a priori and a posteriori analysis of finite volume discretizations of darcy s equations this paper is devoted to the numerical analysis of some finite volume discretizations of darcy s equations we propose two finite volume schemes on unstructured meshes and prove their equivalence with either conforming or nonconforming finite element discrete problems this leads to optimal a priori error estimates in view of mesh adaptivity we exhibit residual type error indicators and prove estimates which allow to compare them with the error in a very accurate way

a posteriori error estimation based on potential and flux reconstruction for the heat equation we derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous galerkin various finite volume and mixed finite element methods in space and the backward euler scheme in time extensions to conforming and nonconforming finite element spatial discretizations are also outlined our estimates are based on a h 1 conforming reconstruction of the potential continuous and piecewise affine in time and a locally conservative mathbf h mathrm div conforming reconstruction of the flux piecewise constant in time they yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative local in time lower bounds are also derived for nonconforming methods on time varying meshes the lower bounds require a mild parabolic type constraint on the meshsize

adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics in this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method the error estimates are based on the solution of an appropriately defined adjoint problem we present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain the primal subdomains are discretized using either an interior penalty discontinuous galerkin method or a continuous galerkin method with weakly imposed dirichlet conditions the mixed subdomains are discretized using raviart thomas mixed finite elements the a posteriori error estimate also accounts for the errors due to adjoint inconsistent subdomain discretizations the coupling between the subdomain discretizations is achieved via a mortar space we show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive

guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems abstract we derive a posteriori error estimates for a class of second order monotone quasi linear diffusion type problems approximated by piecewise affine continuous finite elements our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual as well as a global error lower bound up to a generic constant independent of the nonlinear operator they are thus fully robust with respect to the nonlinearity thanks to the choice of the error measure they are also locally efficient albeit in a different norm and hence suitable for adaptive mesh refinement moreover they allow to distinguish estimate separately and compare the discretization and linearization errors hence the iterative newton raphson fixed point linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error this can lead to important computational savings as performing an excessive number of unnecessary linearization iterations can be avoided a strategy combining the linearization stopping criterion and adaptive mesh refinement is proposed and numerically tested for the p laplacian

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

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

computational survey on a posteriori error estimators for nonconforming finite element methods for the poisson problem this paper compares different a posteriori error estimators for nonconforming first order crouzeix raviart finite element methods for simple second order partial differential equations all suggested error estimators yield a guaranteed upper bound of the discrete energy error up to oscillation terms with explicit constants novel equilibration techniques and an improved interpolation operator for the design of conforming approximations of the discrete nonconforming finite element solution perform very well in an error estimator competition with six benchmark examples

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 strict error bound with separated contributions of the discretization and of the iterative solver in non overlapping domain decomposition methods this paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non overlapping domain decomposition method feti or bdd we propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solver

robust a posteriori error control and adaptivity for multiscale multinumerics and mortar coupling we consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains termed multinumerics coupled using the mortar technique the grids need not match along the interfaces we are also interested in the multiscale setting where the subdomains are partitioned by a mesh of size h whereas the interfaces are partitioned by a mesh of much coarser size h and where lower order polynomials are used in the subdomains and higher order polynomials are used on the mortar interface mesh we derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm our estimates are also locally efficient and one of them is robust with respect to the ratio h h under an assumption of sufficient regularity of the weak solution the present approach allows bounding separately and comparing mutually the subdomain and interface errors a subdomain interface adaptive refinement s

energy norm a posteriori error estimates for mixed finite element methods this paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations it is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method the analysis is performed in two different ways under a saturation assumption and using a helmholtz decomposition for vector fields

robust a posteriori estimator for advection diffusion reaction problems we propose an almost robust residual based a posteriori estimator for the advection diffusion reaction model problem the theory is developed in the one dimensional setting the numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators in particular the mentioned norm possesses features that allow us to obtain a meaningful a posteriori estimator robust up to a log pe factor where pe is the global peclet number of the problem various numerical tests are performed in one dimension to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature we also consider a possible two dimensional extension of our result and only present a few basic numerical tests indicating that the estimator seems to preserve the good features of the one dimensional setting

