residual based a posteriori error estimation for multipoint flux mixed finite element methods a novel residual type a posteriori error analysis technique is developed for multipoint flux mixed finite element methods for flow in porous media in two or three space dimensions the derived a posteriori error estimator for the velocity and pressure error in l 2 l2 norm consists of discretization and quadrature indicators and is shown to be reliable and efficient the main tools of analysis are a locally postprocessed approximation to the pressure solution of an auxiliary problem and a quadrature error estimate numerical experiments are presented to illustrate the competitive behavior of the estimator

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

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 numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

a coupled multipoint stress multipoint flux mixed finite element method for the biot system of poroelasticity abstract we present a mixed finite element method for a five field formulation of the biot system of poroelasticity that reduces to a cell centered pressure displacement system on simplicial and quadrilateral grids a mixed stress displacement rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity pressure darcy formulation the spatial discretization is based on combining the multipoint stress mixed finite element msmfe method for elasticity and the multipoint flux mixed finite element mfmfe method for darcy flow it uses the lowest order brezzi douglas marini mixed finite element spaces for the poroelastic stress and darcy velocity piecewise constant displacement and pressure and continuous piecewise linear or bilinear rotation a vertex quadrature rule is applied to the velocity stress and stress rotation bilinear forms which block diagonalizes the corresponding matrices and allows for local velocity stress and rotation elimination this leads to a cell centered positive definite system for pressure and displacement at each time step we perform error analysis for the semidiscrete and fully discrete formulations establishing first order convergence for all variables in their natural norms the numerical tests confirm the theoretical convergence rates and illustrate the locking free property of the method

a general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids the overlapping covolume case we present a general framework for constructing and analyzing finite volume methods applied to the mixed formulation of second order elliptic problems on quadrilateral grids the control volumes or covolumes in the grids overlap an overlapping finite volume method of this type was first introduced by russell in t f russell tech report 3 reservoir simulation research corp tulsa ok 1995 and was tested for a variety of problems on rectangular and quadrilateral grids in z cai et al comput geosci 1 1997 pp 289 315 later in s h chou and d y kwak siam j numer anal 37 2000 pp 758 771 chou and kwak reformulated it as their mixed covolume method and proved optimal order error estimates using the covolume methodology from s h chou math comp 66 1997 pp 85 104 and s h chou and d y kwak siam j numer anal 35 1998 pp 494 507 however their treatment was restricted to the case of diagonal coefficient tensor and rectangular grids since a different approach was needed for the quadrilateral distorted rectangular case in this paper we give a new framework which can handle not only the rectangular anisotropic case but also the anisotropic and irregular grid cases in which the locally supported test functions are images of the natural unit coordinate vectors under the piola transformation our theory sheds light on how to create new test functions using quadratures and now covers russell s quadrilateral case

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

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

a multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs abstract in coupled flow and poromechanics phenomena representing hydrocarbon production or co 2 sequestration in deep subsurface reservoirs the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs the typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock the former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs in order to address these challenges we augment the fixed stress split iterative scheme with upscaling and downscaling operators to enable modeling flow and mechanics on overlapping nonmatching hexahedral grids flow is solved on a finer mesh using a multipoint flux mixed finite element method and mechanics is solved on a coarse mesh using a conforming galerkin method the multiscale operators are constructed using a procedure that involves singular value decompositions a surface intersections algorithm and delaunay triangulations we numerically demonstrate the convergence of the augmented scheme using the classical mandel s problem solution

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

h div finite elements based on nonaffine meshes for 3d mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

