least squares mixed finite element methods for non selfadjoint elliptic problems i error estimates a least squares mixed finite element method for general second order non selfadjoint elliptic problems in two and three dimensional domains is formulated and analyzed the finite element spaces for the primary solution approximation u h and the flux approximation sigma h consist of piecewise polynomials of degree k and r respectively the method is mildly nonconforming on the boundary the cases k r and k 1 r are studied it is proved that the method is not subject to the lbb condition optimal l 2 and h 1 error estimates are derived for regular finite element partitions numerical experiments confirming the theoretical rates of convergence are presented

a new class of higher order mixed finite volume methods for elliptic problems we introduce a new class of higher order mixed finite volume methods for elliptic problems we start from the usual way of changing the given equation into a mixed system using the darcy s law bf u cal k nabla p by integrating the system of equations with some judiciously chosen test spaces on each element we define new mixed finite volume methods of higher order we show that these new schemes are equivalent to the nonconforming finite element spaces used to define them the darcy velocity can be locally recovered from the solution of nonconforming finite element method hence our work opens a way to make higher order mixed method more practicable three dimensional extensions to parallelepiped elements are also presented this work can be viewed as a generalization of earlier works which were devoted to reducing the lowest order mixed finite element method to a corresponding nonconforming finite element method an optimal error analysis is carried out and numerical results are presented w

a mixed formulation for the brinkman problem the brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium where fractures bubbles or channels alternate inside a porous matrix in this work we explore a novel mixed formulation of the brinkman problem based on the hodge decomposition of the vector laplacian introducing the flow s vorticity as an additional unknown this formulation allows for a uniformly stable and conforming discretization by standard finite elements nedelec raviart thomas piecewise discontinuous a priori error estimates for the discretization error in the h rm curl omega h rm div omega l 2 omega norm of the solution which are optimal with respect to the approximation properties of finite element spaces are obtained the theoretical results are illustrated with numerical experiments finally the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for h rm curl and h

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

self avoiding walks on scale free networks several kinds of walks on complex networks are currently used to analyze search and navigation in different systems many analytical and computational results are known for random walks on such networks self avoiding walks saw s are expected to be more suitable than unrestricted random walks to explore various kinds of real life networks here we study long range properties of random saw s on scale free networks characterized by a degree distribution p k ensuremath sim k ensuremath ensuremath gamma in the limit of large networks system size n ensuremath rightarrow ensuremath infty the average number s n of saw s starting from a generic site increases as ensuremath mu n with ensuremath mu k 2 k ensuremath 1 for finite n s n is reduced due to the presence of loops in the network which causes the emergence of attrition of the paths for kinetic growth walks the average maximum length l increases as a power of the system size l ensuremath sim n ensuremath alpha with an exponent ensuremath alpha increasing as the parameter ensuremath gamma is raised we discuss the dependence of ensuremath alpha on the minimum allowed degree in the network a similar power law dependence is found for the mean self intersection length of nonreversal random walks simulation results support our approximate analytical calculations

a global attractor for the elastic waveguide model in rn abstract this paper studies the longtime behavior of solutions to the elastic waveguide model in r n u t t u u t t 2 u u t u t u g x u f x it proves that when the space dimension n 5 under rather mild conditions on g and f the related solution semigroup possesses in phase space with low regularity x h 2 h 1 h 2 h with 0 1 a global attractor which is connected

the viscous cahn hilliard equation with inertial term abstract we consider a hyperbolic relaxation of the viscous cahn hilliard equation this equation describes the early stages of spinodal decomposition in certain glasses we establish the existence of families of exponential attractors and inertial manifolds which are continuous at any parameter of viscosity 0 continuity properties of the global attractors are also examined

a parallel domain decomposition algorithm of mixed element equation for second order elliptic dirichlet boundary value problem a parallel domain decomposition algorithm of schwarz type is introduced to solve the mixed finite element equation for the dirichlet boundary value problem of a second order partial differential equation of elliptic type and convergence of this algorithm is analyzed

superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data a semidiscrete mixed finite element approximation to parabolic initial boundary value problems is introduced and analyzed superconvergence estimates for both pressure and velocity are obtained the estimates for the errors in pressure and velocity depend on the smoothness of the initial data including the limiting cases of data in l 2 and data in h r for r sufficiently large because of the smoothing properties of the parabolic operator these estimates for large time levels essentially coincide with the estimates obtained earlier for smooth solutions however for small time intervals we obtain the correct convergence orders for nonsmooth data