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

a saddle point least squares approach for primal mixed formulations of second order pdes abstract we present a saddle point least squares spls method for discretizing second order elliptic problems written as primal mixed variational formulations a stability lbb condition and a data compatibility condition at the continuous level are automatically satisfied the proposed discretization method follows a general spls approach and has the advantage that a discrete inf sup condition is automatically satisfied for standard choices of the test and trial spaces for the proposed iterative processes a nodal basis for the trial space is not required efficient preconditioning techniques that involve inversion only on the test space can be considered stability and approximation properties for two choices of discrete spaces are investigated applications of the new approach include discretization of second order problems with highly oscillatory coefficient interface problems and higher order approximation of the flux for elliptic problems with smooth coefficients

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

polynomial degree robust a posteriori estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed discretizations we present equilibrated flux a posteriori error estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed finite element discretizations of the two dimensional poisson problem relying on the equilibration by the mixed finite element solution of patchwise neumann problems the estimates are guaranteed locally computable locally efficient and robust with respect to polynomial degree maximal local overestimation is guaranteed as well numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous galerkin scheme

the numerical solutions of interface problems by infinite element method in this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems furthermore approximations of stress intensity factors are given the infinite element method may be considered as a certain scheme of mesh refinement but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently and that an approximate solution which has a singularity at the singular point can be also obtained

guaranteed a posteriori error estimator for mixed finite element methods of linear elasticity with weak stress symmetry in this paper we propose an a posteriori error estimator for the mixed finite element methods of the linear elasticity problem with the symmetry condition weakly imposed on the stress tensor the error estimator is constructed by making a proper decomposition of the stress error and using an argument similar to the hypercircle method it is shown that the resulting estimator yields a guaranteed upper bound on the stress error which relies on computable upper bounds of the constants in the first and second korn inequalities we also establish the local lower bound by using the discrete friedrichs inequality our approach is equivalent to the helmholtz decomposition of the stress error but requires assumptions neither on the regularity of the solution nor the geometry of the domain numerical results are provided to illustrate the efficiency of our error estimator

non uniform order mixed fem approximation implementation post processing computable error bound and adaptivity the present work provides a straightforward and focused set of tools and corresponding theoretical support for the implementation of an adaptive high order finite element code with guaranteed error control for the approximation of elliptic problems in mixed form the work contains details of the discretisation using non uniform order mixed finite elements of arbitrarily high order a new local post processing scheme for the primary variable the use of the post processing scheme in the derivation of new fully computable bounds for the error in the flux variable and an hp adaptive refinement strategy based on the a posteriori error estimator numerical examples are presented illustrating the results obtained when the procedure is applied to a challenging problem involving a ten pole electric motor with singularities arising from both geometric features and discontinuities in material properties the procedure is shown to be capable of producing high accuracy numerical approximations with relatively modest numbers of unknowns

analysis of an upwind mixed finite element method for nonlinear contaminant transport equations in this paper the numerical approximation of a nonlinear diffusion equation arising in contaminant transport is studied the equation is characterized by advection diffusion and adsorption assuming the adsorption term is modeled by a freundlich isotherm it can be nonlinear in concentration and nondifferentiable as the concentration approaches zero we consider the approximation of this equation by a method which upwinds the advection and incorporates diffusion using a mixed finite element method error estimates for a semidiscrete formulation are derived and numerical results for a fully discrete formulation are given

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 posteriori error analysis of nonconforming finite element methods for convection diffusion problems abstract a unified framework is established for the a posteriori error analysis of nonconforming finite element approximations to convection diffusion problems under some certain conditions the theory assures the semi robustness of residual error estimates in the usual energy norm and the robustness in a modified norm and applies to several nonconforming finite elements such as the crouzeix raviart triangular element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element etc based on the general error decomposition in different norms we show that the key ingredients of error estimation are the existence of a bounded linear operator v h c v h nc with some elementary properties and the estimation on the consistency error related to the particular numerical scheme the numerical results are presented to illustrate the practical behavior of the error estimator and check the theoretical predictions

an a posteriori driven adaptive mixed high order method with application to electrostatics in this work we propose an adaptive version of the recently introduced mixed high order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism the latter include in particular the numerical modeling of comb drive and mems devices mesh adaptation is driven by newly derived residual based error estimators the resulting method has several advantageous features it supports fairly general meshes it enables arbitrary approximation orders and has a moderate computational cost thanks to hybridization and static condensation the a posteriori driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions allowing to fully exploit the high order of approximation

