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

superconvergence of fully discrete rectangular mixed finite element methods of parabolic control problems in this paper we investigate the superconvergence property of the numerical solution of a quadratic parabolic optimal control problem by using fully discrete mixed finite element methods the space discretization of the state variable is done using usual mixed finite elements whereas the time discretization is based on difference methods the state and the co state are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions we derive the superconvergence results for the control and the state approximation some numerical examples are presented to confirm the theoretical investigations

a nonlinear mixed finite eelement method for a degenerate parabolic equation arising in flow in porous media we study a model nonlinear degenerate advection diffusion equation having application in petroleum reservoir and groundwater aquifer simulation the main difficulty is that the true solution is typically lacking in regularity therefore we consider the problem from the point of view of optimal approximation through time integration we develop a mixed variational form that respects the known minimal regularity and then we develop and analyze two versions of a mixed finite element approximation a simpler semidiscrete time continuous version and a fully discrete version our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error the exceptional term is a nonstandard approximation error term we also consider our new formulation for the nondegenerate problem showing the usual optimal l 2 error bounds moreover superconvergence is obtained under special circumstances

multigrid on the interface for mortar mixed finite element methods for elliptic problems we consider mixed finite element approximations of second order elliptic equations on domains that can be described as a union of subdomains or blocks we assume that the subdomain grids are locally defined and need not match across the block boundaries specially chosen mortar finite element spaces are introduced on the interfaces for approximating the scalar variable pressure the mortars also serve as lagrange multipliers for imposing flux matching conditions the method is implemented by reducing the algebraic system to a positive definite interface problem in the mortar spaces this problem is then solved using a multigrid on the interface with conjugate gradient smoothing the algorithm is very efficient in a distributed parallel computing environment as only subdomain solves are required on each conjugate gradient iteration the standard variational assumptions for the multigrid are not satisfied since the interface bilinear forms vary from level to level we present theoretical results for the convergence of the v cycle and the w cycle computational results in two and three dimensions are given to illustrate and confirm the theory

on the finite volume reformulation of the mixed finite element method for elliptic and parabolic pde on triangles abstract a new resolution of parabolic and elliptic partial differential equations pdes based on the mixed finite element approximation on triangles has been recently developed 24 25 this new approach reduces the number of unknowns from fluxes or lagrange multiplier defined on edges to a single unknown per element in this paper we analyze this transformation mathematically and describe in details how to handle singular elements and singular edges for these singular elements the standard mixed method on triangles can always be made equivalent to a finite volume formulation where the finite volumes are obtained by aggregation of finite elements across singular edges the positive definiteness of the system matrix obtained with the new formulation is analyzed in details a criterion is given concerning the property of this matrix which show that its conditioning is related to the shape of the triangle and the contrast in parameters from one element to the adjacent ones numerical experiments are performed for elliptic and parabolic pdes the comparisons between an iterative solver pcg and a direct solver unifrontal multifrontal show that the direct solver is more efficient moreover its performance is not correlated with the system matrix conditioning it appears that the new formulation requires significantly less cpu time for elliptic pdes and is competitive for parabolic pdes the new formulation remains also accurate enough even in nearly singular situations

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

the condition number of the schur complement in domain decomposition it is shown that for elliptic boundary value problems of order 2m the condition number of the schur complement matrix that appears in nonoverlapping domain decomposition methods is of order d 1 h 2m 1 where d measures the diameters of the subdomains and h is the mesh size of the triangulation the result holds for both conforming and nonconforming finite elements

design of time frequency representations using a multiform tiltable exponential kernel a novel cohen s 1981 class time frequency representation with a tiltable generalized exponential kernel capable of attaining a wide diversity of shapes in the ambiguity function plane is proposed for improving the time frequency analysis of multicomponent signals the first advantage of the proposed kernel is its ability to generate a wider variety of passband shapes e g rotated ellipses generalized hyperbolas diamonds rectangles parallel strips at arbitrary angles crosses snowflakes etc and narrower transition regions than conventional cohen s class kernels this versatility enables the new kernel to suppress undesirable cross terms in a broader variety of time frequency scenarios the second advantage of the new kernel is that closed form design equations can now be easily derived to select kernel parameters that meet or exceed a given set of user specified passband and stopband design criteria in the ambiguity function plane thirdly it is shown that simple constraints on the parameters of the new kernel can be used to guarantee many desirable properties of time frequency representations the well known choi williams 1989 exponential kernel the generalized exponential kernel and nuttall s 1990 tilted gaussian kernel are special cases of the proposed kernel