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

a time stepping procedure based on convolution for the mixed finite element approximation for porous media flow abstract a time stepping procedure is established and analyzed for the problem of miscible displacement in porous medium the fluid velocity based on mixed element method is post processed by the convolution with bramble schatz kernel convergence analysis shows that the property of superconvergence originally held for the approximation of velocity can be retained for the approximation of concentration variable compared with the traditional results in references the constraint conditions between parameters in this paper are improved essentially

a priori estimates for mixed finite element approximations of second order hyperbolic equations with absorbing boundary conditions optimal order l infty in time l 2 in space a priori error estimates are derived for mixed finite element approximations for both displacement and stress for a second order hyperbolic equation with first order absorbing boundary conditions continuous in time explicit in time and implicit in time procedures are formulated and analyzed

superconvergence of both the crouzeix raviart and morley elements in this paper a new method is proposed to prove the superconvergence of both the crouzeix raviart and morley elements the main idea is to fully employ equivalences with the first order raviart thomas element and the first order hellan herrmann johnson element respectively in this way some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements finally the superconvergence of one and a half order by postprocessing is proved for both nonconforming elements

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

optimal decay rate of solutions for cahn hilliard equation with inertial term in multi dimensions abstract the cauchy problem of the cahn hilliard equation with inertial term is considered based on green s function method together with energy estimates we get the global in time existence and optimal decay rate of solutions furthermore the viscous case is investigated in the last section

uniqueness and stability of an inverse problem for a phase field model using data from one component we study an inverse problem of determining a spatial varying coefficient in a parabolic hyperbolic phase field model with the following observation data of only one component the order parameter in a subdomain w satisfying email protected email protected email protected along a sufficiently large time interval and at a suitable time over the whole spatial domain based on a carleman estimate for the parabolic hyperbolic phase field system we prove the lipschitz stability and uniqueness for this inverse problem

search in patchy media exploitation exploration tradeoff how to best exploit patchy resources we introduce a minimal exploitation migration model that incorporates the coupling between a searcher s trajectory modeled by a random walk and ensuing depletion of the environment by the searcher s consumption of resources the searcher also migrates to a new patch when it takes mathcal s consecutive steps without finding resources we compute the distribution of consumed resources f t at time t for this non markovian searcher and show that consumption is maximized by exploring multiple patches in one dimension we derive the optimal strategy to maximize f t this strategy is robust with respect to the distribution of resources within patches and the criterion for leaving the current patch we also show that f t has an optimum in the ecologically relevant case of two dimensional patchy environments

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

a family of rectangular mixed elements with a continuous flux for second order elliptic problems we present a family of mixed finite element spaces for second order elliptic equations in two and three space dimensions our spaces approximate the vector flux by a continuous function our spaces generalize certain spaces used for approximation of stokes problems the finite element method incorporates projections of the dirichlet data and certain low order terms the method is locally conservative on the average suboptimal convergence is proven and demonstrated numerically the key result is to construct a flux pi projection operator that is bounded in the sobolev space h1 preserves a projection of the divergence and approximates optimally moreover the corresponding raviart thomas flux preserving pi projection operator is an l2 projection when restricted to this family of spaces

