermore the maximur norm error in the gradient is shown to be of first order similar results hold for the parabolic problems

the characteristic finite volume element method for the nonlinear convection dominated diffusion problem in modern numerical simulation of prospecting and exploiting oil gas resources and environmental science it is important to consider a numerical method for nonlinear convection dominated diffusion problems based on actual conditions such as the three dimensional characteristics of large scale science engineering computation we present a kind of characteristic finite volume element method some techniques such as calculus of variations commutating operators the theory of prior estimates and techniques are adopted suboptimal order error estimate in l 2 norm and optimal order error estimate in h 1 norm are derived to determine the errors for the approximate solution numerical results are presented to verify the performance of the scheme

on the construction and analysis of high order locally conservative finite volume type methods for one dimensional elliptic problems locally conservative finite volume type methods based on continuous piecewise polynomial functions of degree r ge 2 are introduced and analyzed in the context of indefinite elliptic problems in one space dimension the new methods extend and generalize the classical finite volume method based on piecewise linear functions we derive a priori error estimates in the l2 h1 and l infty norm and discuss superconvergence effects for the error and its derivative explicit residual based a posteriori error bounds in the l2 and energy norm are also derived we compute the experimental order of convergence and show the results of an adaptive algorithm based on the a posteriori error estimates

bridging multiple structural scales with a generalized finite element method summary this paper presents a generalized fem based on the solution of interdependent coarse scale global and fine scale local problems in order to resolve multiscale effects due to fine scale heterogeneities overall structural behavior is captured by the global problem while local problems focus on the resolution of fine scale solution features in regions where material heterogeneities may govern the structural response fine scale problems are accurately solved in parallel and to address the intrinsic coupling of scales these solutions are embedded into the global solution space using a partition of unity approach this method is demonstrated on representative heat transfer examples in order to examine its accuracy efficiency and flexibility copyright 2014 john wiley sons ltd

a monolithic multi time step computational framework for first order transient systems with disparate scales abstract developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering a desirable feature to have in a numerical formulation for multiscale transient problems is to be able to employ different time steps multi time step coupling and different time integrators and different numerical formulations mixed methods in different regions of the computational domain to this end we present two new monolithic multi time step mixed coupling methods for first order transient systems we shall employ unsteady advection diffusion reaction equation with linear decay as the model problem which offers several unique challenges in terms of non self adjoint spatial operator and rich features in the solutions we shall employ the dual schur domain decomposition technique to split the computational domain into an arbitrary number of subdomains it will be shown that the governing equations of the decomposed problem after spatial discretization will be differential algebraic equations this is a crucial observation to obtain stable numerical results two different methods of enforcing compatibility along the subdomain interface will be used in the time discrete setting a systematic theoretical analysis which includes numerical stability influence of perturbations bounds on drift along the subdomain interface will be performed the first coupling method ensures that there is no drift along the subdomain interface but does not facilitate explicit implicit coupling the second coupling method allows explicit implicit coupling with controlled but non zero drift in the solution along the subdomain interface several canonical problems will be solved to numerically verify the theoretical predictions and to illustrate the overall performance of the proposed coupling methods finally we shall illustrate the robustness of the proposed coupling methods using a multi time step transient simulation of a fast bimolecular advective diffusive reactive system

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

analysis of mixed finite element methods on locally refined grids we consider the mixed finite element method for locally refined triangulations a local projection operator hat pi h is defined to satisfy the commutation property that is required in the general theory of mixed methods our results can be applied to every known space of arbitrary order over rectangles or triangles optimal order error estimates and superconvergence for the flux along the gauss lines are established

