a posteriori error estimates for mixed finite element approximations of elliptic problems we derive residual based a posteriori error estimates of the flux in l 2 norm for a general class of mixed methods for elliptic problems the estimate is applicable to standard mixed methods such as the raviart thomas nedelec and brezzi douglas marini elements as well as stabilized methods such as the galerkin least squares method the element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order

a characterization of hybridized mixed methods for second order elliptic problems in this paper we give a new characterization of the approximate solution given by hybridized mixed methods for second order self adjoint elliptic problems we apply this characterization to obtain an explicit formula for the entries of the matrix equation for the lagrange multiplier unknowns resulting from hybridization we also obtain necessary and sufficient conditions under which the multipliers of the raviart thomas and the brezzi douglas marini methods of similar order are identical

a multiscale direct solver for the approximation of flows in high contrast porous media abstract we consider a non overlapping domain decomposition approach to approximate the solution of elliptic boundary value problems with high contrast in their coefficients we propose a method such that initially local solutions subject to robin boundary conditions in each primal subdomain are constructed with locally conservative finite element or finite volume methods then a novel approach is introduced to obtain a discontinuous global solution in terms of linear combination of the local subdomain solutions in the proposed algorithm the computation of local solutions for unions of subdomains are localized at nearest neighbor subdomain boundaries thus avoiding the solution of global interface problems we remove discontinuities in a smoothing step that is defined on a staggered grid or dual subdomains the resulting algorithm is naturally parallelizable and can be employed as a parallel direct solver offering great potential for the numerical solution of large problems in fact subdomains can be considered small enough to fit well in gpus and the proposed procedure can handle adaptive in space simulations effectively numerical simulations are presented and discussed we demonstrate the effectiveness of the proposed approach with two and three dimensional high contrast and channelized coefficients that lead to challenging approximation problems the new procedure although designed for parallel processing is also of value for serial calculations

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

subgrid upscaling and mixed multiscale finite elements second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale epsilon can be approximated on coarse meshes of spacing h gg epsilon only if one uses special techniques the mixed variational multiscale method also called subgrid upscaling can be used and this method is extended to allow oversampling of the local subgrid problems the method is shown to be equivalent to the multiscale finite element method when one uses the lowest order raviart thomas spaces and provided that there are no fine scale components in the source function f in the periodic setting a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is o epsilon hm sqrt epsilon h where m 1 moreover m 2 if one uses the second order brezzi douglas marini or brezzi douglas duran fortin spaces and no oversampling the error bounding constant depends only on the hm 1 norm of f and so is independent of small scales when m 1 when oversampling is not used a superconvergence result for the pressure approximation is shown

w stability and local input output stability results a new type of input output stability is defined based on the use of a sobolev space w w is well suited like the lebesgue space l sub 2 to obtain stability characterizations in the time and frequency domains moreover if compared with l sub 2 w has additional properties which enable one to establish local stability results a local version of the small gain theorem is established in this framework as well as some consequences of this result in particular local versions of the passivity theorem and of the circle criterion the relationship between w stability and asymptotic stability is studied

stochastic multiscale flux basis for stokes darcy flows abstract three algorithms are developed for uncertainty quantification in modeling coupled stokes and darcy flows the porous media may consist of multiple regions with different properties the permeability is modeled as a non stationary stochastic variable with its log represented as a sum of local karhunen loeve kl expansions the problem is approximated by stochastic collocation on either tensor product or sparse grids coupled with a multiscale mortar mixed finite element method for the spatial discretization a non overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem which is solved by an iterative solver for each stochastic realization in the traditional domain decomposition implementation each subdomain solves a local dirichlet or neumann problem in every interface iteration to reduce this cost two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced the basis consists of the local flux or velocity trace responses from each mortar degree of freedom it is computed for each subdomain independently before the interface iteration begins the use of the multiscale flux basis avoids the need for subdomain solves on each iteration the deterministic basis is computed at each stochastic collocation and used only at this realization the stochastic basis is formed by further looping over all local realizations of a subdomain s kl region before the stochastic collocation begins it is reused over multiple realizations numerical tests are presented to illustrate the performance of the three algorithms with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms