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

superconvergence for discontinuous galerkin finite element methods by l2 projection methods a general superconvergence of discontinuous galerkin dg finite element method for the elliptic problem is established by using l 2 projection method regularity assumptions for the elliptic problem with regular partitions are required numerical experiments are given to verify the theoretical results

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

superconvergence by l2 projection for a stabilized finite volume method for the stationary navier stokes equations a superconvergence result is established for the stationary navier stokes equations by a stabilized finite volume method and l 2 projection on a coarse mesh like other results in the family of l 2 projection methods the superconvergence presented in this paper is based on some regularity assumption for the navier stokes problem and is applicable to the stabilized finite volume method with quasi uniform partitions

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

superconvergence and extrapolation for mixed finite element methods on rectangular domains asymptotic expansions for the rt raviart thomas mixed finite element approximation by the lowest order rectangular element associated with a second order elliptic equation on a rectangular domain are derived superconvergence for the vector field along the gauss lines is obtained as a result of the expansion a procedure of postprocessed extrapolation is presented for the scalar field as well as procedures of pure richardson extrapolation for both the vector and the scalar fields

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

derivative superconvergence of rectangular finite elements for the reissner mindlin plate abstract the finite element method with rectangular meshes for the reissner mindlin plate is analyzed for a plate with thickness bounded below by a positive constant moderately thick plate derivative superconvergence of the finite element solution at the gaussian points is justified and optimal error estimates for both rotation and displacement are established

global superconvergence for maxwell s equations in this paper the global superconvergence is analysed on two schemes a mixed finite element scheme and a finite element scheme for maxwell s equations in r 3 such a supercovergence analysis is achieved by means of the technique of integral identity which has been used in the supercovergence analysis for many other equations and schemes on a rectangular mesh and then are generalized into more general domains and problems with the variable coefficients besides being more direct our analysis generalizes the results of monk

decoupling mixed finite elements on hierarchical triangular grids for parabolic problems in this paper we propose a numerical method for the solution of time dependent flow problems in mixed form such problems can be efficiently approximated on hierarchical grids obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements if these elements are considered as subdomains we can formulate a non overlapping domain decomposition method based on the lowest order raviart thomas elements properly enhanced with lagrange multipliers on the boundaries of each subdomain excluding the dirichlet edges a suitable choice of mixed finite element spaces and quadrature rules yields a cell centered scheme for the pressures with a local 10 point stencil the resulting system of differential algebraic equations is integrated in time by the crank nicolson method which is known to be a stiffly accurate scheme as a result we obtain independent subdomain linear systems that can be solved in parallel the behavior of the algorithm is illustrated on a variety of numerical experiments

on some mixed finite element methods with numerical integration in this paper a new family of mixed finite volume methods is analyzed for the approximation of a reaction diffusion problem all the methods are obtained starting from the dual mixed formulation of the problem and then employing the lowest order raviart thomas finite element spaces plus a suitable quadrature formula for the mass matrix this allows for the use of different averages of the inverse diffusion coefficient to enforce the constitutive law for the fluxes at the interelement boundary in a finite volume fashion the soundness of the methods is supported by an error analysis which shows optimal cal o h convergence rate with respect to the standard mixed finite element norm in the case of both smooth and piecewise smooth coefficients numerical results on test problems with both smooth and nonsmooth coefficients support the theoretical estimates