hybrid simulations of reaction diffusion systems in porous media hybrid or multiphysics algorithms provide an efficient computational tool for combining micro and macroscale descriptions of physical phenomena their use becomes imperative when microscale descriptions are too computationally expensive to be conducted in the whole domain while macroscale descriptions fail in a small portion of the computation domain we present a hybrid algorithm to model a general class of reaction diffusion processes in granular porous media which includes mixing induced mineral precipitation on or dissolution of the porous matrix these processes cannot be accurately described using continuum darcy scale models the pore scale darcy scale hybrid is constructed by coupling solutions of the reaction diffusion equations rde at the pore scale with continuum darcy level solutions of the averaged rdes the resulting hybrid formulation is solved numerically by employing a multiresolution meshless discretization based on the smoothed particle hydrodynamics method this ensures seamless noniterative coupling of the two components of the hybrid model computational examples illustrate the accuracy and efficiency of the hybrid algorithm

conservative load transfer along curved fluid solid interface with non matching meshes we investigate the effect of curvature on the accuracy of schemes used to transfer loads along the interface in coupled fluid solid simulations involving non matching meshes we analyze two types of load transfer schemes for the coupled system a point to element projection schemes and b common refinement schemes the accuracy of these schemes over the curved interface is assessed with the aid of static and transient problems we show that the point to element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution the common refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non matching meshes we show theoretically that the accurate transfer preserves the stability of the coupled system while maintaining the energy conservation over a reference interface finally we introduce simple analytical error functions which correlate well with the numerical errors of the load transfer schemes

superconvergent analysis of a nonconforming mixed finite element method for non stationary conduction convection problem abstract in this paper a backward euler mixed finite element mfe approximation scheme is presented for the non stationary conduction convection problem in which the constrained nonconforming rotated cnr q 1 element is used to approximate the velocity u the temperature t and the q 0 element to the pressure p based on the characters of these elements and some special skills i e mean value skill and a new transforming skill with respect to the superclose results of order o h 2 for u t in the broken h 1 norm and p in l 2 norm are deduced respectively here h is the subdivision parameter and the time step furthermore the global superconvergent results are obtained through the interpolation postprocessing technique finally numerical results are provided to confirm the theoretical analysis

a hybrid multi time step framework for pore scale and continuum scale modeling of solute transport in porous media abstract in this paper we propose a computational framework which is based on a domain decomposition technique to employ both finite element method which is a popular continuum modeling approach and lattice boltzmann method which is a popular pore scale modeling approach in the same computational domain to bridge the gap across the disparate length and time scales we first propose a new method to enforce continuum scale boundary conditions i e dirichlet and neumann boundary conditions onto the numerical solution from the lattice boltzmann method this method is based on maximization of entropy and preserve the non negativity of discrete distributions under the lattice boltzmann method the proposed computational framework allows different grid sizes orders of interpolation and time steps in different subdomains this allows for different desired resolutions in the numerical solution in different subdomains through numerical experiments the effect of grid and time step refinement disparity of time steps in different subdomains domain partitioning and the number of iteration steps on the accuracy and rate of convergence of the proposed methodology are studied finally to showcase the performance of this framework in porous media applications we use it to simulate the dissolution of calcium carbonate in a porous structure

proper orthogonal decomposition based reduced order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible boussinesq equations abstract in this study we employ a proper orthogonal decomposition pod method to establish a pod based reduced order stabilized mixed finite volume element smfve extrapolating model with very few degrees of freedom for the nonstationary incompressible boussinesq equations we provide error estimates of the pod based reduced order smfve solutions and the algorithm implementation for the pod based reduced order smfve extrapolating model in addition we present some numerical experiments that verify the validity and reliability of the pod based reduced order smfve extrapolating model

penalty combinations of the ritz galerkin and finite difference methods for singularity problems abstract penalty combination of the ritz galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities the superconvergence rate o h2 of solution derivatives by the combination can be achieved while using quasiuniform rectangular difference grids where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number it is due to its simplicity that the penalty combination of the ritz galerkin and finite difference methods is highly recommended for solving the complicated problems with multiple singularities and multiple interfaces

superconvergence of finite volume methods for the second order elliptic problem abstract this paper derives a superconvergence result for finite volume approximations of the second order elliptic problem by using a l2 projection post processing technique the superconvergence result is applicable to different kind of finite volume methods and to general quasi uniform meshes

