a multigrid method for the pseudostress formulation of stokes problems the purpose of this paper is to develop and analyze a multigrid solver for the finite element discretization of the pseudostress system associated with the differential operator mathcal a gamma mathbf grad mathbf div over 2 times 2 matrix valued functions this system is derived from the pseudostress velocity formulation z cai b lee and p wang siam j numer anal 42 2004 pp 843 859 of two dimensional stokes problems through the penalty method or natural time discretization for the respective steady or unsteady state problems here gamma 0 is a constant associated with either the penalty parameter or the time step size and mathcal a is a singular map in this paper we develop a multigrid solver for the discrete problem using both the raviart thomas rt and the brezzi douglas marini bdm finite elements we show that the multigrid convergence rate is o frac 1 gamma 2 1 gamma 2 m where m is the number of smoothings this convergence rate is independent of the mesh size and the number of levels used in multigrid moreover numerical results presented in this paper show that the multigrid convergence rate for the bdm elements does not depend on the parameter gamma

h adaptive least squares finite element methods for the 2d stokes equations of any order with optimal convergence rates abstract the adaptive least squares finite element method ls fem for the stokes equations has recently been based on alternative error estimators in bringmann and carstensen 2016 for the lowest order case since the first order div ls fem measures the flux errors in h div the data resolution error measures the l 2 norm of the right hand side f minus the piecewise polynomial approximation k f without a mesh size factor this enforces separate marking with an overall abstract theory carstensen and rabus 2016 this paper contributes a a discussion of the scaling of the ls fem b optimal rates of the adaptive h version of any order k and c comparing numerical results in 2d with all details on inhomogeneous dirichlet data covered by the analysis

stabilized finite element methods for the velocity pressure stress formulation of incompressible flows abstract formulated in terms of velocity pressure and the extra stress tensor the incompressible navier stokes equations are discretized by stabilized finite element methods the stabilized methods proposed are analyzed for a linear model and extended to the navier stokes equations the numerical tests performed confirm the good stability characteristics of the methods these methods are applicable to various combinations of interpolation functions including the simplest equal order linear and bilinear elements

time domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method the acoustical wave propagator awp scheme involves an effective time domain calculation of sound propagation using the combination of chebyshev polynomial expansion and the fourier pseudospectral method the accuracy of this scheme degrades when the media has discontinuities due to the well known gibbs phenomenon in this paper several issues concerning awp are addressed including an analysis of the effect of gibbs phenomenon on the accuracy a mapped pseudospectral method is proposed wherein the grid points are redistributed with the emphasis across the media discontinuities by a pre determined smooth mapping curve then the spatial derivatives are calculated through a modified fourier pseudospectral method using this method the influence of the gibbs phenomenon is effectively alleviated while the computational efficiency of awp is still maintained the superiority of this improved awp scheme is illustrated by three one dimensional 1 d numerical examples

quasi optimal adaptive pseudostress approximation of the stokes equations the pseudostress velocity formulation of the stationary stokes problem allows some quasi optimal raviart thomas mixed finite element formulation for any polynomial degree the adaptive algorithm employs standard residual based explicit a posteriori error estimation from carstensen kim and park siam j numer anal 49 2011 pp 2501 2523 for the lowest order raviart thomas finite element functions in a simply connected lipschitz domain this paper proves optimal convergence rates in terms of the number of unknowns of the adaptive mesh refining algorithm based on the concept of approximation classes the proofs use some novel equivalence to first order nonconforming crouzeix raviart discretization plus a particular helmholtz decomposition of deviatoric tensors

mixed finite element methods for incompressible flow stationary navier stokes equations in z cai c tong p s vassilevski and c wang numer methods partial differential equations to appear the authors developed and analyzed a mixed finite element method for the stationary stokes equations based on the pseudostress velocity formulation the pseudostress and the velocity are approximated by a stable pair of finite elements raviart thomas elements of index k geq0 and discontinuous piecewise polynomials of degree k geq0 respectively this paper extends the method to the stationary incompressible navier stokes equations under appropriate assumptions we show that the pseudostress velocity formulation of the navier stokes equation and its discrete counterpart have branches of nonsingular solutions and error estimates of the mixed finite element approximations are established as well

receiver operator characteristic roc curves and non normal data an empirical study this paper evaluates the performance of several diagnostic kits for assessing levels of serum prostatic acid phosphatase on patients at different stages of cancer of the prostate each patient was studied with several kits we compare results obtained for receiver operator characteristic roc curve methodology with data assumed to follow a normal distribution with log transformed data assumed to follow a normal distribution and when neither of these assumptions holds there were important differences between the results of the different approaches for these data the normal distribution assumption should be used with extreme caution the log transformed based results compared favourably with the non parametric but unconsidered application of the method should be avoided