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

new elliptic projections and a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations in this paper we discuss a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions both semidiscrete and fully discrete schemes are considered based on some new elliptic projections we derive a priori error estimates for the control variable the state variables and the adjoint state variables the related a priori error estimates for the new projections error are also established a numerical example is given to demonstrate the theoretical results

mixed finite element approximation of a degenerate elliptic problem we present a mixed finite element approximation of an elliptic problem with degenerate coefficients arising in the study of the electromagnetic field in a resonant structure with cylindrical symmetry optimal error bounds are derived

diffuse interface model for rapid phase transformations in nonequilibrium systems a thermodynamic approach to rapid phase transformations within a diffuse interface in a binary system is developed assuming an extended set of independent thermodynamic variables formed by the union of the classic set of slow variables and the space of fast variables we introduce finiteness of the heat and solute diffusive propagation at the finite speed of the interface advancing to describe transformations within the diffuse interface we use the phase field model which allows us to follow steep but smooth changes of phase within the width of the diffuse interface governing equations of the phase field model are derived for the hyperbolic model a model with memory and a model of nonlinear evolution of transformation within the diffuse interface the consistency of the model is proved by the verification of the validity of the condition of positive entropy production and by outcomes of the fluctuation dissipation theorem a comparison with existing sharp interface and diffuse interface versions of the model is given

enhanced accuracy by post processing for finite element methods for hyperbolic equations we consider the enhancement of accuracy by means of a simple post processing technique for finite element approximations to transient hyperbolic equations the post processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes if the mesh is locally translation invariant the support of the kernel is a cube whose edges are of size of the order of x only for example when polynomials of degree k are used in the discontinuous galerkin dg method and the exact solution is globally smooth the dg method is of order k 1 2 in the l2 norm whereas the post processed approximation is of order 2k 1 if the exact solution is in l2 only in which case no order of convergence is available for the dg method the post processed approximation converges with order k 1 2 in l2 0 where 0 is a subdomain over which the exact solution is smooth numerical results displaying the sharpness of the estimates are presented

pruned enriched rosenbluth method simulations of θ polymers of chain length up to 1 000 000 we present an algorithm for simulating flexible chain polymers it combines the rosenbluth rosenbluth method with recursive enrichment although it can be applied also in more general situations it is most efficient for three dimensional theta polymers on the simple cubic lattice there it allows high statistics simulations of chains of length up to n 10 sup 6 for storage reasons this is feasable only for polymers in a finite volume for free theta polymers in infinite volume we present very high statistics runs with n 10000 these simulations fully agree with previous simulations made by hegger and grassberger j chem phys bold 102 6681 1995 with a similar but less efficient algorithm showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory but the finite volume simulations show that the density inside a collapsed globule scales with the distance from the theta point as predicted by mean field theory in contrast to claims in the work mentioned above in addition to the simple cubic lattice we also studied two versions of the bond fluctuation model but with much shorter chains finally we show that our method can be applied also to off lattice models and illustrate this with simulations of amore model studied in detail by freire ital et al macromolecules bold 19 452 1986 and later work copyright ital 1997 ital the american physical society less

convergence to equilibrium for a parabolic hyperbolic phase field system with dynamical boundary condition abstract this paper is concerned with the well posedness and the asymptotic behavior of solutions to the following parabolic hyperbolic phase field system 0 1 t 0 t t t 0 in 0 subject to the neumann boundary condition for 0 2 0 on 0 the dynamical boundary condition for 0 3 t 0 on 0 and the initial conditions 0 4 0 0 0 0 t 0 1 in where is a bounded domain in r 3 with smooth boundary is the outward normal direction to the boundary and is a real analytic function in this paper we first establish the existence and uniqueness of a global strong solution to 0 1 0 4 then we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate

supraconvergence and supercloseness in quasilinear coupled problems the aim of this paper is to study a finite difference method for quasilinear coupled problems of partial differential equations that presents numerically an unexpected second order convergence rate the error analysis presented allows us to conclude that the finite difference method is supraconvergent as the method studied in this paper can be seen as a fully discrete piecewise linear finite element method we conclude the supercloseness of our approximations