exceptional points and non hermitian degeneracy of resonances in a two channel model we study the mixing and degeneracy of two unbound energy eigenstates resonances in a two coupled channel model of scattering and reactions we derive the necessary and sufficient conditions for existence of an exceptional point in the extended spectrum of bound and resonance energy eigenvalues in this model and show that these are not the same as in the single channel case when these conditions are satisfied in the complex energy plane the two simple resonance poles of the scattering matrix merge into one double pole at the exceptional point in parameter space the surface of the eigenenergies has a branch point of square root type and branch cuts in its real and imaginary parts that start at the exceptional point and extend in opposite directions the rich phenomenology of crossings and anticrossings of energies and widths of the doublet of unbound states as well as the changes of identity of the poles of the scattering matrix observed when one control parameter is varied while the other is kept constant is fully explained in terms of sections of the eigenenergy surfaces

a reduced order crank nicolson finite volume element formulation based on pod method for parabolic equations in this paper a proper orthogonal decomposition pod method is used to deal with a classical crank nicolson finite volume element cnfve method for two dimensional parabolic equations a reduced order cnfve formulation with lower dimensions and sufficiently high accuracy based on pod technique is established the error estimates between reduced order cnfve solutions based on the pod method and classical cnfve solutions are provided and the extrapolation algorithm for solving reduced order cnfve formulation is implemented some numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions moreover it is shown that the reduced order cnfve formulation based on pod method is feasible and efficient for solving two dimensional parabolic equations

slow non hermitian cycling exact solutions and the stokes phenomenon for non hermitian hamiltonians with an isolated degeneracy exceptional point a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of bessel functions floquet solutions returning exactly to their initial states up to a constant are found as well as exact expressions for the adiabatic multipliers when the evolving states are represented as a superposition of eigenstates of the instantaneous hamiltonian adiabatically i e for slow cycles the multipliers of exponentially subdominant eigenstates can vary wildly unlike those driven by hermitian operators which change little these variations are explained as an example of the stokes phenomenon of asymptotics improved superadiabatic approximations tame the variations of the multipliers but do not eliminate them

least squares finite element lattice boltzmann method a new numerical model of the lattice boltzmann method utilizing least squares finite element in space and crank nicolson method in time is presented the new method is able to solve problem domains that contain complex or irregular geometric boundaries by using finite element method s geometric flexibility and numerical stability while employing efficient and accurate least squares optimization for the pure advection equation on a uniform mesh the proposed method provides for fourth order accuracy in space and second order accuracy in time with unconditional stability in the time domain accurate numerical results are presented through two dimensional incompressible poiseuille flow and couette flow

proximity of degeneracies and chiral points for complex 2 2 matrices m points of degeneracy d and chiral points c where the eigenvectors are circularly polarized coincide whenm is symmetric and often lie close together when m is not symmetric this proximity is explained statistically by averaging over an ensemble in which the elements of m are random complex numbers the standard deviation of the separation between c and d points is less than half the separation of the two d points if the departure from symmetry is treated as a perturbation of strength e t he separation between c and d points is of order e 2 the probability distribution of the separation is calculated

the mmoc and mmocaa schemes for the finite volume element method of convection diffusion problems we present and analyze the modified method of characteristics mmoc and the modified method of characteristics with adjusted advection mmocaa for the finite volume element fve method of convection diffusion problems these two schemes maintain the advantages of both the mmoc and the fve method and the mmocaa scheme discussed herein conserves the conservation law globally at a minor additional computational cost optimal order error estimates in the h 1 norm are proved for these schemes a numerical example is presented to confirm the estimates