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

an efficient cell centered multigrid method for problems with discontinuous coefficients on semi structured triangular grids this paper is focused on the numerical solution of elliptic equations with discontinuous coefficients in particular the design of efficient geometric multigrid methods for cell centered finite volume schemes for this kind of problems is dealt with in this work we propose a block wise multigrid algorithm on semi structured triangular grids for solving piecewise constant diffusivity problems on relatively complex domains appropriate novel smoothers for cell centered discretizations are considered on each structured patch of the mesh the difficulties appearing when highly varying coefficients occur are overcome by the use of a modified galerkin coarse grid approximation numerical experiments are presented to illustrate the good behavior of the proposed multigrid method which achieves an h independent convergence rate

discretization schemes on triangular grids abstract physical applications posed on irregular domains have caused difficulties in the use of many otherwise effective discretization schemes in two dimensional examples we consider discretization schemes on triangles to approximate irregular domains although finite element mixed finite element and finite volume schemes for triangles are well understood efficient solvers are essential for their effective use in this paper we consider certain cell centered finite difference schemes on triangles both for their theoretical properties on regular grids and as effective preconditioners for other discretization schemes for more general triangulations two cell centered finite difference schemes four point star stencils are derived from the simplification of the lowest order raviart thomas mixed finite element discretization on certain triangular grids optimal and superconvergence error estimates for both the solution and the gradient of the solution are established the accuracy of these schemes is also discussed in the case of discontinuous coefficients computational experiments verify the error estimates finally the use of these methods as preconditioners is also discussed

uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids the lowest order raviart thomas rectangular element is considered for solving the singular perturbation problem mbox div a nabla p bp f where the diagonal tensor a varepsilon 2 1 or a varepsilon 2 varepsilon 2 global uniform convergence rates of o n 1 for both p and a1 2 nabla p in the l2 norm are obtained in both cases where n is the number of intervals in either direction the pointwise interior away from the boundary layers convergence rates of o n 1 for p are also proved superconvergence i e o n 2 at special points and o n 2 global l2 estimate for both p and a 1 2 nabla p are obtained by a local postprocessing numerical results support our theoretical analysis moreover numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh

multidisciplinary interactions in energy and environmental modeling abstract mathematical models have been widely used to understand predict or optimize many complex physical processes from semiconductor or pharmaceutical design to large scale applications such as global weather models to astrophysics in particular simulation of environmental effects of air polution is extensive here we address the need for using similar models to understand the fate and transport of groundwater contaminants and to design in situ remediation strategies three basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination one must first obtain an effective model to describe the complex fluid fluid and fluid rock interactions that control the transport of contaminants in groundwater this includes the problem of obtaining accurate reservoir descriptions at various length scales and modeling the effects of this heterogeneity in the reservoir simulators next one must develop accurate discretization techniques that retain the important physical properties of the continuous models finally one should develop efficient numerical solution algorithms that utilize the potential of the emerging computing architectures we will discuss recent advances in each of these three areas

on the convergence of a dual primal substructuring method in the dual primal feti method introduced by farhat et al 1999 the domain is decomposed into non overlapping subdomains but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint the continuity of the remaining degrees of freedom on subdomain interfaces is enforced by lagrange multipliers and all degrees of freedom are eliminated the resulting dual problem is solved by preconditioned conjugate gradients we give an algebraic bound on the condition number assuming only a single inequality in discrete norms and use the algebraic bound to show that the condition number is bounded by c 1 log 2 h h for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements as well as for a wide class of finite elements for the reissner mindlin plate model

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

superconvergence and a posteriori error estimation in triangular mixed finite elements in this paper we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of raviart thomas type 17 on uniform triangulations are used i e that the h mbox div omega distance between the approximate solution and a suitable projection of the real solution is of higher order than the h mbox div omega error we prove results for both dirichlet and neumann boundary conditions recently duran 9 proved similar results for rectangular mixed finite elements and superconvergence along the gauss lines for rectangular mixed finite elements was considered by douglas ewing lazarov and wang in 11 8 and 18 the triangular case however needs some extra effort using the superconvergence results a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the l 2 omega error in the approximation of the vector variable