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

a k space method for coupled first order acoustic propagation equations a k space method for large scale simulation of ultrasonic pulse propagation is presented the present method which solves the coupled first order differential equations for wave propagation in inhomogeneous media is derived in a simple form analogous to previous finite difference methods with staggered spatial and temporal grids like k space methods based on second order wave equations the present method is exact for homogeneous media unconditionally stable for slow c r c0 media and highly accurate for general weakly scattering media in addition unlike previous k space methods the form of the method allows straightforward inclusion of relaxation absorption and perfectly matched layer pml nonreflecting boundary conditions numerical examples illustrate the capabilities of the present k space method for weakly inhomogeneous media accurate results are obtained using coarser temporal and spatial steps than possible with comparable finite difference and pseudospectral methods the low dispers

a residual based a posteriori error estimator for a fully mixed formulation of the stokes darcy coupled problem abstract in this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2d flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we consider dual mixed formulations in both media which yields the pseudostress and the velocity in the fluid together with the velocity and the pressure in the porous medium and the traces of the porous media pressure and the fluid velocity on the interface as the resulting unknowns the set of feasible finite element subspaces includes raviart thomas elements of lowest order and piecewise constants for the velocities and pressures respectively in both domains together with continuous piecewise linear elements for the traces we derive a reliable and efficient residual based a posteriori error estimator for the coupled problem the proof of reliability makes use of the global inf sup condition helmholtz decompositions in both media and local approximation properties of the clement interpolant and raviart thomas operator on the other hand inverse inequalities the localization technique based on element bubble and edge bubble functions and known results from previous works are the main tools for proving the efficiency of the estimator finally some numerical results confirming the theoretical properties of this estimator and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution are reported

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

a unifying theory of a posteriori finite element error control residual based a posteriori error estimates are derived within a unified setting for lowest order conforming nonconforming and mixed finite element schemes the various residuals are identified for all techniques and problems as the operator norm l of a linear functional of the form in the variable of a sobolev space v the main assumption is that the first order finite element space is included in the kernel ker l of l as a consequence any residual estimator that is a computable bound of l can be used within the proposed frame without further analysis for nonconforming or mixed fe schemes applications are given for the laplace stokes and navier lame equations

convergence analysis of an adaptive nonconforming finite element method an adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations the analysis is carried out for the lowest order crouzeix raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels important tools in the convergence proof are a discrete local efficiency and a quasi orthogonality property the proof does neither require regularity of the solution nor uses duality arguments as a consequence on the data control no particular mesh design has to be monitored

an efficient high order algorithm for acoustic scattering from penetrable thin structures in three dimensions this paper presents a high order accelerated algorithm for the solution of the integral equation formulation of volumetric scattering problems the scheme is particularly well suited to the analysis of thin structures as they arise in certain applications e g material coatings in addition it is also designed to be used in conjunction with existing low order fft based codes to upgrade their order of accuracy through a suitable treatment of material interfaces the high order convergence of the new procedure is attained through a combination of changes of parametric variables to resolve the singularities of the green function and partitions of unity to allow for a simple implementation of spectrally accurate quadratures away from singular points accelerated evaluations of the interaction between degrees of freedom on the other hand are accomplished by incorporating two face equivalent source approximations on cartesian grids a detailed account of the main algorithmic components of the scheme are presented together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results

compatible spectral approximations for the velocity pressure stress formulation of the stokes problem the numerical approximation of the mixed velocity pressure stress formulation of the stokes problem using spectral methods is considered in addition to the compatibility condition between the discrete velocity and pressure spaces a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well posed problem the theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational the discrete problem is analyzed and error estimates are derived a comparison between the mixed approach and the standard velocity pressure formulation is made in terms of the condition number of the resulting systems

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

an augmented mixed finite element method for the vorticity velocity pressure formulation of the stokes equations this paper deals with the numerical approximation of the stationary two dimensional stokes equations formulated in terms of vorticity velocity and pressure with non standard boundary conditions here by introducing a galerkin least squares term we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem we propose two families of mixed finite elements to solve the discrete problem in the first family the unknowns are approximated by piecewise continuous and quadratic elements brezzi douglas marini and piecewise constant finite elements respectively while in the second family the unknowns are approximated by piecewise linear and continuous raviart thomas and piecewise constant finite elements respectively the wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the babuska brezzi theory we establish a priori error estimates in the natural norms and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes additional examples of cases not covered by our theory are also presented