stability analysis of lattice boltzmann methods the lattice boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered although the equation is a derivative of lattice gas automata it may be interpreted as a lagrangian finite difference method for the numerical simulation of the discrete velocity boltzmann equation that makes use of a bgk collision operator as a result it is not surprising that numerical instability of lattice boltzmann methods have been frequently encountered by researchers we present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant density uniform mean flow the linear stability depends on the following parameters the distribution of the mass at a site between the different discrete speeds the bgk relaxation time the mean velocity and the wavenumber of the perturbations this parameter space is too large to compute the complete stability characteristics we report some stability results for a subset of the parameter space for a 7 velocity hexagonal lattice a 9 velocity square lattice and a 15 velocity cubic lattice results common to all three lattices are 1 the bgk relaxation time must be greater than 12 corresponding to positive shear viscosity 2 there exists a maximum stable mean velocity for fixed values of the other parameters and 3 as is increased from 12 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger

a finite volume element formulation and error analysis for the non stationary conduction convection problem abstract the non stationary conduction convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element fve method a fully discrete fve formulation and the error estimates between the fully discrete fve solutions and the accuracy solution are provided it is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that the fve method is feasible and efficient for finding the numerical solutions of the non stationary conduction convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the fve formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non stationary conduction convection problem

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

a reduced order extrapolated crank nicolson finite spectral element method based on pod for the 2d non stationary boussinesq equations abstract in this paper we mainly utilize proper orthogonal decomposition pod to reduce the order for the coefficient vector of the classical crank nicolson finite spectral element ccnfse method of the two dimensional 2d non stationary boussinesq equations about vorticity stream functions so that the reduced order method maintains all the advantages of the ccnfse method toward this end we first establish a ccnfse format with the second order time accuracy for the two dimensional 2d non stationary boussinesq equations about vorticity stream functions and analyze the existence stability and convergence of the ccnfse solutions and then by pod we establish a reduced order extrapolated crank nicolson finite spectral element roecnfse method and analyze the existence stability and convergence of the roecnfse solutions as well as offer the flowchart for seeking roecnfse solutions finally we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the roecnfse method are further verified both theory and method in this paper is new and completely different from the existing reduced order methods

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

a fully discrete stabilized mixed finite volume element formulation for the non stationary conduction convection problem abstract at first a semi discrete formulation with respect to time for the non stationary conduction convection problem is recalled then a fully discrete stabilized mixed finite volume element smfve formulation based on two local gauss integrals and parameter free is established directly from the semi discrete formulation with respect to time for the non stationary conduction convection problem following this the error estimates for the fully discrete smfve solutions are derived by means of the standard mixed finite element method finally some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results the degrees of freedom of the fully discrete smfve formulation are far fewer than those of the classical finite volume element fve formulation without any stabilization and its numerical solutions are more stable than those of the classical fve formulation without any stabilization thus validating that the fully discrete smfve formulation is feasible and efficient for finding the numerical solutions of the non stationary conduction convection problem

exceptional points of non hermitian operators exceptional points associated with non hermitian operators i e operators being non hermitian for real parameter values are investigated the specific characteristics of the eigenfunctions at the exceptional point are worked out within the domain of real parameters the exceptional points are the points where eigenvalues switch from real to complex values these and other results are exemplified by a classical problem leading to exceptional points of a non hermitian matrix

cubic superconvergent finite volume element method for one dimensional elliptic and parabolic equations in this paper a cubic superconvergent finite volume element method based on optimal stress points is presented for one dimensional elliptic and parabolic equations for elliptic problem it is proved that the method has optimal third order accuracy with respect to h 1 norm and fourth order accuracy with respect to l 2 norm we also obtain that the scheme has fourth order superconvergence for derivatives at optimal stress points for parabolic problem the scheme is given and error estimate is obtained with respect to l 2 norm finally numerical examples are provided to show the effectiveness of the method

the l p d e m a mixed finite difference method for elliptic problems a mixed finite difference method is analyzed for solving certain elliptic problems this method called l p d e m locally exact partial differential equation method was initially proposed in the frame of hydrodynamic lubrication convergence is obtained relations between this scheme and homogenization theory are also discussed for a one dimensional elliptic equation with no zero order term and in conservative form this method is an exact one some numerical results will also be given