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

a supraconvergent scheme for the korteweg de vries equation a finite difference method for the integration of the korteweg de vries equation on irregular grids is analyzed under periodic boundary conditions the method is shown to be supraconvergent in the sense that though being inconsistent it is second order convergent however such a convergence only takes place on grids with an odd number of points per period when a grid with an even number of points is used the inconsistency of the method leads to divergence numerical results backing the analysis are presented

godunov mixed methods for advection diffusion equations in multidimensions time split methods for multidimensional advection diffusion equations are considered in these methods advection is approximated by a godunov type procedure and diffusion is approximated by a low

a sparse finite element method with high accuracy part i in this paper we develop and analyze a new finite element method called the sparse finite element method for second order elliptic problems this method involves much fewer degrees of freedom than the standard finite element method we show nevertheless that such a sparse finite element method still possesses the superconvergence and other high accuracy properties same as those of the standard finite element method the main technique in our analysis is the use of some integral identities

a constrained weighted least squares approach for time frequency distribution kernel design in most applications of time frequency t f distributions the t f kernel is of finite extent and applied to discrete time signals this paper introduces a matrix based approach for t f distribution kernel design in this new approach the optimum kernel is obtained as the solution of a linearly constrained weighted least squares minimization problem in which the kernel is vectorial and the constraints form a linear subspace similar to fir temporal and spatial constrained least squares ls design methods the passband stopband and transition band of an ideal kernel are first specified the optimum kernel that best approximates the ideal kernel in the ls error sense and simultaneously satisfies the multiple linear constraints is then obtained using closed form expressions this proposed design method embodies a well structured procedure for obtaining fixed and data dependent kernels that are difficult to obtain using other design approaches

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

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

interior superconvergence in mortar and non mortar mixed finite element methods on non matching grids we establish interior velocity superconvergence estimates for mixed finite element approximations of second order elliptic problems on non matching rectangular and quadrilateral grids both mortar and non mortar methods for imposing the interface conditions are considered in both cases it is shown that a discrete l 2 error in the velocity in a compactly contained subdomain away from the interfaces con1 2 higher than the error in the whole domain for the non mortar method we also establish pressure superconvergence which is needed in the velocity analysis numerical results are presented in confirmation of the theory

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

the generalized exponential time frequency distribution time frequency distributions tfd are joint time and frequency signal representations that among other properties maintain the true support of a signal s energy in both time and frequency in addition to their mathematical elegance tfds can provide simultaneous resolution in time and frequency that exceeds that of the common spectrogram in general however tfds also exhibit certain peculiarities that arise in part from the bilinear structure of the fundamental tfd form l cohen s class perhaps most notable is the presence of spectral cross term artifacts a kind of spectral chaff that tends to impede visual understanding and interpretation of tfds as instantaneous power spectrums several researchers have proposed and demonstrated a variety of tfds which through cohen s 1989 form are defined through a kernel spl phi spl theta spl tau particularly notable among these is choi and williams 1989 exponential distribution in which spl phi spl theta spl tau exp spl theta sup 2 spl tau sup 2 spl sigma of all distributions investigated the exponential distribution is relatively immune to spectral cross term generation and yet maintains high simultaneous time frequency resolution in a generalization of choi and williams work the author introduces the broader class of exponential distributions defined by the kernel exp spl theta sup p spl tau sup q spl sigma and investigate its properties in particular he shows that this generalized exponential distribution can exceed the time frequency resolution performance of the exponential distribution and get also remain relatively free from spectral cross term distortion

natural superconvergence points in three dimensional finite elements a systematic and analytic process is conducted to identify natural superconvergence points of high degree polynomial c 0 finite elements in a three dimensional setting this identification is based upon explicitly constructing an orthogonal decomposition of local finite element spaces derivative and function value superconvergence points are investigated for both the poisson and the laplace equations superconvergence results are reported for hexahedral pentahedral and tetrahedral elements up to certain degrees