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

a posteriori error estimates for the mortar mixed finite element method several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived a residual based estimator provides optimal upper and lower bounds for the pressure error an efficient and reliable estimator for the velocity and mortar pressure error is also derived which is based on solving local element problems in a higher order space the interface flux jump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only

interpolating stabilized moving least squares mls approximation for 2d elliptic interface problems abstract the main aim of the current paper is to propose a new truly meshless numerical technique to solve the one and two dimensional elliptic interface problems the employed meshless approach is based on a new class of mls approximation the numerical procedure is based on the interpolating stabilized mls ismls approximation the new shape functions that have been made by the ismls technique have the kronecker property thus the dirichlet boundary conditions can be applied directly in the current investigation we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ismls technique several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure moreover test problems show the acceptable accuracy and efficiency of the proposed scheme

high order finite difference schemes for elliptic problems with intersecting interfaces in this work we derive high order second and fourth of convergence but with first and third order local truncation error respectively compact finite difference schemes for elliptic equations with intersecting interfaces the approach uses the differential equation and the jump interface relations as additional identities which can be differentiated to eliminate higher order local truncation errors numerical experiments are carried out to demonstrate the high order accuracy and to show that our method is effective to sharp contrast in the diffusion coefficients of the problems

a posteriori error analysis for hybridizable discontinuous galerkin methods for second order elliptic problems we present a unified a posteriori error analysis for hybridizable discontinuous galerkin methods for second order elliptic partial differential equations the main feature of this analysis is that it reduces the task of establishing the reliability and efficiency of the estimator to the verification of three simple conditions this approach allows us not only to derive new estimates for hybridizable discontinuous galerkin methods but also to recover well known a posteriori error estimates for mixed methods and for the continuous galerkin method

a priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations we study the mixed finite element approximation of the second order elliptic problem with gradient nonlinearities in two and three space dimensions existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates in l m omega are obtained also reliable and efficient a posteriori error estimators measured in the l m omega norm are derived numerical examples are provided to illustrate the performance of the proposed estimator

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

fundamental relationship between bilateral filtering adaptive smoothing and the nonlinear diffusion equation in this paper the relationship between bilateral filtering and anisotropic diffusion is examined the bilateral filtering approach represents a large class of nonlinear digital image filters we first explore the connection between anisotropic diffusion and adaptive smoothing and then the connection between adaptive smoothing and bilateral filtering previously adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation we extend adaptive smoothing to make it consistent thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation

the sensitivity and accuracy of fourth order finite difference schemes on nonuniform grids in one dimension abstract we construct local fourth order finite difference approximations of first and second derivatives on nonuniform grids in one dimension the approximations are required to satisfy symmetry relationships that come from the analogous higher dimensional fundamental operators the divergence the gradient and the laplacian for example we require that the discrete divergence and gradient be negative adjoint of each other div grad and the discrete laplacian is defined as lap divgrad the adjointness requirement on the divergence and gradient guarantees that the laplacian is a symmetric negative operator the discrete approximations we derive are fourth order on smooth grids but the approach can be extended to create approximations of arbitrarily high order we analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids

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

a simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows this paper develops an posteriori error estimate for lowest order locally conservative methods on meshes consisting of very general polytopal elements we focus on the ease of implementation of the methodology based on h1 conforming potential reconstruction and h div conforming flux reconstruction in particular the evaluation of our estimates merely consists in some local matrix vector multiplications where on each mesh element the matrices are either directly inherited from the given numerical method or trivially constructed from the element geometry while the vectors are the degrees of freedom on the given element we then apply this methodology to unsteady nonlinear coupled degenerate problems describing complex multiphase flows in porous media here on each step of the time marching scheme linearization procedure and linear algebraic solver we distinguish the corresponding error components this leads to an easy to implement and fast to run adaptive algorithm with simultaneously guaranteed overall precision and optimal efficiency ensured through the use of adaptive stopping criteria together with adaptive space and time mesh refinements numerous numerical experiments on practical problems in two and three space dimensions illustrate the performance of our methodology

a priori and a posteriori error analysis of a staggered discontinuous galerkin method for convection dominant diffusion equations abstract an upwind staggered discontinuous galerkin upwind sdg method for convection dominant diffusion problems is developed optimal a priori error estimates can be achieved for both the scalar and vector functions approximated by the method to efficiently capture the layer problems we propose a robust a posteriori error estimator for upwind sdg method measured in a natural norm and a semi norm associated with the convective derivative the semi norm can be neglected when the mesh peclet number is sufficiently small the key is to bound the conforming contribution by the dual norm of the residual and the robustness arises from the incorporation of the semi norm associated with the convective derivative finally various numerical examples are tested to illustrate the performance of the robust a posteriori error estimator the results indicate that optimal convergence rates can be achieved and the singularity can be well captured by the proposed error estimator