the optical singularities of birefringent dichroic chiral crystals using a new formalism involving projection from the sphere of directions to the stereographic plane and associated complex variables explicit formulae are obtained for the two refractive indices and polarizations in optically anisotropic crystals that are both dichroic absorbing and chiral optically active this enables three types of polarization singularity to be classified and explored singular axes which are degeneracies where the two refractive indices are equal and which for a transparent non chiral crystal condense pairwise onto the optic axes c points where the polarization is purely circular right or left handed with topological index 1 or and whose positions are independent of the chirality and l lines where the polarization is purely linear dividing direction space into regions with right and left handedness a local model captures essential features of the general theory interference figures generated by slabs of crystal viewed directly or through a polarizer and or analyser enable the singularities to be displayed directly

on multi time step monolithic coupling algorithms for elastodynamics we present a way of constructing multi time step monolithic coupling methods for elastodynamics the governing equations for constrained multiple subdomains are written in dual schur form and the continuity of velocities is enforced at system time levels the resulting equations will be in the form of differential algebraic equations to crystallize the ideas we shall employ newmark family of time stepping schemes the proposed method can handle multiple subdomains and allows different time steps as well as different time stepping schemes from the newmark family in different subdomains we shall use the energy method to assess the numerical stability and quantify the influence of perturbations under the proposed coupling method two different notions of energy preservation are introduced and employed to assess the performance of the proposed method several numerical examples are presented to illustrate the accuracy and stability properties of the proposed method we shall also compare the proposed multi time step coupling method with some other methods available in the literature

superconvergent biquadratic finite volume element method for two dimensional poisson s equations in this paper a kind of biquadratic finite volume element method is presented for two dimensional poisson s equations by restricting the optimal stress points of biquadratic interpolation as the vertices of control volumes the method can be effectively implemented by alternating direction technique it is proved that the method has optimal energy norm error estimates the superconvergence of numerical gradients at optimal stress points is discussed and it is proved that the method has also superconvergence displacement at nodal points by a modified dual argument technique finally a numerical example verifies the theoretical results and illustrates the effectiveness of the method

a finite difference scheme for two dimensional semiconductor device of heat conduction on composite triangular grids abstract the momentary state of a semiconductor device of heat conduction is described by a system of four nonlinear partial differential equations one elliptic equation is for the electrostatic two parabolic equations are for the electron concentration and the hole concentration and one heat exchange equation is for the temperature according to the necessary of practical numerical simulations and based on the balance equation finite difference schemes for two dimensional transient behavior of a semiconductor device of heat conduction on composite triangular grids are constructed studying their stability and convergence properties the error estimate in the energy norm is obtained finally a numerical example is given

unfolding a diabolic point a generalized crossing scenario the typical avoided crossings for hermitian quantum systems depending on parameters the diabolic crossing scenario are generalized to the non hermitian case e g for resonances two types of crossings appear for type i the real parts show an avoided and the imaginary parts a true crossing of the eigenenergies and for type ii the opposite is found a simple symmetric non hermitian two state matrix hamiltonian is analysed in detail the diabolic point bifurcates into two exceptional ones on exceptional lines where the matrices are defective the adiabatic transport of eigenvectors and eigenstates in parameter space is discussed in this generalized diabolic crossing scenario in particular the geometric berry phases for a cyclic variation of system parameters depending on the topology of the closed curves with respect to the exceptional lines

stability and stabilization of the lattice boltzmann method we revisit the classical stability versus accuracy dilemma for the lattice boltzmann methods lbm our goal is a stable method of second order accuracy for fluid dynamics based on the lattice bhatnager gross krook method lbgk the lbgk scheme can be recognized as a discrete dynamical system generated by free flight and entropic involution in this framework the stability and accuracy analysis are more natural we find the necessary and sufficient conditions for second order accurate fluid dynamics modeling in particular it is proven that in order to guarantee second order accuracy the distribution should belong to a distinguished surface the invariant film up to second order in the time step this surface is the trajectory of the quasi equilibrium distribution surface under free flight the main instability mechanisms are identified the simplest recipes for stabilization add no artificial dissipation up to second order and provide second order accuracy of the method two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blowup demonstration of the proposed stable lbgk schemes are provided by the numerical simulation of a one dimensional 1d shock tube and the unsteady 2d flow around a square cylinder up to reynolds number mathrm re ensuremath sim 20 phantom rule 0 2em 0ex 000