global attractor for the generalized double dispersion equation abstract the paper studies the existence of global attractor for the generalized double dispersion equation arising in elastic waveguide model u t t u u t t 2 u u t g u f x the main result is concerned with nonlinearities g u with supercritical growth in that case we construct a subclass g of the limit solutions and show that it has a weak global attractor especially in non supercritical case the weak topology becomes strong the further regularity of the global attractor is obtained and the exponential attractor is established in natural energy space

superconvergence in the pressure in the simulation of miscible displacement a modification of a finite element method of douglas ewing and wheeler for approximating the solution of the equations describing miscible displacement in porous media is proposed and analyzed in a periodic setting the pressure is approximated by a mixed finite element procedure using a raviart thomas space of index k over a uniform grid the resulting darcy velocity field is post processed by convolution with a bramble schatz kernel and this enhanced velocity is used in the evaluation of the coefficients in the galerkin procedure for the concentration if the concentration space is of local degree l then the error as measured in l 2 omega in the concentration is o h c l 1 h p 2k 2 i e an optimal reflection of the superconvergent velocity approximation

solution of parabolic equations by backward euler mixed finite element methods on a dynamically changing mesh we develop and analyze methods based on combining the lowest order mixed finite element method with backward euler time discretization for the solution of diffusion problems on dynamically changing meshes the methods developed are shown to preserve the optimal rate error estimates that are well known for static meshes the novel aspect of the scheme is the construction of a linear approximation to the solution which is used in projecting the solution from one mesh to another extensions to advection diffusion equations are discussed where the advection is handled by upwinding numerical results validating the theory are also presented

convergence and domain decomposition algorithm for nonconforming and mixed methods for nonselfadjoint and indefinite problems in this paper the existence uniqueness and convergence of the solutions of nonconforming and mixed methods with non quasi uniform partition are proven for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption moreover uniform convergence of the solution of nonconforming method is obtained meanwhile the domain decomposition algorithm which suits non quasi uniform partition is proposed for nonconforming and mixed methods

mixed finite element and finite volume discretization for heavy brine simulations in groundwater recently a new theory of high concentration brine transport in groundwater has been developed this approach is based on two nonlinear mass conservation equations one for the fluid flow equation and one for the salt transport equation both having nonlinear diffusion terms in this paper we present and analyze a numerical technique for the solution of such a model the approach is based on the mixed hybrid finite element method for the discretization of the diffusion terms in both the flow and transport equations and a high resolution tvd finite volume scheme for the convective term this latter technique is coupled to the discretized diffusive flux by means of a time splitting approach a commonly used benchmark test elder problem is used to verify the robustness and nonoscillatory behavior of the proposed scheme and to test the validity of two different formulations one based on using pressure head and concentration c as dependent variables and one using pressure p and mass fraction as dependent variables it is found that the latter formulation gives more accurate and reliable results in particular at large times the numerical model is then compared against a semi analytical solution and the results of a laboratory test these tests are used to verify numerically the performance and robustness of the proposed numerical scheme when high concentration gradients i e the double nonlinearity are present

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint abstract in this paper we shall investigate the superconvergence property of quadratic elliptical optimal control problems by triangular mixed finite element methods the state and co state are approximated by the order k 1 raviart thomas mixed finite elements and the control is discretized by piecewise constant functions we prove the superconvergence error estimate of h 2 in l 2 norm between the approximated solution and the interpolation of the exact control variable moreover by postprocessing technique we find that the projection of the discrete adjoint state is superclose in order h 2 to the exact control variable