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

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

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

nonconcave penalized likelihood with a diverging number of parameters a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by fan and li to simultaneously estimate parameters and select important variables they demonstrated that this class of procedures has an oracle property when the number of parameters is finite however in most model selection problems the number of parameters should be large and grow with the sample size in this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators furthermore the consistency of the sandwich formula of the covariance matrix is demonstrated nonconcave penalized likelihood ratio statistics are discussed and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions the asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary

a multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra in this paper we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell centered finite difference scheme a special non symmetric quadrature rule is employed that yields a positive definite cell centered system for the pressure by eliminating local velocities the method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids including hexahedra with non planar faces theoretical and numerical results indicate first order convergence for the pressure and face fluxes

optimal geostatistical model selection in many fields of science predicting variables of interest over a study region based on noisy data observed at some locations is an important problem two popular methods for the problem are kriging and smoothing splines the former assumes that the underlying process is stochastic whereas the latter assumes it is purely deterministic kriging performs better than smoothing splines in some situations but is outperformed by smoothing splines in others however little is known regarding selecting between kriging and smoothing splines in addition how to perform variable selection in a geostatistical model has not been well studied in this article we propose a general methodology for selecting among arbitrary spatial prediction methods based on approximately unbiased estimation of mean squared prediction errors using a data perturbation technique the proposed method accounts for estimation uncertainty in both kriging and smoothing spline predictors and is shown to be optimal in terms of two mean sq

fixed domain asymptotic properties of tapered maximum likelihood estimators when the spatial sample size is extremely large which occurs in many environmental and ecological studies operations on the large covariance matrix are a numerical challenge covariance tapering is a technique to alleviate the numerical challenges under the assumption that data are collected along a line in a bounded region we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator mle for the microergodic parameter in the matern covariance function by establishing the fixed domain asymptotic distribution of the exact mle and that of the tapered mle our results imply that under some conditions on the taper the tapered mle is asymptotically as efficient as the true mle for the microergodic parameter in the matern model

a multipoint flux mixed finite element method on hexahedra we develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell centered finite differences the paper is an extension of our earlier paper for quadrilateral and simplicial grids m f wheeler and i yotov siam j numer anal 44 2006 pp 2082 2106 the construction is motivated by the multipoint flux approximation method and it is based on an enhancement of the lowest order brezzi douglas duran fortin bddf mixed finite element spaces on hexahedra in particular there are four fluxes per face one associated with each vertex a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate first order convergence for pressures and subface fluxes on sufficiently regular grids as well as second order convergence for pressures at the cell centers second order convergence for face fluxes is also observed computationally

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

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

from mixed finite elements to finite volumes for elliptic pdes in two and three dimensions the link between mixed finite element mfe and finite volume fv methods applied to elliptic partial differential equations has been investigated by many authors recently a fv formulation of the mixed approach has been developed this approach was restricted to 2d problems with a scalar for the parameter used to calculate fluxes from the state variable gradient this new approach is extended to 2d problems with a full parameter tensor and to 3d problems the objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy this is achieved by defining one new variable per element for the 2d case with full parameter tensor this new formulation exists for any kind of triangulation it allows the reduction of the number of unknowns to the number of elements instead of the number of edges no additional assumptions are required concerning the averaging of the parameter in hetero geneous domains for 3d problems we demonstrate that the new formulation cannot exist for a general 3d tetrahedral discretization unlike in the 2d problem however it does exist when the tetrahedrons are regular or deduced from rectangular parallelepipeds and allows reduction of the number of unknowns numerical experiments and comparisons between both formulations in 2d show the efficiency of the new formulation copyright 2003 john wiley sons ltd

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

one step sparse estimates in nonconcave penalized likelihood models fan and li propose a family of variable selection methods via penalized likelihood using concave penalty functions the nonconcave penalized likelihood estimators enjoy the oracle properties but maximizing the penalized likelihood function is computationally challenging because the objective function is nondifferentiable and nonconcave in this article we propose a new unified algorithm based on the local linear approximation lla for maximizing the penalized likelihood for a broad class of concave penalty functions convergence and other theoretical properties of the lla algorithm are established a distinguished feature of the lla algorithm is that at each lla step the lla estimator can naturally adopt a sparse representation thus we suggest using the one step lla estimator from the lla algorithm as the final estimates statistically we show that if the regularization parameter is appropriately chosen the one step lla estimates enjoy the oracle properties with good initial estimators computationally the one step lla estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood we conduct some monte carlo simulation to assess the finite sample performance of the one step sparse estimation methods the results are very encouraging