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

a two grid method for finite volume element approximations of second order nonlinear hyperbolic equations the two grid method is studied for solving a two dimensional second order nonlinear hyperbolic equation using finite volume element method the method is based on two different finite element spaces defined on one coarse grid with grid size h and one fine grid with grid size h respectively the nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid a prior error estimate in the h 1 norm is proved to be o h h 3 lnh for the two grid semidiscrete finite volume element method with these proposed techniques solving such a large class of second order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation finally a numerical example is presented to validate the usefulness and efficiency of the method

mixed covolume methods on rectangular grids for elliptic problems we consider a covolume method for a system of first order pdes resulting from the mixed formulation of the variable coefficient matrix poisson equation with the neumann boundary condition the system may be used to represent the darcy law and the mass conservation law in anisotropic porous media flow the velocity and pressure are approximated by the lowest order raviart thomas space on rectangles the method was introduced by russell rigorous block centered discretizations on irregular grids improved simulation of complex reservoir systems reservoir simulation research corporation denver co 1995 as a control volume mixed method and has been extensively tested by jones a mixed finite volume elementary method for accurate computation of fluid velocities in porous media university of colorado at denver 1995 and cai et al computational geosciences 1 1997 pp 289 345 we reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures

the upwind finite volume element method based on straight triangular prism partition for nonlinear convection diffusion problem the upwind finite volume element method based on straight triangular prism partition is put forward for the nonlinear convection diffusion problems some techniques such as calculus of variations commutating operator and prior estimates are adopted optimal order error estimate in l 2 is derived to determine the error in approximate solution numerical experiment shows that the method is effective for avoiding numerical diffusion and nonphysical oscillations and reducing work time

on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

higher order finite volume methods ii inf sup condition and uniform local ellipticity the main purpose of this paper is to study the construction of higher order finite volume methods fvms of triangle meshes we investigate the relationship of the three theoretical notions crucial in the construction of fvms the uniform ellipticity of the family of its discrete bilinear forms its inf sup condition and its uniform local ellipticity both the uniform ellipticity of the family of the discrete bilinear forms and its inf sup condition guarantee the unique solvability of the fvm equations and the optimal error estimate of the approximate solution we characterize the uniform ellipticity in terms of the inf sup condition and a geometric condition on the bijective operator mapping from the trial space onto the test space involved in the construction of fvms we present a geometric interpretation of the inf sup condition moreover since the uniform local ellipticity is a convenient sufficient condition for the uniform ellipticity we further provide sufficient conditions and necessary conditions of the uniform local ellipticity

hamiltonian and linear space structure for damped oscillators ii critical points the eigenvector expansion developed in the preceding paper for a system of damped linear oscillators is extended to critical points where eigenvectors merge and the time evolution operator h assumes a jordan block structure the representation of the bilinear map is obtained in this basis perturbations h around an mth order critical point generically lead to eigenvalue shifts 1 m dependent on only one matrix element with the m eigenvalues splitting in equiangular directions in the complex plane small denominators near criticality are shown to cancel

finite volume element method with interpolated coefficients for two point boundary value problem of semilinear differential equations abstract in this paper we extend the idea of interpolated coefficients for semilinear problems to the finite volume element methods at first we introduce linear finite volume element method with interpolated coefficients for two point boundary value problem of semilinear differential equations next we derive convergence estimate in h 1 norm and superconvergence of derivative finally an example is given to illustrate the effectiveness of the proposed method