numerical multiscale methods for a reaction dominated model a galerkin enriched finite element method gem is proposed for the singularly perturbed reaction diffusion equation this new method is an improvement on the petrov galerkin enriched method pgem where now the standard piecewise bi linear test space incorporates fine scales this appears as the fundamental ingredient for suppressing oscillations in the numerical solutions also new parameter free stabilized finite element methods derived from both the gem and the pgem are driven by local generalized eigenvalue problems in the process jump stabilizing terms belonging to the class of cip methods emerge as a result of the enriching procedure interestingly numerical results indicate that jump based stabilizations are unnecessary and sometimes undesirable when treating reaction dominated problems finally we establish relationships with more standard enriched and stabilized methods and show that the proposed methods outperform them numerically

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

mixed finite element approximations based on 3d hp adaptive curved meshes with two types of h div conforming spaces summary two stable approximation space configurations are treated for the mixed finite element method for elliptic problems based on curved meshes their choices are guided by the property that in the master element the image of the flux space by the divergence operator coincides with the potential space by using static condensation the sizes of global condensed matrices which are proportional to the dimension of border fluxes are the same in both configurations the meshes are composed of different topologies tetrahedra hexahedra or prisms simulations using asymptotically affine uniform meshes exactly fitting a spherical like region and constant polynomial degree distribution k show l2 errors of order k 1 or k 2 for the potential variable while keeping order k 1 for the flux in both configurations the first case corresponds to rt k and bdfm k 1 spaces for hexahedral and tetrahedral meshes respectively but holding for prismatic elements as well the second case further incrementing the order of approximation of the potential variable holds for the three element topologies the case of hp adaptive meshes is considered for a problem modelling a porous media flow around a cylindrical horizontal well with elliptical drainage area the effect of parallelism and static condensation in cpu time reduction is illustrated this article is protected by copyright all rights reserved

multiscale hybrid mixed method for the stokes and brinkman equations the method abstract the multiscale hybrid mixed mhm method is extended to the stokes and brinkman equations with highly heterogeneous coefficients the approach is constructive we first propose an equivalent dual hybrid formulation of the original problem using a coarse partition of the heterogeneous domain faces may be not aligned with jumps in the data then the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent stokes or brinkman problems at the continuous level owing to this decomposition the one level mhm method stems from the standard galerkin approach for the lagrange multiplier space basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation they are the exact solution of the local problems with prescribed neumann boundary conditions on faces driven by the lagrange multipliers we make the mhm method effective by adopting the unusual stabilized finite element method to solve the local problems approximately as such equal order interpolation turns out to be an option for the velocity the pressure and the lagrange multipliers the numerical solutions share the important properties of the continuum such as local equilibrium with respect to external forces and local mass conservation several academic and highly heterogeneous tests infer that the method achieves super convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms

the multiscale hybrid mixed method in general polygonal meshes this work extends the general form of the multiscale hybrid mixed mhm method for the second order laplace darcy equation to general non conforming polygonal meshes the main properties of the mhm method i e stability optimal convergence and local conservation are proven independently of the geometry of the elements used for the first level mesh more precisely it is proven that piecewise polynomials of degree k and k 1 k ge 0 for the lagrange multipliers flux along with continuous piecewise polynomial interpolations of degree k 1 posed on second level sub meshes are stable if the latter is fine enough with respect to the mesh for the lagrange multiplier we provide an explicit sufficient condition for this restriction also we prove that the error converges with order k 1 and k 2 in the broken h 1 and l 2 norms respectively under usual regularity assumptions and that such estimates also hold for non convex or even non simply connected elements numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides

the multiscale robin coupled method for flows in porous media abstract a multiscale mixed method aiming at the accurate approximation of velocity and pressure fields in heterogeneous porous media is proposed the procedure is based on a new domain decomposition method in which the local problems are subject to robin boundary conditions the domain decomposition procedure is defined in terms of two independent spaces on the skeleton of the decomposition corresponding to interface pressures and fluxes that can be chosen with great flexibility to accommodate local features of the underlying permeability fields the well posedness of the new domain decomposition procedure is established and its connection with the method of douglas et al 1993 12 is identified also allowing us to reinterpret the known procedure as an optimized schwarz or two lagrange multiplier method the multiscale property of the new domain decomposition method is indicated and its relation with the multiscale mortar mixed finite element method mmmfem and the multiscale hybrid mixed mhm finite element method is discussed numerical simulations are presented aiming at illustrating several features of the new method initially we illustrate the possibility of switching from mmmfem to mhm by suitably varying the robin condition parameter in the new multiscale method then we turn our attention to realistic flows in high contrast channelized porous formations we show that for a range of values of the robin condition parameter our method provides better approximations for pressure and velocity than those computed with either the mmmfem and the mhm this is an indication that our method has the potential to produce more accurate velocity fields in the presence of rough realistic permeability fields of petroleum reservoirs

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

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

multivariate approximation theory and applications this introductory paper describes the main topics of this special issue dedicated to leonardo traversoni known at the international level as the promoter of the conference series multivariate approximation theory and applications to celebrate his 60th birthday

multiscale finite element methods for high contrast problems using local spectral basis functions in this paper we study multiscale finite element methods msfems using spectral multiscale basis functions that are designed for high contrast problems multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem this local spectral problem strongly depends on the choice of initial partition of unity functions the resulting space enriches the initial multiscale space using eigenvectors of local spectral problem the eigenvectors corresponding to small asymptotically vanishing eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small asymptotically vanishing eigenvalues we present a convergence study that shows that the convergence rate in energy norm is proportional to h 1 2 where is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space thus we would like to reach to a larger eigenvalue with a smaller coarse space this is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems numerical results are presented to back up our theoretical results and to show higher accuracy of msfems with spectral multiscale basis functions we also present a hierarchical construction of the eigenvectors that provides cpu savings

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

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

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

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

two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence abstract the purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal potential variables one for triangles and the other one for quadrilateral elements to be used in discrete versions of the mixed finite element method for elliptic problems they can be interpreted as enriched versions of b d f m k 1 spaces based on triangles and of r t k spaces for quadrilateral elements the new flux approximations are incremented with properly chosen internal shape functions with vanishing normal components on the edges of degree k 2 and matching primal functions of degree k 1 higher than the border fluxes which are kept of degree k in all these cases the divergence of the flux space coincide with the primal approximation space on the master element producing stable simulations using static condensation the global condensed system to be solved in the enriched cases has same dimension and structure of the original ones which is proportional to the space dimension of the border fluxes for each element geometry measuring the errors with l 2 norms the enriched space configurations give higher convergence rate of order k 2 for the primal variable while keeping the order k 1 for the flux for affine meshes the divergence errors have the same improved accuracy rate as for the error in the primal variable for quadrilateral non affine meshes for instance trapezoidal elements the divergence error has order k 1 one unit more than the order k occurring for r t k spaces on this kind of deformed meshes this fact also holds for a b f k elements but for them the potential order of accuracy does not improve keeping order k 1

a petrov galerkin enriched method a mass conservative finite element method for the darcy equation starting from the non stable p1 p0 discretization we build enhanced methods for the darcy equation which are stable and locally mass conservative the methods are derived in a petrov galerkin framework where both velocity and pressure trial spaces are enriched with multiscale functions these functions solve local problems correcting the residuals of the strong equations in each element and interior edge which leads to a velocity space enhanced with functions belonging to the lowest order raviart thomas space several numerical tests validate the methods