mixed volume element combined with characteristic mixed finite volume element method for oil water two phase displacement problem abstract as the basic of numerical simulation of energy science the displacement of three dimensional oil water two phase in porous media is discussed in this paper for incompressible miscible displacement the pressure is described by a flow equation in an elliptic mathematical formulation and the saturation is defined by a convection diffusion equation the pressure exists in the saturation equation by darcy velocity and controls the whole flow we develop and improve substantially the work of arbogast and wheeler then put forward a mixed volume element combined with characteristic mixed finite volume element method for two phase displacement problem the flow equation is discretized by the conservative mixed volume element method which could improve the accuracy of an order for computing darcy velocity the saturation equation is solved by a characteristic mixed volume element method where mixed volume element method is used to compute the diffusion term and the method of characteristics is adopted to discretize the convection term the method of characteristics can confirm high stability of numerical simulation at the fronts avoid numerical dispersion and nonphysical oscillation and can adopt large time steps obtain smaller time truncation error and improve the computation accuracy mixed volume element can solve the convection term confirm conservation of mass at each element and approximate the saturation and the adjoint vector meanwhile it is most important in numerical computation of seepage mechanics optimal second order estimates in l 2 norm are derived by theory and special techniques of priori estimates finally numerical experiments are shown to illustrate the efficiency and practicability and to solve the international problem successfully

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

analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method ii abstract this is the first in a series of two papers in which the patch recovery technique proposed by zienkiewicz and zhu is analyzed in this work it is proved that the recovered derivative by the least squares fitting is superconvergent for the two point boundary value problems further the a posteriori error estimator based on the recovery technique is shown to be asymptotically exact this confirms the computational results obtained in 1 2

superconvergence of the h p version of the finite element method in one dimension in this paper we investigate the superconvergence properties of the h p version of the finite element method fem for two point boundary value problems a postprocessing technique for the h p finite element approximation is analyzed the analysis shows that the postprocess improves the order of convergence furthermore we obtain asymptotically exact a posteriori error estimators based on the postprocessing results numerical examples are included to illustrate the theoretical analysis

generalizations and accelerations of lions nonoverlapping domain decomposition method for linear elliptic pde in this paper a one parameter generalization of lions nonoverlapping domain decomposition method for linear elliptic pdes is proposed and studied the generalized methods are shown to be descent direction methods for minimizing an interface bias functional iteration convergence of both the continuous and finite element versions of the proposed methods is established it is theoretically and numerically demonstrated that for generic choices of the parameter the generalized methods converge faster than lions original method algorithms are given and numerical results are presented

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

generalization of the choi williams distribution and the butterworth distribution for time frequency analysis we provide a generalization to the choi williams expo nential distribution and introduce the new butterworth distribution we compute design equations and curves for the optimal choices of their parameters and for the optimal value of c of the exponential dis tribution we also provide examples that demonstrate the superior performance of our method over the choi williams our new distri butions can achieve cross term reduction and autoterm preservation simultaneously whereas the choi williams distribution suffers from an inherent tradeoff between the two

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

full order convergence of a mixed finite element method for fourth order elliptic equations abstract by using a special interpolation operator and an elaborate element analysis in this paper we improve the classical error estimates to full order for a mixed finite element method for the fourth order elliptic equations on the rectangular mesh therefore we obtain the truly optimal error estimates in view of the interpolation space for the first time

superconvergence for control volume mixed finite element methods on rectangular grids we consider control volume mixed finite element methods for the approximation of second order elliptic problems on rectangular grids these methods associate control volumes covolumes with the vector variable as well as the scalar obtaining local algebraic representation of the vector equation e g darcy s law as well as the scalar equation e g conservation of mass we establish o h 2 superconvergence for both the scalar variable in a discrete l 2 norm and the vector variable in a discrete h rm div norm the analysis exploits a relationship between control volume mixed finite element methods and the lowest order raviart thomas mixed finite element methods