error analysis of piecewise constant pressure approximations of darcy s law abstract in this paper we present a short discussion on some finite element formulations for linear elliptic problems for the sake of simplicity we consider the poisson equation p f taking the notation from darcy s law among the zillions of such methods we shall concentrate our attention on fem leading to a final system of linear algebraic equations mp f where each unknown pi represents the constant value of the approximated pressure ph in a single element it is indeed well known that for some applications there is a certain demand for these types of schemes

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

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational results

a mourre estimate and related bounds for hyperbolic manifolds with cusps of non maximal rank we consider the laplace operator on quotients of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries with parabolic subgroups of non maximal rank using methods developed by the first two authors we prove a mourre estimate and commutator estimates on the laplacian which imply absolute continuity of the spectrum and quantitative resolvent estimates these estimates will be used elsewhere to study the scattering matrix and eisenstein series and their meromorphic continuations and should be useful in studying trace formulas for these discrete groups

variable selection for binary spatial regression penalized quasi likelihood approach we consider the problem of selecting covariates in a spatial regression model when the response is binary penalized likelihood based approach is proved to be effective for both variable selection and estimation simultaneously in the context of a spatially dependent binary variable an uniquely interpretable likelihood is not available rather a quasi likelihood might be more suitable we develop a penalized quasi likelihood with spatial dependence for simultaneous variable selection and parameter estimation along with an efficient computational algorithm the theoretical properties including asymptotic normality and consistency are studied under increasing domain asymptotics framework an extensive simulation study is conducted to validate the methodology real data examples are provided for illustration and applicability although theoretical justification has not been made we also investigate empirical performance of the proposed penalized quasi likelihood approach for spatial count data to explore suitability of this method to a general exponential family of distributions

tuning parameter selectors for the smoothly clipped absolute deviation method summary the penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method it not only automatically and consistently selects the important variables but also produces estimators which are as efficient as the oracle estimator however these attractive features depend on appropriate choice of the tuning parameter we show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily with a nonignorable overfitting effect in the resulting model in addition we propose a bic tuning parameter selector which is shown to be able to identify the true model consistently simulation studies are presented to support theoretical findings and an empirical example is given to illustrate its use in the female labor supply data

additive model building for spatial regression summary spatial regression is an important predictive tool in many scientific applications and an additive model provides a flexible regression relationship between predictors and a response variable we develop a regularized variable selection technique for building a spatial additive model we find that the methods developed for independent data do not work well for spatially dependent data this motivates us to propose a spatially weighted error norm with a group lasso type of penalty to select additive components in spatial additive models we establish the selection consistency of the approach proposed where the penalty parameter depends on several factors such as the order of approximation of additive components characteristics of the spatial weight and spatial dependence an extensive simulation study provides a vivid picture of the effects of dependent data structure and choice of a spatial weight on selection results as well as the asymptotic behaviour of the estimators as an illustrative example the method is applied to lung cancer mortality data over the period of 2000 2005 obtained from the surveillance epidemiology and end results programme national cancer institute usa

robust convergence of multi point flux approximation on rough grids this paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids by rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit in contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space and not on a reference cell as a consequence less regular grids are allowed however the extra cost is that the symmetry of the method is lost

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

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

on semiparametric inference of geostatistical models via local karhunen loeve expansion summary we develop a semiparametric approach to geostatistical modelling and inference in particular we consider a geostatistical model with additive components where the form of the covariance function of the spatial random error is not prespecified and thus is flexible a novel local karhunen loeve expansion is developed and a likelihood based method is devised for estimating the model parameters and statistical inference a simulation study demonstrates sound finite sample properties and a real data example is given for illustration finally the theoretical properties of the estimates are explored and in particular consistency results are established