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

a multiscale hdg method for second order elliptic equations part i polynomial and homogenization based multiscale spaces we introduce a finite element method for numerical upscaling of second order elliptic equations with highly heterogeneous coefficients the method is based on a mixed formulation of the problem and the concepts of the domain decomposition and the hybrid discontinuous galerkin methods the method utilizes three different scales 1 the scale of the partition of the domain of the problem 2 the scale of partition of the boundaries of the subdomains related to the corresponding space of lagrange multipliers and 3 the fine grid scale that is assumed to resolve the scale of the heterogeneous variation of the coefficients our proposed method gives a flexible framework that 1 couples independently generated multiscale basis functions in each coarse patch 2 provides a stable global coupling independent of local discretization physical scales and contrast and 3 allows avoiding any constraints arbogast et al multiscale model simul 6 2007 pp 319 346 on coarse spaces in this paper we

multiscale model reduction for transport and flow problems in perforated domains abstract convection dominated transport phenomenon is important for many applications in these applications the transport velocity is often a solution of heterogeneous flow problems which results to a coupled flow and transport phenomena in this paper we consider a coupled flow stokes problem and transport unsteady convection diffusion problem in perforated domains perforated domains see fig 1 represent void space outside hard inclusions as in porous media filters and so on we construct a coarse scale solver based on generalized multiscale finite element method gmsfem for a coupled flow and transport the main idea of the gmsfem is to develop a systematic approach for computing multiscale basis functions we use a mixed formulation and appropriate multiscale basis functions for both flow and transport to guarantee a mass conservation for the transport problem we use petrov galerkin mixed formulation which provides a stability as a first approach we use the multiscale flow solution in constructing the basis functions for the transport equation in the second approach we construct multiscale basis functions for coupled flow and transport without solving global flow problem the novelty of this approach is to construct a coupled multiscale basis function numerical results are presented for computations using offline basis we also present an algorithm for adaptively adding online multiscale basis functions which are computed using the residual information numerical examples using online gmsfem show the speed up of convergence

error analysis of variable degree mixed methods for elliptic problems via hybridization a new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree raviart thomas method for second order elliptic problems the approach gives error estimates for the lagrange multipliers without using error estimates for the other variables error estimates for the primal and flux variables then follow from those for the lagrange multipliers in contrast traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the lagrange multipliers the new approach not only gives new error estimates for the new variable degree raviart thomas method but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates the error analysis is achieved by using a variational characterization of the lagrange multipliers wherein the other unknowns do not appear this approach can be applied to other hybridized mixed methods as well

zeroth and first order homogenized approximations to nonlinear diffusion through block inclusions by an analytical approach abstract approximate solutions to nonlinear diffusion systems are useful for many applications in computational science when the heterogeneous nonlinear diffusion coefficient has high contrast values an average solution given by upscaling the diffusion coefficient provides the average behavior of the fine scale solution which sometimes is infeasible to compute this is also related to a problem that occurs during numerical simulations when it is necessary to coarsen meshes and an upscale coefficient is needed in order to build the data from the fine mesh to the coarse mesh in this paper we present a portable and computationally attractive procedure for obtaining not only the upscaled coefficient and the zeroth order approximation of nonlinear diffusion systems but also the first order approximation which captures fine scale features of the solution these are possible by considering a correction to an approximate solution to the well known periodic cell problem obtained by a two scale asymptotic expansion of the respective nonlinear diffusion equation the correction allows one to obtain analytically the upscale diffusion coefficient when the heterogeneous coefficient is periodic and rapidly oscillating describing inclusions in a main matrix the approximate solutions provide a set of analytical basis functions used to construct the first order approximation and also an estimate for the upper bound error implied in using the upscaled approximations we demonstrate agreement with theoretical and published numerical results for the upscale coefficient when heterogeneous coefficients are described by step functions as well as convergence properties of the approximations corroborating with classical results from homogenization theory even though the results can be generalized the emphasis is for conductivity functions of the form k x u x k s x k r u x widely used for simulating flows in reservoirs