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

a posteriori error estimates for a discontinuous galerkin method applied to elliptic problems log number r74 abstract a posteriori error estimates for locally mass conservative methods for subsurface flow are presented these methods are based on discontinuous approximation spaces and referred to as discontinuous galerkin methods in the case where penalty terms are added to the bilinear form one obtains the nonsymmetric interior penalty galerkin methods in a previous work we proved optimal rates of convergence of the methods applied to elliptic problems here h adaptivity is investigated for flow problems in 2d we derive global explicit estimators of the error in the l 2 norm and we numerically investigate an implicit indicator of the error in the energy norm model problems with discontinuous coefficients are considered

a modified finite volume approximation of second order elliptic equations with discontinuous coefficients a modified finite difference approximation for interface problems in rn n 1 2 3 is presented the essence of the modification falls in the simultaneous discretization of any two normal components of the flux at the opposite faces of the finite volume in this way the continuous normal component of the flux through an interface is approximated by finite differences with second order consistency the derived scheme has a minimal 2n 1 point stencil for problems in rn second order convergence with respect to the discrete h1 norm is proved for a class of interface problems second order pointwise convergence is observed in a series of numerical experiments with one dimensional 1 d two dimensional 2 d and three dimensional 3 d interface problems the numerical experiments presented demonstrate advantages of the new scheme compared with the known schemes which use arithmetic and harmonic averaging of the discontinuous diffusion coefficient

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

error indicators for incompressible darcy flow problems using enhanced velocity mixed finite element method abstract local mesh adaptivity serves as a practical tool in numerical simulations to accurately capture features of interest while reducing computational time and memory requirements in this work we suggest a refinement strategy based on pressure and flux error estimates for numerical simulation of an incompressible single phase flow and transport process in the subsurface porous media we derive a posteriori error estimates for an enhanced velocity mixed finite element method evmfem as a spatial domain decomposition approach we note that the flux errors play an important role in coupled flow and transport systems later demonstrated using numerical experiments a comparison between explicit residual based error estimators and an implicit error estimator based upon the post processing proposed by arbogast and chen 1995 shows that the latter performs better a residual based error estimator for pressure was found to be both computationally efficient while sufficiently indicating the large error subdomains numerical studies are also presented that confirm our theoretical derivations while demonstrating the advantages of post processing in detecting velocity errors

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

the semidiscrete finite volume element method for nonlinear convection diffusion problem abstract in this work a semidiscrete finite volume element method for the nonlinear convection diffusion problem is considered under some regular assumptions we obtain the l2 and h1 norm error estimates of numerical solution furthermore we investigate two grid finite volume element method for the considered equations compared with the standard method the two grid method is of the same order as the standard method in the h1 norm as long as the mesh sizes satisfy h o h 1 3 however the two grid method involves much less work than the standard method finally some numerical results are provided to verify the established theoretical analysis

a posteriori error estimation for interior penalty finite element approximations of the advection reaction equation in this note we consider residual based a posteriori error estimation for finite element approximations of the transport equation for the discretization we use piecewise affine continuous or discontinuous finite elements and symmetric stabilization of interior penalty type the lowest order discontinuous galerkin method using piecewise constant approximation is included as a special case the key elements in the analysis are a saturation assumption and an approximation result for interpolation between discrete spaces

the one machine sequencing problem let m 1 and m 3 be non bottleneck machines and m 2 a bottleneck machine processing only one job at a time suppose that n jobs have to be processed on m 1 m 2 and m 1 in that order and job i has to spend a time a on m 1 d 1 on m 2 and q 1 on m 3 we want to minimize the makespan this problem is important since its resolution provides a bound on the makespan of complicated systems such as job shops it is np hard in the strong sense however efficient branch and bound methods exist and we describe one of them our bound for the tree search is very close to the bound used by florian et al but the principle of branching is quite different at every node we construct by an o n log n algorithm a schrage schedule then we define a critical job c a critical set j and consider two subsets of schedules the schedules when c procedes every job of j and the schedules when c follows every job of j we give the results of this method and prove that the difference between the optimum and the schrage schedule is less than d 1

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations by a well known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned cg iterations using a multilevel preconditioner in the spirit of bramble pasciak and xu designed for standard nonconforming approximations local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results the performance of the preconditioner and the error estimator is illustrated by several numerical examples