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains three iterative domain decomposition methods are considered simultaneous updates on all subdomains additive schwarz method flow directed sweeps and double sweeps by using some techniques of formal language theory we obtain a unique criterion of convergence for the three methods the convergence rate is a function of the criterion and depends on the algorithm

analysis of the parametric models of passive scalar transport used in the lattice boltzmann method abstract the paper is devoted to the analysis of passive scalar transport models used in the lattice boltzmann method the case of the pure advection process without physical diffusion is considered the models proposed by other authors the modifications of these models and new high order finite difference schemes based on the extended runge kutta like formulae are analyzed the attention is focused on the stability analysis analysis of the fictitious numerical effects and on the investigation of the sensitivity of accuracy order to the parameter values the stability analysis is based on the von neumann method the influence of the parameter values on the stability is demonstrated stability conditions are obtained numerical dispersion and diffusion are analyzed test problems with discontinuous and smooth initial conditions are considered the sensitivity of the accuracy order on the parameter values is analyzed

lattice boltzmann equation linear stability analysis thermal and athermal models although several thermal lattice boltzmann models have been proposed this method has not yet been shown to be able to describe nonisothermal fully compressible flows in a satisfactory manner mostly due to the presence of important deviations from the advection diffusion macroscopic equations and also due to numerical instabilities in this context this paper presents a linear stability analysis for some lattice boltzmann models that were recently derived as discrete forms of the continuous boltzmann equation p c philippi l a hegele jr l o e dos santos and r surmas phys rev e 63 056702 2006 in order to investigate the sources of instability and to find for each model the upper and lower limits for the macroscopic variables between which it is possible to ensure a stable behavior the results for two dimensional 2d lattices with 9 17 25 and 37 velocities indicate that increasing the order of approximation of the lattice boltzmann equation enhances stability results are also presented for an athermal 2d nine velocity model the accuracy of which has been improved with respect to the standard d2q9 model by adding third order terms in the lattice boltzmann equation

two grid methods for characteristic finite volume element solution of semilinear convection diffusion equations abstract two grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection dominated diffusion equations the methods are based on the method of characteristics two grid method and the finite volume element method the nonsymmetric and nonlinear iterations are only executed on the coarse grid with grid size h and the fine grid solution with grid size h can be obtained by a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid the two grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 3

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

finite difference schemes for two dimensional miscible displacement flow in porous media on composite triangular grids considering two dimensional miscible displacement flow in porous media the local grid refinement method of a coupled system on triangular cell centered grids with local refinement in space is studied based on the balance equation finite difference schemes of the coupled equations on composite grids are constructed studying their stability and convergence properties the error estimate in the energy norm is obtained finally a numerical example is given

a reduced order extrapolation algorithm based on sfve method and pod technique for non stationary stokes equations in this paper a reduced order extrapolation algorithm roea based on proper orthogonal decomposition pod technique and classical stabilized finite volume element sfve method for non stationary stokes equations is established the error estimates between the roea solutions and the classical sfve solutions and the implementation for solving roea are provided some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that roea based on sfve formulation and pod method is feasible and efficient for solving non stationary stokes equations

analysis of the cell centred finite volume method for the diffusion equation abstract this paper presents a rigorous theoretical analysis of the cell centred finite volume method for poisson s equation we review the traditional taylor series expansion technique which suggests that the cell centred method is inconsistent on nonuniform grids which is not confirmed by numerical experiments we then present an analysis of the method which confirms that the solution error does indeed reduce as the cell size is reduced this is supported by numerical calculations

unified analysis of finite volume methods for the stokes equations we establish a general framework for analyzing the class of finite volume methods for the stokes equations under the framework optimal order convergence for velocity and pressure in various sobolev norms are obtained in a natural and systematic way specifically optimal l 2 error estimates for velocity are obtained for the first time for several different finite volume methods