hierarchical high order finite element bases for h div spaces based on curved meshes for two dimensional regions or manifolds the mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential primal variable and of the flux field dual variable this work focuses on new h div conforming finite element spaces which are suitable for flux approximations based on curved meshes of a planar region or a manifold domain embedded in r 3 the adopted methodology for the construction of h div bases consists in using hierarchical h 1 conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure they are classified as being of edge or internal type the normal component of an edge function coincides on the corresponding edge with the associated scalar shape function and vanishes over the other edges and the normal components of an internal shape function vanishes on all element edges these properties are fundamental for the global assembly of h div conforming functions locally defined by these vectorial shape functions for applications to the mixed formulation the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular cylindrical and spherical regions demonstrating stable convergence with optimal convergence rates coinciding for primal and dual variables innovative two dimensional hdiv approximation spaces for curved quadrilateral and triangular elements hdiv bases properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure stable convergence with optimal convergence rates coinciding for primal and dual variables study of number of condensed equations as a function of the polynomial order of the shape functions

multiscale empirical interpolation for solving nonlinear pdes in this paper we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations the proposed method combines empirical interpolation techniques and local multiscale methods such as the generalized multiscale finite element method gmsfem to solve nonlinear equations the gmsfem is used to represent the solution on a coarse grid with multiscale basis functions computed offline computing the gmsfem solution involves calculating the system residuals and jacobians on the fine grid we use empirical interpolation concepts to evaluate these residuals and jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse scale problem rather than the fully resolved fine scale one the empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times the coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system the proposed multiscale empirical interpolation techniques 1 divide computing the nonlinear function into coarse regions 2 evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced order representation of the solution and 3 introduce multiscale proper orthogonal decomposition techniques to find appropriate interpolation vectors we demonstrate the effectiveness of the proposed methods on several nonlinear multiscale pdes that are solved with newton s methods and fully implicit time marching schemes our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale pdes on a coarse grid with bounded error and significant computational cost reduction

robust nonlinear control associating robust feedback linearization and h sub spl infin control in this note a robust nonlinear controller for a nonlinear system subject to model uncertainties is proposed such a controller associates a robust feedback linearization with a robust linear h sub spl infin controller the robust feedback linearization exactly transforms the nonlinear system into a linear system equal to the linear approximation of the original nonlinear system around a nominal operating point the robustness of the resulting overall nonlinear controller is proved by theoretical arguments and illustrated through an application example the control of a magnetic bearing system

a multiscale hybrid method for darcy s problems using mixed finite element local solvers abstract multiscale hybrid mixed mhm method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions for fluid flow normal fluxes multiplier over macro element boundaries and coarse piecewise constant potential approximations in each macro element are computed upscaling then small details are resolved by local problems using fine representations inside the macro elements setting the multiplier as neumann boundary conditions downscaling in this work a variant of the method is developed denoted by mhm h div adopting mixed finite elements at the downscaling stage instead of continuous finite elements used in all previous publications of the method thus this alternative mhm method inherits improvements typical of mixed methods as better flux accuracy and local mass conservation at the micro scale level inside the macro elements which are important properties for multi phase flows in rough heterogeneous media different two scale stable space settings are considered vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries in each macro element the internal flux components with vanishing normal traces and the potential approximations may be enriched in different extents with respect to internal mesh size internal polynomial degree or both the choice being determined by the problem at hands a unified general error analysis of the mhm h div method is presented for all these two scale space scenarios both mhm versions are compared for 2d test problems with smooth solutions for convergence rates verification and for darcy s flow in heterogeneous media mhm h div 3d simulations are presented for a known singular darcy s solution using adaptive macro partitions and for an oscillatory permeability scenario

perfect single error correcting codes in the johnson scheme delsarte conjectured in 1973 that there are no nontrivial pefect codes in the johnson scheme etzion and schwartz recently showed that perfect codes must be k regular for large k and used this to show that there are no perfect codes correcting single errors in j n w for n les 50 000 in this correspondence we show that there are no perfect single error correcting codes for n les 2250