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

analysis of a velocity pressure pseudostress formulation for the stationary stokes equations abstract we consider a non standard mixed approach for the stokes problem in which the velocity the pressure and the pseudostress are the main unknowns alternatively the pressure can be eliminated from the original equations thus yielding an equivalent formulation with only two unknowns in this paper we develop a priori and a posteriori error analyses of both approaches we first apply the babuska brezzi theory to prove the well posedness of the continuous and discrete formulations in particular we show that raviart thomas elements of order k 0 for the pseudostresses and piecewise polynomials of degree k for the velocities and the pressures ensure unique solvability and stability of the associated galerkin schemes then we derive reliable and efficient residual based a posteriori error estimators for both schemes without and with pressure unknown finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimators and showing the behaviour of the associated adaptive algorithms

an optimal adaptive mixed finite element method various applications in fluid dynamics and computational continuum mechanics motivate the development of reliable and efficient adaptive algorithms for mixed finite element methods in order to save degrees of freedom not all but just some selected set of finite element domains are refined hence the fundamental question of convergence as well as the question of optimality require new mathematical arguments the presented adaptive algorithm for raviart thomas mixed finite element methods solves the poisson model problem with optimal convergence rate chen holst and xu presented convergence and optimality of adaptive mixed finite element methods 2008 following arguments of rob stevenson for the conforming finite element method their algorithm reduces oscillations separately before approximating the solution by some adaptive algorithm in the spirit of w dorfler 1996 the algorithm proposed here appears more natural in switching to either reduction of the edge error estimator or of the oscillations

dual mixed finite element approximation of stokes and nonlinear stokes problems using trace free velocity gradients in this work a finite element method for a dual mixed approximation of stokes and nonlinear stokes problems is studied the dual mixed structure which yields a twofold saddle point problem arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning the method approximates the velocity its gradient and the total stress tensor but avoids the explicit computation of the pressure which can be recovered through a simple postprocessing technique this method improves an existing approach for these problems and uses raviart thomas elements and discontinuous piecewise polynomials for approximating the unknowns existence uniqueness and error results for the method are given and numerical experiments that exhibit the reduced computational cost of this approach are presented

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

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

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

an error estimate for two dimensional stokes driven cavity flow discontinuous velocity boundary data for the lid driven cavity flow has long been causing difficulties in both theoretical analysis and numerical simulations in finite element methods the variational form for the driven cavity flow is not valid since the velocity is not in h 1 hence standard error estimates do not work by using only w 1 r 1 r 2 regularity and constructing a continuous approximation to the boundary data here we present error estimates for both the velocity pressure formulation and the pseudostress velocity formulation of the two dimensional stokes driven cavity flow

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

mixed finite element methods for quasilinear second order elliptic problems a mixed finite element method is developed to approximate the solution of a quasilinear second order elliptic partial differential equation the existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived

optimal adaptive nonconforming fem for the stokes problem this paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi optimal complexity for the stokes equations with respect to natural approximation classes the proof does not explicitly involve the pressure variable and follows from a novel discrete helmholtz decomposition of deviatoric functions

first order system least squares for the stress displacement formulation linear elasticity this paper develops a least squares finite element method for linear elasticity in both two and three dimensions the least squares functional is based on the stress displacement formulation with the symmetry condition of the stress tensor imposed in the first order system for the respective displacement and stress using the crouzeix raviart and raviart thomas finite element spaces our least squares finite element method is shown to be optimal in the broken h1 and h div norms uniform in the incompressible limit

a priori and a posteriori error analyses of a velocity pseudostress formulation for a class of quasi newtonian stokes flows abstract in this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear stokes models arising in quasi newtonian fluids the methods are based on a non standard mixed approach in which the velocity the pressure and the pseudostress are the original unknowns however we use the incompressibility condition to eliminate the pressure and set the velocity gradient as an auxiliary unknown which yields a twofold saddle point operator equation as the resulting dual mixed variational formulation in addition a suitable augmented version of the latter showing a single saddle point structure is also considered we apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well posed in particular we show that raviart thomas elements of order k 0 for the pseudostress and piecewise polynomials of degree k for the velocity and its gradient ensure the well posedness of the associated galerkin schemes moreover we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation then we derive a reliable and efficient residual based a posteriori error estimator for each scheme finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithms

mixed methods for stationary navier stokes equations based on pseudostress pressure velocity formulation in this paper we develop and analyze mixed finite element methods for the stokes and navier stokes equations our mixed method is based on the pseudostress pressure velocity formulation the pseudostress is approximated by the raviart thomas brezzi douglas marini or brezzi douglasfortin marini elements the pressure and the velocity by piecewise discontinuous polynomials of appropriate degree it is shown that these sets of finite elements are stable and yield optimal accuracy for the stokes problem for the pseudostress pressure velocity formulation of the stationary navier stokes equations the well posedness and error estimation results are established by eliminating the pseudostress variables in the resulting algebraic system we obtain cell centered finite volume schemes for the velocity and pressure variables that preserve local balance of momentum

an adaptive least squares fem for the stokes equations with optimal convergence rates this paper introduces the first adaptive least squares finite element method ls fem for the stokes equations with optimal convergence rates based on the newest vertex bisection with lowest order raviart thomas and conforming p 1 p1 discrete spaces for the divergence least squares formulation in 2d although the least squares functional is a reliable and efficient error estimator the novel refinement indicator stems from an alternative explicit residual based a posteriori error control with exact solve particular interest is on the treatment of the data approximation error which requires a separate marking strategy the paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non linear approximation class it extends and generalizes the approach of carstensen and park siam j numer anal 53 43 62 2015 from the poisson model problem to the stokes equations

miscellanea maximum likelihood estimation of misclassification rates of a binomial regression summary binomial regression is commonly used to construct a binary classification rule from a set of observable covariates the accuracy of the rule is summarised by empirical type 1 and type 2 misclassification rates as the threshold of the rule varies however the empirical misclassification rates are not the maximum likelihood estimators of the true rates if the binomial regression model is assumed true we give a simple algorithm for the maximum likelihood estimators of the misclassification rates for any given threshold statistical calibration of these estimates is best provided by bootstrap

a unified analysis of quasi optimal convergence for adaptive mixed finite element methods in this paper we present a unified analysis of both convergence and optimality of adaptive mixed finite element methods for a class of problems when the finite element spaces and corresponding a posteriori error estimates under consideration satisfy five hypotheses we prove that these five conditions are sufficient for convergence and optimality of the adaptive algorithms under consideration the main ingredient for the analysis is a new method to analyze both discrete reliability and quasi orthogonality this new method arises from an appropriate and natural choice of the norms for both the discrete displacement and stress spaces namely a mesh dependent discrete h 1 norm for the former and an l 2 norm for the latter and a newly defined projection operator from the discrete stress space on the coarser mesh onto the discrete divergence free space on the finer mesh as applications we prove these five hypotheses for the raviart thomas and brezzi douglas marini elements of the poisson and stokes

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

a priori and a posteriori pseudostress velocity mixed finite element error analysis for the stokes problem the pseudostress velocity formulation of the stationary stokes problem allows a raviart thomas mixed finite element formulation with quasi optimal convergence and some superconvergent reconstruction of the velocity this local postprocessing gives rise to some averaging a posteriori error estimator with explicit constants for reliable error control standard residual based explicit a posteriori error estimation is shown to be reliable and efficient and motivates adaptive mesh refining algorithms numerical experiments confirm our theoretical findings and illustrate the accuracy of the guaranteed upper error bounds even with reduced regularity

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

semiparametric methods for evaluating risk prediction markers in case control studies the performance of a well calibrated risk model for a binary disease outcome can be characterized by the population distribution of risk and displayed with the predictiveness curve better performance is characterized by a wider distribution of risk since this corresponds to better risk stratification in the sense that more subjects are identified at low and high risk for the disease outcome although methods have been developed to estimate predictiveness curves from cohort studies most studies to evaluate novel risk prediction markers employ case control designs here we develop semiparametric methods that accommodate case control data the semiparametric methods are flexible and naturally generalize methods previously developed for cohort data applications to prostate cancer risk prediction markers illustrate the methods

a posteriori error control for dpg methods a combination of ideas in least squares finite element methods with those of hybridized methods recently led to discontinuous petrov galerkin dpg finite element methods they minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard locally computable dual norm this paper establishes a general a posteriori error analysis for the natural norms of the dpg schemes under conditions equivalent to a priori stability estimates it is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors the presented abstract framework for a posteriori error analysis applies to known dpg discretizations of laplace and lame equations and to a novel dpg method for the stress velocity formulation of stokes flow with symmetric stress approximations since the error control does not rely on the discrete equations it applies to inexactly computed or otherwise perturbed solutions within the discrete space