an impes scheme for a two phase flow in heterogeneous porous media using a structured grid abstract we develop a numerical scheme for a two phase immiscible flow in heterogeneous porous media using a structured grid finite element method which has been successfully used for the computation of various physical applications involving elliptic interface equations li et al 2003 2004 chou et al 2010 kwak et al 2010 chang and kwak 2011 the proposed method is based on the implicit pressure explicit saturation procedure to solve the pressure equation we use an ifem based on the rannacher turek rannacher and turek 1992 nonconforming space which is a modification of the work in kwak et al 2010 where broken p 1 nonconforming element of crouzeix raviart 1973 1 was developed for the darcy velocity we apply the mixed finite volume method studied in chou et al 2003 and kwak et al 2010 on the basis of immersed finite element method ifem in this way the darcy velocity of the flow can be computed cheaply locally after we solve the pressure equation the computed darcy velocity is used to solve the saturation equation explicitly the whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media in fact we apply a new version of multigrid algorithm to solve the pressure equation for which the cpu time grows like o n numerical tests for analytic problems show that our method is almost optimal in all variables problem with sources sinks is also computed and it seems to work well

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

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

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

application of superconvergence to problems in the simulation of miscible displacement abstract application of superconvergence along gauss lines is considered in the context of problems in the simulation of the miscible displacement of one fluid by another in a porous medium a simple computational procedure allows the superconvergence property of the fluid velocity to be retained in our total algorithm

kinetic growth walks on complex networks kinetically grown self avoiding walks on various types of generalized random networks have been studied networks with short and long tailed degree distributions p k were considered k degree or connectivity including scale free networks with p k k the long range behaviour of self avoiding walks on random networks is found to be determined by finite size effects the mean self intersection length of non reversal random walks l scales as a power of the system size n l n with an exponent 0 5 for short tailed degree distributions and 0 5 for scale free networks with 3 the mean attrition length of kinetic growth walks l scales as l n with an exponent which depends on the lowest degree in the network results of approximate probabilistic calculations are supported by those derived from simulations of various kinds of networks the efficiency of kinetic growth walks to explore networks is largely reduced by inhomogeneity in the degree distribution as happens for scale free networks

convergence and quasi optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems recently an adaptive nonconforming finite element method anfem has been developed by carstensen and hoppe in numer math 103 251 266 2006 in this paper we extend the result to some nonsymmetric and indefinite problems the main tools in our analysis are a posteriori error estimators and a quasi orthogonality property in this case we need to overcome two main difficulties one stems from the nonconformity of the finite element space the other is how to handle the effect of a nonsymmetric and indefinite bilinear form an appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order crouzeix raviart element it is shown that the anfem is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops moreover quasi optimality in the sense of quasi optimal algorithmic complexity can be shown for the anfem the results of numerical experiments confirm the theoretical findings

superconvergence of triangular raviart thomas mixed finite element methods for a bilinear constrained optimal control problem in this paper we analyze the superconvergence of the bilinear constrained elliptic optimal control problem by triangular raviart thomas mixed finite element methods the state and the co state are approximated by the order k 1 raviart thomas mixed finite elements and the control is approximated by piecewise constant functions we obtain the superconvergence property between average l 2 projection and the approximation of the control variable and the convergence order is h 2 two numerical examples are presented for illustrating the superconvergence results

attractors for the cahn hilliard equation with memory in 2d abstract we deal with the memory relaxation of the cahn hilliard equation proposed by rotstein et al as a model for the phenomenological description of phase transition which contains as a particular case the hyperbolic version of the equation due to the absence of viscosity effects existence and uniqueness of solutions are far from being trivial in space dimension n 1 this paper is devoted to the analysis of energy solutions for the 2d model after discussing the well posedness of the problem we study its long term dynamics and we prove the existence and regularity of the global attractor provided that the relaxation parameter of the memory kernel is small enough

uniform convergence and preconditioning methods for projection nonconforming and mixed methods for nonselfadjoint and indefinite problems abstract the purpose of this paper is to prove the existence uniqueness and uniform convergence of the solutions of so called projection nonconforming and mixed element methods and the equivalence between projection nonconforming element method and mixed element method with nonquasi uniform partition for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption meanwhile the optimal error estimate of the solution of mixed element method is obtained under h 2 smoothness hypothesis for nonselfadjoint and indefinite elliptic problems without h 2 regularity finally the discrete compactness result for nonconforming element space with nonquasi uniform partition is proven and some preconditioning methods for projection nonconforming and mixed element methods with nonquasi uniform partition are given it is proven that the h 1 condition number of preconditioned operator is uniformly bounded and its singular values cluster in a relatively small finite interval