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

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy abstract there are different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal pressure variables to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations three cases shall be studied and compared for discretized three dimensional formulations based on tetrahedral hexahedral and prismatic meshes the principle guiding the constructions is the property that the divergence of the dual space and the primal approximation space should coincide while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable there is the classic case of b d m k spaces based on tetrahedral meshes and polynomials of total degree k for the dual variable and k 1 for the primal variable showing stable simulations with optimal convergence rates of orders k 1 and k respectively another case is related to r t k and b d f m k 1 spaces for hexahedral and tetrahedral meshes respectively but holding for prismatic elements as well it gives identical approximation order k 1 for both primal and dual variables an improvement in accuracy obtained by increasing the degree of primal functions to k and by enriching the dual space with some properly chosen internal shape functions of degree k 1 while keeping degree k for the border fluxes a new type of approximation is proposed by further incrementing the order of some internal flux functions to k 2 and matching primal functions to k 1 higher than the border fluxes of degree k thus higher convergence rate of order k 2 is obtained for the primal variable using static condensation the global condensed system to be solved in all the cases has same dimension and structure which is proportional to the space dimension of the border fluxes for each element geometry illustrating results comparing the three different space configurations are presented for simulations based on hierarchical high order shape functions for h div conforming spaces which are specially constructed for affine tetrahedral hexahedral and prismatic meshes expected convergence rates are obtained for the flux pressure and divergence variables

a two level enriched finite element method for a mixed problem the simplest pair of spaces is made inf sup stable for the mixed form of the darcy equation the key ingredient is to enhance the finite element spaces inside a petrov galerkin framework with functions satisfying element wise local darcy problems with right hand sides depending on the residuals over elements and edges the enriched method is symmetric locally mass conservative and keeps the degrees of freedom of the original interpolation spaces first we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms then a low cost two level finite element method is proposed to effectively obtain enhancing basis functions the approach lays on a two scale numerical analysis and shows that well posedness and optimality is kept despite the second level numerical approximation several numerical experiments validate the theoretical results and compares favourably in some cases our results with the classical raviart thomas element

the existence of perfect codes in doob graphs we solve the problem of existence of perfect codes in the doob graph it is shown that 1 perfect codes in the doob graph d m n exist if and only if 6 m 3 n 1 is a power of 2 that is if the size of a 1 ball divides the number of vertices

towards multiscale functions enriching finite element spaces with local but not bubble like functions in this paper we propose a novel way via finite elements to treat problems that can be singular perturbed a reaction diffusion equation in our case we enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem as in the residual free bubble rfb setting but do not require these functions to vanish on each element edge a departure from the rfb paradigm such multiscale functions have an analytic expression for triangles and rectangles bubbles are the choice for the test functions allowing static condensation thus our method is of petrov galerkin type we perform several numerical validations which confirm the good performance of the method 2004 elsevier b v all rights reserved

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

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

estimation of local modeling error and goal oriented adaptive modeling of heterogeneous materials i error estimates and adaptive algorithms abstract a theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented these quantities may for example represent averaged stresses on the surface of inclusions or mollifications of pointwise stresses or displacements or in general local features of the fine scale solution characterized by continuous linear functionals these estimators are used to construct goal oriented adaptive procedures in which models of the microstructure are adapted to deliver local features to a preset level of accuracy algorithms for implementing these procedures are discussed and preliminary numerical results are given the analysis is restricted to linear static heterogeneous elastic materials

detecting perfect powers in essentially linear time this paper 1 gives complete details of an algorithm to compute approximate kth roots 2 uses this in an algorithm that given an integer n 1 either writes n as a perfect power or proves that n is not a perfect power 3 proves using loxton s theorem on multiple linear forms in logarithms that this perfect power decomposition algorithm runs in time log n 1 o 1