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

an a posteriori error estimate for the variable degree raviart thomas method we propose a new a posteriori error analysis of the variable degree hybridized version of the raviart thomas method for second order elliptic problems on conforming meshes made of simplexes we establish both the reliability and eciency of the estimator for the l2 norm of the error of the ux we also nd the explicit dependence of the estimator on the order of the local spaces k 0 the only constants that are not explicitly computed are those depending on the shape regularity of the simplexes in particular the constant of the local eciency in equality is proven to behave like k 2 3 2 however we present numer ical experiments suggesting that such constant is actually independent of k

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

error control for the localized reduced basis multiscale method with adaptive on line enrichment in this contribution we consider localized robust and efficient a posteriori error estimation of the localized reduced basis multiscale lrbms method for parametric elliptic problems with possibly heterogeneous diffusion coefficient the numerical treatment of such parametric multiscale problems is characterized by a high computational complexity arising from the multiscale character of the underlying differential equation and the additional parameter dependence the lrbms method can be seen as a combination of numerical multiscale methods and model reduction using reduced basis rb methods to efficiently reduce the computational complexity with respect to the multiscale as well as the parametric aspect of the problem simultaneously in contrast to the classical residual based error estimators currently used in rb methods we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solut

some new a priori estimates for second order elliptic and parabolic interface problems abstract we present some new a priori estimates of the solutions to the second order elliptic and parabolic interface problems the novelty of these estimates lies in the explicit appearance of the discontinuous coefficients and the jumps of coefficients across the interface

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a posteriori error estimation for the discrete duality finite volume discretization of the laplace equation an efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the laplace equation on very general two dimensional meshes the main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

core instances for testing a case study abstract data generation for computational testing of optimization algorithms is a key topic in experimental algorithmics recently concern has arisen that many published computational experiments are inadequate with respect to the way test instances are generated in this paper we suggest a new research direction that might be useful to cope with the possible limitations of data generation the basic idea is to select a finite set of instances which represent the whole set of instances we propose a measure of the representativeness of an instance which we call e representativeness for a minimization problem an instance xe is e representative of another instance x if a 1 e approximate solution to x can be obtained by solving xe focusing on a strongly np hard single machine scheduling problem we show how to map the infinite set of all instances into a finite set of e representative core instances we propose to use this finite set of e representative core instances to test heuristics

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

closed form numerical solutions of variable coefficient linear second order elliptic problems abstract in this work we develop an alternative numerical technique which allows to construct a numerical solution in closed form of variable coefficient linear second order elliptic problems with dirichlet boundary conditions the elliptic partial differential equation is approximated by a consistent explicit difference scheme and using a discrete separation of the variables method we determine a closed form solution of the two resulting discrete boundary value problems with the separated variables avoiding to have to solve large algebraic systems one of these boundary value problems is a discrete sturm liouville problem which guarantees the qualitative properties of the exact solution of elliptic problem a constructive procedure for the computation of the numerical solution is given and an illustrative example is included

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

estimateur d erreur a posteriori hierarchique application aux elements finis mixtes on propose une generalisation de l estimateur a posteriori hierarchique de bank weiser aux formulation mixtes dans le cas d approximations conformes et non conformes avec ou sans integration numerique on presente comme exemples d application la formulation mixte duale du probleme de dirichlet pour le laplacien avec et sans integration numerique

an improved kernel regression method based on taylor expansion many regression functions obtained by nonparametric regression method often appear inconsonance between smoothness and fitness this phenomenon is extremely outstanding near the vertex regions how to improve the fitness and smoothness simultaneously becomes an important problem in the nonparametric regression field in this paper an improved kernel regression is proposed by introducing second derivative estimation into kernel regression function based on taylor expansion theorem experimental results on regression problems show that this new method is feasible and enables us to get regression function that is both smooth and well fitting the application of the method to grey image enhancement indicates that this approach is fruitful to the enhancement of weak information in the images