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

a convergent nonconforming adaptive finite element method with quasi optimal complexity in this paper we prove convergence and quasi optimal complexity of a simple adaptive nonconforming finite element method in each step of the algorithm the iterative solution of the discrete system is controlled by an adaptive stopping criterion and the local refinement is based on either a simple edge residual or a volume term depending on an adaptive marking strategy we prove that this marking strategy guarantees a strict reduction of the error augmented by the volume term and an additional oscillation term and quasi optimal complexity of the generated sequence of meshes

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

a priori and a posteriori error analyses of a pseudostress based mixed formulation for linear elasticity in this paper we present the a priori and a posteriori error analyses of a non standard mixed finite element method for the linear elasticity problem with non homogeneous dirichlet boundary conditions more precisely the approach introduced here is based on a simplified interpretation of the pseudostress displacement formulation originally proposed in arnold and falk 1988 which does not require symmetric tensor spaces in the finite element discretization in addition physical quantities such as the stress the strain tensor of small deformations and the rotation are computed through a simple postprocessing in terms of the pseudostress variable furthermore we also introduce a second element by element postprocessing formula for the stress which yields an optimally convergent approximation of this unknown with respect to the broken h div norm we apply the classical babuska brezzi theory to prove that the corresponding continuous and discrete schemes are well posed in particular raviart thomas spaces of order k 0 for the pseudostress and piecewise polynomials of degree k for the displacement can be utilized moreover we remark that in the 3d case the number of unknowns behaves approximately as 9 times the number of elements tetrahedra of the triangulation when k 0 this factor increases to 12 5 when one uses the classical peers next we derive a reliable and efficient residual based a posteriori error estimator for the mixed finite element scheme finally several numerical results illustrating the performance of the method confirming the theoretical properties of the estimator and showing the expected behaviour of the associated adaptive algorithm are provided

pseudo analytical finite difference elastic wave extrapolation based on the k space method abstractcost effective elastic wave modeling is the key to practical elastic reverse time migration and full waveform inversion implementations we have developed an efficient elastic pseudo analytical finite difference pafd scheme for elastic wave extrapolation the elastic pafd scheme is based on a modified pseudo spectral method k space method in which a pseudo analytical operator is used to ensure the high accuracy of elastic wave extrapolation however the k space method is motivated for a pure wave mode and thus its application in coupled first order elastic wave equations may cause the elastic pseudo analytical operators to suffer from crosstalk between the p and s wavefields the approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal this is done first by performing a predictor corrector strategy in first order elastic wave equations to eliminate those errors during wave extrapolation considering the massive computational cost in

augmented mixed finite element method for the oseen problem a priori and a posteriori error analyses abstract we propose a new augmented dual mixed method for the oseen problem based on the pseudostress velocity formulation the stabilized formulation is obtained by adding to the dual mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations we prove that for appropriate values of the stabilization parameters the new variational formulation and the corresponding galerkin scheme are well posed and a cea estimate holds for any finite element subspaces we also provide the rate of convergence when each row of the pseudostress is approximated by raviart thomas or brezzi douglas marini elements and the velocity is approximated by continuous piecewise polynomials moreover we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient finally we include several numerical experiments that support the theoretical results

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

the k space formulation of the scattering problem in the time domain an improved single propagator formulation the storage allocation requirement and solution speed of the k space formulation of the scattering problem in the time domain developed by this author n bojarski j acoust soc am 72 570 584 1982 were improved by a factor of 2 and 4 respectively by the replacement of the earlier two temporal propagators by a single more accurate propagator and the utilization of the special case real to complex fft algorithm it is shown that for problems of the size of about one million spatially discretized cells this method is about five times slower than an ideal solution and requires about five times as much in core storage allocation as an ideal solution it is shown that for a modern array processor this solution is i o bound it is pointed out that this method is ideally suited for parallel array processor implementation resulting in even greater speed

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

a first order k space model for elastic wave propagation in heterogeneous media a pseudospectral model of linear elastic wave propagation is described based on the first order stress velocity equations of elastodynamics k space adjustments to the spectral gradient calculations are derived from the dyadic green s function solution to the second order elastic wave equation and used to a ensure the solution is exact for homogeneous wave propagation for timesteps of arbitrarily large size and b also allows larger time steps without loss of accuracy in heterogeneous media the formulation in k space allows the wavefield to be split easily into compressional and shear parts a perfectly matched layer pml absorbing boundary condition was developed to effectively impose a radiation condition on the wavefield the staggered grid which is essential for accurate simulations is described along with other practical details of the implementation the model is verified through comparison with exact solutions for canonical examples and further examples are given to show the efficiency of the method for practical problems the efficiency of the model is by virtue of the reduced point per wavelength requirement the use of the fast fourier transform fft to calculate the gradients in k space and larger time steps made possible by the k space adjustments