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

systematic and generic construction of shape functions for p adaptive meshes of multidimensional finite elements this paper presents a methodology for generating high order shape functions for the complete family of finite elements the geometric entities presented are the point line triangle quadrilateral tetrahedron pyramid prism and hexahedron the shape functions constructed are hierarchical and generate continuous approximation spaces the order of interpolation of the shape function can be determined independently for each edge face and volume the space of interpolation of order p for each element is complete this definition allows for the construction of hybrid meshes with any combination of these elements including elements of distinct dimension and or interpolation order the reader will observe that our systematic way of constructing shape functions can be extended to elements of higher dimension

a new procedure for the construction of hierarchical high order hdiv and hcurl finite element spaces this paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for hdiv and hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains the principle is to choose an appropriate set of vectors based on the geometry of each element which are multiplied by an available set of h 1 hierarchical scalar basic functions this strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces properties that characterise functions in hdiv or hcurl respectively we also present a numerical study to evaluate the correct balancedness of the resulting hdiv spaces of degree k and l 2 spaces of degree k 1 on the resolution of the mixed formulation for a steklov eigenvalue problem

perfect constant weight codes in his pioneering work from 1973 delsarte conjectured that there are no nontrivial perfect codes in the johnson scheme many attempts were made during the years which followed to prove delsarte s conjecture but only partial results have been obtained we survey all these attempts and prove some new results having the same flavor we also present a new method taking a different approach which we hope can lead to the settling of this conjecture we show how this new method rules out sets of parameters as well as specific given parameters

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

multiscale hybrid mixed method this work presents a priori and a posteriori error analyses of a new multiscale hybrid mixed method mhm for an elliptic model specially designed to incorporate multiple scales into the construction of basis functions this finite element method relaxes the continuity of the primal variable through the action of lagrange multipliers while assuring the strong continuity of the normal component of the flux dual variable as a result the dual variable which stems from a simple postprocessing of the primal variable preserves local conservation we prove existence and uniqueness of a solution for the mhm method as well as optimal convergence estimates of any order in the natural norms also we propose a face residual a posteriori error estimator and prove that it controls the error of both variables in the natural norms several numerical tests assess the theoretical results

a generalized multiscale finite element method for the brinkman equation in this paper we consider the numerical upscaling of the brinkman equation in the presence of high contrast permeability fields we develop and analyze a robust and efficient generalized multiscale finite element method gmsfem for the brinkman model in two dimensions in the fine grid we use mixed finite element method with the velocity and pressure being continuous piecewise quadratic and piecewise constant finite element spaces respectively using the gmsfem framework we construct suitable coarse scale spaces for the velocity and pressure that yield a robust mixed gmsfem we develop a novel approach to construct a coarse approximation for the velocity snapshot space and a robust small offline space for the velocity space the stability of the mixed gmsfem and a priori error estimates are derived a variety of two dimensional numerical examples are presented to illustrate the effectiveness of the algorithm

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

analysis of the multiscale finite element method for nonlinear and random homogenization problems in this paper we study the convergence of the multiscale finite element method for nonlinear and random homogenization problems error estimates similar to those for linear homogenization problems are obtained here

calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media we present an analytical and numerical investigation of the finite difference computation of the equivalent conductivity of heterogeneous porous media the customary harmonic scheme to evaluate finite difference internodal transmissivities produces a systematic bias in the numerical results unless an extremely fine grid is used in order to quantify such effects we have developed an analytical approach in the form of a series expansion of the equivalent numerical conductivity in powers of the conductivity variance this leads to an expression of the numerical answer as a function of the grid block size the calculation confirms the existence of a strong bias and of a very slow convergence we propose a simple method to correct it which is well suited for upscaling numerical experiments performed with more contrasting heterogeneous media show similar results this allows the use of coarser gridding and consequently an appreciable speedup in the numerical simulation approach