a posteriori error estimates including algebraic error and stopping criteria for iterative solvers for the finite volume discretization of a second order elliptic model problem we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system we show that the algebraic error can be bounded by constructing an equilibrated raviart thomas nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector next claiming that the discretization error and the algebraic one should be in balance we construct stopping criteria for iterative algebraic solvers an attention is paid in particular to the conjugate gradient method which minimizes the energy norm of the algebraic error using this convenient balance we also prove the efficiency of our a posteriori estimates i e we show that they also represent a lower bound up to a generic constant for the overall energy error a local version of this result is also stated this makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems abstract in this work we analyze a posteriori error estimator for a general class of mixed finite element methods of elliptic problems which guarantees an upper bound on the vector error thus extending the recent result obtained for the lowest order triangular raviart thomas mixed finite element method in m ainsworth a posteriori error estimation for lowest order raviart thomas mixed finite elements siam j sci comput 30 2007 08 189 204 the error estimator is constructed through a variant of stenberg s postprocessing procedure and the guaranteed upper bound is readily established by making use of the argument similar to the hypercircle method however the proof of the lower bound given in the above reference does not seem to apply to other kinds of mixed elements so we employ a different technique using the discrete friedrichs inequality to establish the lower bound of the error estimator

robust a posteriori error estimation for nonconforming finite element approximation the equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media the estimator is shown to provide a computable upper bound on the error and up to a constant depending only on the geometry provides two sided bounds on the error the robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

analysis of a modified second order mixed hybrid bdm_1 finite element method for transport problems in divergence form we prove optimal second order convergence of a modified lowest order brezzi douglas marini bdm 1 mixed finite element scheme for advection diffusion problems in divergence form if advection is present it is known that the total flux is approximated only with first order accuracy by the classical bdm 1 mixed method which is suboptimal since the same order of convergence is obtained if the computationally less expensive raviart thomas rt 0 element is used the modification that was first proposed by brunner et al adv water res 35 2012 pp 163 171 is based on the hybrid problem formulation and consists in using the lagrange multipliers for the discretization of the advective term instead of the cellwise constant approximation of the scalar unknown

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

robust a posteriori error estimates for conforming and nonconforming finite element methods for convection diffusion problems a posteriori error estimation is carried out within a unified framework for various conforming and nonconforming finite element methods for convection diffusion problems our main contribution is finding an appropriate norm to measure the error which incorporates a discrete energy norm a discrete dual semi norm of the convective derivative and jumps of the approximate solution over element faces edges in two dimensions the error estimator is shown to be robust with respect to the peclet number in the sense of the modified norm based on a general error decomposition we show that the key ingredient of error estimation is the estimation on the consistency error related to the particular numerical scheme and the remaining terms can be bounded in a unified way the numerical results are presented to illustrate the robustness and practical performance of the estimator in an adaptive refinement strategy

robust a posteriori error estimates for subgrid stabilization of non stationary convection dominated diffusive transport abstract we derive a robust residual a posteriori error estimator for time dependent convection diffusion reaction problem stabilized by subgrid viscosity in space and discretized by crank nicolson scheme in time the estimator yields upper bounds on the error which are global in space and time and lower bounds that are global in space and local in time numerical experiments illustrate the theoretical performance of the error estimator

needed an empirical science of algorithms deductive algorithmic science has reached a high level of sophistication but its worst case and average case results seldom tell us how well an algorithm is actually going to work in practice i argue that an empirical science of algorithms is a viable alternative i respond to misgivings about an empirical approach including the prevalent notion that only a deductive treatment can be theoretical or sophisticated np completeness theory for instance is interesting partly because it has significant if unacknowledged empirical content an empirical approach requires not only rigorous experimental design and analysis but also the invention of empirically based explanatory theories i give some examples of recent work that partially achieves this aim

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part i low order conforming nonconforming and mixed fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper their reliablility is shown for conforming nonconforming and mixed low order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

restoration of images corrupted by gaussian and uniform impulsive noise many approaches to image restoration are aimed at removing either gaussian or uniform impulsive noise this is because both types of degradation processes are distinct in nature and hence they are easier to manage when considered separately nevertheless it is possible to find them operating on the same image which produces a hard damage this happens when an image already contaminated by gaussian noise in the image acquisition procedure undergoes impulsive corruption during its digital transmission here we propose a principled method to remove both types of noise it is based on a bayesian classification of the input pixels which is combined with the kernel regression framework

robust a posteriori error estimates for stationary convection diffusion equations we analyze a posteriori error estimators for finite element discretizations of convection dominated stationary convection diffusion equations using locally refined isotropic meshes the estimators are based on either the evaluation of local residuals or the solution of discrete local problems with dirichlet or neumann boundary conditions all estimators yield global upper and lower bounds for the error measured in a norm that incorporates the standard energy norm and a dual norm of the convective derivative they are fully robust in the sense that the ratio of the upper and lower bounds is uniformly bounded with respect to the size of the convection the estimates are also uniform with respect to the size of the zero order reaction term and also hold for the limit case of vanishing reaction