superconvergence for the gradient of finite element approximations by l 2 projections a gradient recovery technique is proposed and analyzed for finite element solutions which provides new gradient approximations with high order of accuracy the recovery technique is based on the method of least squares surface fitting in a finite dimensional space corresponding to a coarse mesh it is proved that the recovered gradient has a high order of superconvergence for appropriately chosen surface fitting spaces the recovery technique is robust efficient and applicable to a wide class of problems such as the stokes and elasticity equations

an analysis of monotonicity conditions in the mixed hybrid finite element method on unstructured triangulations in this paper we consider the mixed hybrid finite element method on unstructured triangular grids and evaluate its monotonicity properties by using a non standard set of basis functions for the velocity approximation space the mixed hybrid discretization of the steady state diffusion equation produces a system matrix that depends only on the inner product of the outward normals to the edges of the triangulation and not on the choice of the velocity space basis this property is used to study the characteristics of the system matrix it is well known that this matrix is of type m if the angles of the triangulation are not bigger than 2 an m matrix has a nonnegative inverse i e all the elements are nonnegative this implies the existence of a discrete maximum principle and thus monotonicity of the discretization we show that when the triangulation is of delaunay type and satisfies the property that no circumcenters of boundary elements with dirichlet conditions lie outside the domain the inverse of the final matrix is always positive even in the presence of obtuse angles

superconvergence of mixed finite element methods for optimal control problems t in this paper we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods the state and co state variables are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique we derive l 2 superconvergence properties for the flux functions along the gauss lines and for the scalar functions at the gauss points via mixed projections moreover global l 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique thus based on these superconvergence estimates some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

a local superconvergence analysis of the finite element method for the stokes equations by local projections abstract in this paper we focus on a local superconvergence analysis of the finite element method for the stokes equations by local projections the local and global superconvergence results of finite element solutions are provided for the stokes problem under some corresponding regularity assumptions conclusion can be drawn that the local superconvergence has advantages over the global superconvergence in two important aspects on the one hand it offsets theoretical limitation in practical applications on the other hand interior estimates are derived on the base of local properties of the domain without global smoothness for the exact solution and prior regularity of the problem globally over the whole domain

a non mortar mixed finite element method for elliptic problems on non matching multiblock grids abstract we consider the approximation of second order elliptic equations on domains that can be described as a union of sub domains or blocks we assume that a grid is defined on each block independently so that the resulting grid over the entire domain need not be conforming i e match across the block boundaries several techniques have been developed to approximate elliptic equations on multiblock grids that utilize a mortar finite element space defined on the block boundary interface itself we define a mixed finite element method that does not use such a mortar space the method has an advantage in the case where adaptive local refinement techniques will be used in that there is no mortar grid to refine as is typical of mixed methods our method is locally conservative element by element it is also globally conservative across the block boundaries theoretical results show that the approximate solution converges at the optimal rate to the true solution we present computational results to illustrate and confirm the theory

voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids natural neighbours are based on the voronoi diagram which partitions space into closest point regions the sibson and the laplace non sibsonian interpolants which are based on natural neighbours have shown promise within a galerkin framework for the solution of partial differential equations in this paper we focus on the laplace interpolant with a two fold objective first to unify the previous developments related to the laplace interpolant and to indicate its ties to some well known numerical methods and secondly to propose a voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids a conservation law in integral form is discretized on voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids the proposed scheme can also be viewed as a point collocation technique a detailed study on consistency is conducted and the satisfaction of the discrete maximum principle stability is established owing to symmetry of the laplace weight a symmetric positive definite stiffness matrix is realized which permits the use of efficient linear solvers on a regular rectangular or hexagonal grid the difference scheme reduces to the classical finite difference method numerical examples for the poisson equation with dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme copyright 2003 john wiley sons ltd