the hyperbolic allen cahn equation exact solutions using the first integral method a general set of analytical solutions is obtained for the hyperbolic allen cahn equation the solutions are presented by i the class of continual solutions described by profiles for traveling waves of the order parameter and ii the class of singular solutions which exhibit unbounded discontinuity in the profile of the order parameter at the origin of the coordinate system it is shown that the solutions include the previous analytical results for the parabolic allen cahn equation as a limited class of functions in which the inertial effects are omitted

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

a priori and a posteriori error estimates of h 1 galerkin mixed finite element methods for optimal control problems governed by pseudo hyperbolic integro differential equations abstract in this paper we investigate a priori and a posteriori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by pseudo hyperbolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions based on two new elliptic projections we derive a priori error estimates both for the control variable the state variable and the co state variable the related a priori error estimates for the new projections error are also established moreover a posteriori error estimates for all variables are derived via energy method such a posteriori error estimates which are apparently not available in the literature are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem

the spatial temporal lamellar structures in the confined ideal polymer blends we consider the formation of self organized spatial temporal oscillating structures in symmetric binary polymer blends confined by two flat walls an influence of these walls on the formation of the oscillating volume structures is studied this phenomenon is simulated by an initial boundary value problem for the conserved order parameter or the concentration of one of the components in a binary mixture under a special choice the dynamical puri binder s boundary conditions these structures look like the lamellar structures the behavior of the order parameter is described by the modified cahn hilliard equation which models so called the non fickian diffusion in the symmetric binary polymer blends the nonlinear dynamical boundary conditions correspond to the process of adsorption desorption on the walls as a result these nonlinear surface processes induce into the volume the spatial temporal asymptotically periodic structures of relaxation pre turbulent or turbulent type with finite countable or non countable points of discontinuities on the period correspondingly the frequency of oscillations on the period follows a power law for the relaxation type and increases exponentially in the other cases

mixed finite element methods for linear elasticity with weakly imposed symmetry in this paper we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima tions to both stresses and displacements the methods are based on a modified form of the hellinger reissner variational principle that only weakly imposes the symmetry condition on the stresses although this approach has been previously used by a number of authors a key new ingredient here is a constructive derivation of the elasticity complex starting from the de rham complex by mimicking this construction in the discrete case we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de rham complex these elements appear to be simpler than the ones previously derived for example we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field

a new mixed formulation for elasticity we propose a new mixed variational formulation for the equations of linear elasticity it does not require symmetric tensors and consequently is easy to discretize by adapting mixed finite elements developed for scalar second order elliptic equations

self avoiding walks and connective constants in small world networks long distance characteristics of small world networks have been studied by means of self avoiding walks saw s we consider networks generated by rewiring links in one and two dimensional regular lattices the number of saw s u n was obtained from numerical simulations as a function of the number of steps n on the considered networks the so called connective constant mu lim n infinity u n u n 1 which characterizes the long distance behavior of the walks increases continuously with disorder strength or rewiring probability p for small p one has a linear relation mu mu 0 ap mu 0 and a being constants dependent on the underlying lattice close to p 1 one finds the behavior expected for random graphs an analytical approach is given to account for the results derived from numerical simulations both methods yield results agreeing with each other for small p and differ for p close to 1 because of the different connectivity distributions resulting in both cases

asymptotic behavior of a nonisothermal viscous cahn hilliard equation with inertial term we consider a differential model describing nonisothermal fast phase separation processes taking place in a three dimensional bounded domain this model consists of a viscous cahn hilliard equation characterized by the presence of an inertial term tt being the order parameter which is linearly coupled with an evolution equation for the relative temperature the latter can be of hyperbolic type if the cattaneo maxwell heat conduction law is assumed the state variables and the chemical potential are subject to the homogeneous neumann boundary conditions we first provide conditions which ensure the well posedness of the initial and boundary value problem then we prove that the corresponding dynamical system is dissipative and possesses a global attractor moreover assuming that the nonlinear potential is real analytic we establish that each trajectory converges to a single steady state by using a suitable version of the eojasiewicz simon inequality we also obtain an estimate of the decay rate to equilibrium