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation the paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and hermitian matrices due to an arbitrary complex perturbation near a diabolic point general asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived as a physical application singularities of the surfaces of refractive indices in crystal optics are studied

a stabilized finite volume method for the evolutionary stokes darcy system abstract in this paper we propose a stabilized fully discrete finite volume method based on two local gauss integrals for a non stationary stokes darcy problem this stabilized method is free of stabilized parameters and uses the lowest equal order finite element triples p 1 p 1 p 1 for approximating the velocity pressure and hydraulic head of the stokes darcy model under a modest time step restriction in relation to physical parameters we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements finally a series of numerical experiments are provided to demonstrate the validity of the theoretical results

block iterative solvers for higher order finite volume methods recently new higher order finite volume methods fvm were introduced in z cai j douglas m park development and analysis of higher order finite volume methods over rectangles for elliptic equations adv comput math 19 2003 3 33 where the linear system derived by the hybridization with lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for a pressure variable by an appropriate quadrature rule we study the convergence of an iterative solver for this linear system the conjugate gradient cg method is a natural choice to solve the system but it seems slow possibly due to the non diagonal dominance of the system in this paper we propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order fvm and prove their convergence with a proper ordering each block subproblem can be solved by fast methods such as the multigrid mg method the numerical experiments show that these block iterative methods are much faster than cg

on the observability and asymmetry of adiabatic state flips generated by exceptional points in open quantum systems where the effective hamiltonian is not hermitian it is known that the adiabatic or instantaneous basis can be multivalued by adiabatically transporting an eigenstate along a closed loop in the parameter space of the hamiltonian it is possible to end up in an eigenstate different from the initial eigenstate this adiabatic flip effect is an outcome of the appearance of a degeneracy known as an exceptional point inside the loop we show that contrary to what is expected of the transport properties of the eigenstate basis the interplay between gain loss and non adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect

an assessment of discretizations for convection dominated convection diffusion equations the performance of several numerical schemes for discretizing convection dominated convection diffusion equations will be investigated with respect to accuracy and efficiency accuracy is considered in measures which are of interest in applications the study includes an exponentially fitted finite volume scheme the streamline upwind petrov galerkin supg finite element method a spurious oscillations at layers diminishing sold finite element method a finite element method with continuous interior penalty cip stabilization a discontinuous galerkin dg finite element method and a total variation diminishing finite element method femtvd a detailed assessment of the schemes based on the hemker example will be presented

a discontinuous finite volume method for the stokes problems we develop a new discontinuous finite volume method for solving the stokes equations on both triangular and rectangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

a reduced fve formulation based on pod method and error analysis for two dimensional viscoelastic problem abstract proper orthogonal decomposition pod method has been successfully used in the reduced order modeling of complex systems in this paper we extend the applications of pod method i e combine a classical finite volume element fve method with pod method to establish a reduced fve formulation with lower dimensions and sufficiently high accuracy for two dimensional viscoelastic problem with real practical applied background and analyze the errors between the reduced pod fve solution and the classical fve solution so as to provide scientific theoretic basis for service applications some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions moreover it is also shown that the reduced fve formulation based on pod method is feasible and efficient for solving two dimensional viscoelastic problem

coupling of eigenvalues of complex matrices at diabolic and exceptional points the paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters the cases of weak and strong coupling are distinguished and their geometric interpretation in two and three dimensional spaces is given general asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios two physical examples illustrate effectiveness and accuracy of the presented theory

a covolume method based on rotated bilinears for the generalized stokes problem we introduce a covolume or marker and cell mac method for approximating the generalized stokes problem on an axiparallel domain two grids are needed the primal grid made up of rectangles and the dual grid of quadrilaterals the velocity is approximated by nonconforming rotated bilinear elements with degrees of freedom at midpoints of rectangular elements and the pressure by piecewise constants the error in the velocity in the h 1 h norm and the pressure in the l2 norm are of first order provided that the exact velocity is in h2 and the exact pressure in h1