goal oriented a posteriori error estimation for the travel time functional in porous media flows in this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows the key application of this work is in the safety assessment of radioactive waste facilities in this setting the travel time functional measures the time taken for a nonsorbing radioactive solute transported by groundwater to travel from a potential site deep underground to the biosphere to ensure the computability of the travel time functional we employ a mixed formulation of darcy s law and conservation of mass together with raviart thomas h mathrm div omega conforming finite elements the proposed a posteriori error bound is derived based on a variant of the standard dual weighted residual approximation which takes into account the lack of smoothness of the underlying functional of interest the proposed adaptive refinement strategy is tested on both a simple academic test case and a problem based on the

asymptotic expansions and l error estimates for mixed finite element methods for second order elliptic problems asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and richardson extrapolation can be applied to increase the accuracy of the approximations a new procedure which is called the error corrected method is presented as a further application of the asymptotic error expansion for the first order bdm approximation of the scalar field the key point in deriving the asymptotic expansions for the error is an establishment ofl 1 error estimates for mixed finite element approximations for the regularized green s functions as another application of thel 1 error estimates for the regularized green s functions we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems

a superconvergent linear fe approximation for the solution of an elliptic system of pdes the aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations this nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem the method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations

stability and error analysis of mixed finite volume methods for advection dominated problems abstract we consider a convection diffusion reaction problem and we analyze a stabilized mixed finite volume scheme introduced in 1 the scheme is presented in the format of discontinuous galerkin methods and error bounds are given proving o h convergence in the l 2 norm for the scalar variable which is approximated with piecewise constant elements

iterative solution methods for modeling multiphase flow in porous media fully implicitly we discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell centered finite difference discretization of two and three phase porous media flow every outer nonlinear newton iteration requires solution of a nonsymmetric jacobian linear system two major types of preconditioners supercoarsening multigrid scmg and two stage are developed for the gmres iteration applied to the solution of the jacobian system the scmg reduces the three dimensional system to two dimensions using a vertical aggregation followed by a two dimensional multigrid the two stage preconditioners are based on decoupling the system into a pressure and concentration equations several pressure preconditioners of different types are described extensive numerical results are presented using the integrated parallel reservoir simulator ipars and indicate that these methods have low arithmetical complexity per iteration and good convergence rates

a local post processing technique for improving the accuracy in mixed finite element approximations a general simple post processing technique is given in this paper under certain conditions if an algorithm approximating a function u also provides a good approximation for its gradient and for some locally defined projection pu then by this method a better approximation for u can be easily obtained essentially the leading term of the error between u and its original approximation can be computed it is shown how the method may be applied to various mixed type finite element methods

godunov mixed methods for advective flow problems in one space dimension a time splitting method for solving advection dominated parabolic partial differential equations is presented in this method a higher order godunov procedure approximates advection and a mixed finite element procedure approximates diffusion several variations on the basic scheme are formulated for solving one dimensional quasilinear parabolic problems with dirichlet boundary conditions a maximum principle for one variant of the scheme is demonstrated and l infty l 2 and l 2 l 2 error estimates for the approximate solution and the diffusive flux respectively are derived these estimates indicate that one variant of the scheme is l infty stable in certain situations but possibly suboptimal in error while another variant is optimal and l 2 stable

a domain decomposition method using efficient interface acting preconditioners the conjugate gradient boundary iteration cgbi is a domain decomposition method for symmetric elliptic problems on domains with large aspect ratio high efficiency is reached by the construction of preconditioners that are acting only on the subdomain interfaces the theoretical derivation of the method and some numerical results revealing a convergence rate of 0 04 0 1 per iteration step are given in this article for the solution of the local subdomain problems both finite element fe and spectral chebyshev methods are considered

a block finite difference scheme for second order elliptic problems with discontinuous coefficients a block finite element difference scheme is derived from the next to the lowest order raviart thomas mixed quadrilateral finite element method by using some numerical quadratures element wise this finite difference scheme is much simpler than the mixed finite element method itself with the same order of accuracy superconvergence error estimates of order o h 3 for both the pressure and the darcy velocity are established even for certain discontinuous coefficients numerical experiments are presented to confirm the analysis at the end of this paper