a mixed multiscale finite element method for convex optimal control problems with oscillating coefficients abstract we study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients since the objective functional contains flux we approximate the problems using the mixed finite element methods we first analyze the standard mixed finite element approximation schemes then motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media we use a multiscale mixed finite element method to solve the state equations the multiscale finite element bases are constructed by locally solving neumann boundary value problems the analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic which allows us to use homogenization theory to obtain the asymptotic structure of the solutions although the numerical schemes are designed for the general case

asymptotic expansions and richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods in this paper asymptotic error expansions for mixed finite element approximations of general second order elliptic problems are derived under rectangular meshes and the richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation postprocessing technique the results of this paper provide new asymptotic expansions and new approximate solutions which are one order and a half order higher in accuracy than those obtained in j wang math comp 56 1991 pp 477 503 and h chen r e ewing and r lazarov asymptotic error expansion for the lowest order raviart thomas rectangular mixed finite elements technical report isc 97 01 1997 respectively as a by product we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators for the mixed finite element method

l error estimates for linear elasticity problems two dimensional linear elasticity problems are approximately solved by a mixed finite element method based on raviart thomas nedelec spaces maximum norm error estimates of optimal rate and almost optimal regularity are derived for the displacement field and of optimal regularity and almost optimal rate for the stress tensor the superconvergence of the approximate displacement field to the l2 projection of the actual one is established

first passage properties of the erdos renyi random graph we study the mean time for a random walk to traverse between two arbitrary sites of the erd s renyi random graph we develop an effective medium approximation that predicts that the mean first passage time between pairs of nodes as well as all moments of this first passage time are insensitive to the fraction p of occupied links this prediction qualitatively agrees with numerical simulations away from the percolation threshold near the percolation threshold the statistically meaningful quantity is the mean transit rate namely the inverse of the first passage time this rate varies non monotonically with p near the percolation transition much of this behaviour can be understood by simple heuristic arguments

an efficient algorithm for solving the phase field crystal model we present and discuss the development of an unconditionally stable algorithm used to solve the evolution equations of the phase field crystal pfc model this algorithm allows for an arbitrarily large algorithmic time step as the basis for our analysis of the accuracy of this algorithm we determine an effective time step in fourier space we then compare our calculations with a set of representative numerical results and demonstrate that this algorithm is an effective approach for the study of the pfc models yielding a time step effectively 180 times larger than the euler algorithm for a representative set of material parameters as the pfc model is just a simple example of a wide class of density functional theories we expect this method will have wide applicability to modeling systems of considerable interest to the materials modeling communities

superconvergence of new mixed finite element spaces in this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in etna vol 37 pp 189 201 2010 among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids this space has the smallest unknowns however the scalar variable is only suboptimal in general thus we have employed a post processing technique for the scalar variable as a byproduct we have obtained a superconvergence on a rectangular grid the superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory but that of a scalar variable requires a new technique especially for k 1 numerical experiments are provided to support the theory

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined

a mixed finite element method for a strongly nonlinear second order elliptic problem the approximation of the solution of the first boundary value problem for a strongly nonlinear second order elliptic problem in divergence form by the mixed finite element method is considered existence and uniqueness of the approximation are proved and optimal error estimates in l 2 are established for both the scalar and vector functions approximated by the method error estimates are also derived in l q 2 q

distance distribution in configuration model networks we present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks the results which are based on recursion equations are shown to be in good agreement with numerical simulations for networks with degenerate binomial and power law degree distributions the mean mode and variance of the distribution of shortest path lengths are also evaluated these results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution illuminating the connection between the two distributions