use of cross time frequency estimators for structural identification in non stationary conditions and under unknown excitation abstract this paper introduces a new structural identification method for use in non stationary conditions which applies to structures and systems in normal serviceability conditions under unknown excitation the proposed method uses auto and cross time frequency transforms of accelerometer signals recorded from the structure to identify the vibration modes the transforms considered are those in cohen s class which in addition to possessing valuable properties for the analysis of mechanical signals lend themselves to a clear interpretation in energy terms this method enables modal parameters to be reliably estimated and an earlier technique proposed by the authors which was based on modal filters to be improved it is further shown that the cross correlation based estimators are more effective than techniques based on auto transformations due to their noise filtering properties finally a method to refine the estimate of modal shapes which avoids autocorrelation is proposed the accuracy of the procedure was assessed by means of numerical simulations

an interior estimate of superconvergence for finite element solutions for second order elliptic problems on quasi uniform meshes by local projections this paper establishes some superconvergence estimates for finite element solutions of second order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution the projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares in particular some local superconvergence estimates in the l2 and l infty norms are derived for the local projections of the galerkin finite element solution the results have two prominent features first they are established for any quasi uniform meshes which are of practicalimportance in scientific computation second they are derived on the basis of local properties of the domain and the solution for the second order elliptic problem therefore the result of this paper can be employed to provide useful a posteriori error estimators in practical computing

high accuracy analysis of two nonconforming plate elements we prove the superconvergence of morley element and the incomplete biquadratic nonconforming element for the plate bending problem under uniform rectangular meshes we obtain a superconvergence property at the symmetric points of the elements and a global superconvergent result by a proper postprocessing method

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

analysis of discontinuous finite element methods for ground water surface water coupling we derive and analyze new numerical approaches for modeling coupled ground water surface water flow in this coupled model surface water flow is described by the depth averaged shallow water equations while ground water is modeled by saturated darcy flow the coupling between the two models assumes continuity of pressure and water flux across the ground water surface water interface the coupled model is approximated by a local discontinuous galerkin method for ground water flow and a discontinuous galerkin method for surface water flow a priori error estimates are derived for this approach a closely related approach where the well known mixed finite element method is applied to the ground water flow equations is also described and analyzed one advantage of these approaches is that they allow for the ground water and surface water domains to be meshed independently under some mild restrictions

the simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on n rectangular grids abstract a family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension where the stress field is approximated by symmetric finite element tensors this family of elements has a perfect matching between the stress and the displacement the discrete spaces for the normal stress i i the shear stress i j and the displacement u i are span 1 x i span 1 x i x j and span 1 respectively on rectangular grids in particular the definition remains the same for all space dimensions as a result of these choices the theoretical analysis is independent of the spatial dimension as well in 1d the element is nothing else but the 1d raviart thomas element which is the only conforming element in this family in 2d and higher dimensions they are new elements but of the minimal degrees of freedom the total degrees of freedom per element are 2 plus 1 in 1d 7 plus 2 in 2d and 15 plus 3 in 3d these elements are the simplest element for any space dimension the well posedness condition and the optimal a priori error estimate of the family of finite elements are proved numerical tests in 2d and 3d are presented to show a superiority of the new elements over others as a superconvergence is exhibited and proved

lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions lower bounds for the condition numbers of the preconditioned systems are obtained for the bramble pasciak schatz substructuring preconditioner and the neumann neumann preconditioner in two dimensions they show that the known upper bounds are sharp

convergence of a substructuring method with lagrange multipliers we analyze the convergence of a substructing iterative method with lagrange multipliers propose recently by farhat and roux the method decomposes finite element discretization of an elliptic boundary value problem into neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components for linear conforming elements and preconditioning by the dirichlet problems on the subdomains we prove the asymptotic bound on the condition number c 1 log h h alpha alpha 2 or 3 where h is the characteristic element size h subdomain size and c is independent of the number of subdomains