residual driven online generalized multiscale finite element methods the construction of local reduced order models via multiscale basis functions has been an area of active research in this paper we propose online multiscale basis functions which are constructed using the offline space and the current residual online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators we derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error this error decrease is independent of physical parameters such as the contrast and multiple scales in the problem the offline spaces are constructed using generalized multiscale finite element methods gmsfem we show that if one chooses a sufficient number of offline basis functions one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast we note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects using the residual information we can incorporate the distant information provided the offline approximation satisfies certain properties in the paper theoretical and numerical results are presented our numerical results show that if the offline space is sufficiently large in terms of the dimension such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues then the error reduction by adding online multiscale basis function is independent of the contrast we discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces

isogeometric analysis in electromagnetics b splines approximation abstract we introduce a new discretization scheme for maxwell equations in two space dimension inspired by the new paradigm of isogeometric analysis introduced in hughes et al 2005 16 we propose an algorithm based on the use of bivariate b splines spaces suitably adapted to electromagnetics we construct b splines spaces of variable interelement regularity on the parametric domain these spaces and their push forwards on the physical domain form a de rham diagram and we use them to solve the maxwell source and eigen problem our scheme has the following features i is adapted to treat complex geometries ii is spectral correct iii provides regular e g globally c 0 discrete solutions of maxwell equations

the scaled boundary finite element method alias consistent infinitesimal finite element cell method for diffusion note 264 reference lch article 1999 008doi 10 1002 sici 1097 0207 19990810 45 10 3 0 co 2 eview record in web of science record created on 2007 04 24 modified on 2016 08 08

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

a family of multiscale hybrid mixed finite element methods for the darcy equation with rough coefficients abstract we aim at proposing novel stable finite element methods for the mixed darcy equation with heterogeneous coefficients within a space splitting framework we start from the primal hybrid formulation of the elliptic model for the pressure localization of this infinite dimensional problem leads to element level boundary value problems which embed multiscale and high contrast features in a natural way with neumann boundary conditions driven by the lagrange multipliers such a procedure leads to methods involving the space of piecewise constants for the pressure together with a discretization of the fluxes choosing arbitrarily polynomial interpolations the lowest order raviart thomas element as well as some recent multiscale methods are recovered in addition the methods assure local mass conservation and can be interpreted as stabilized primal hybrid methods extensive numerical validation attests to the accuracy of the new methods on academic and more realistic problems with rough coefficients

a remark concerning divergence accuracy order for h div conforming finite element flux approximations abstract the construction of finite element approximations in h div usually requires the piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region it is known that degradation may occur in convergence order if non affine geometric mappings are used on this point we revisit a general procedure for the improvement of two dimensional flux approximations discussed in a recent paper of this journal comput math appl 74 2017 3283 3295 the starting point is an approximation scheme which is known to provide l 2 errors with accuracy of order k 1 for sufficiently smooth flux functions and of order r 1 for flux divergence an example is r t k spaces on quadrilateral meshes where r k or k 1 if linear or bilinear geometric isomorphisms are applied furthermore the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions the goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy r n 1 as desired for any n 1 the enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level k n while keeping the original border fluxes at level k the case n 1 has been discussed in the mentioned publication for two particular examples general stronger enrichment n 1 shall be analyzed and applied to darcy s flow simulations the global condensed systems to be solved having same dimension and structure of the original scheme

normalized coprime factorizations and balancing for unstable nonlinear systems in this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view in order to do so we introduce the notion of inner and co inner nonlinear systems secondly we deal with balancing for unstable nonlinear systems for linear systems there are several ways to approach this problem and we generalize the ideas of balancing the normalized coprime factorization and of lqg balancing to the nonlinear case lqg balancing can be seen as measuring the influence of a state component on a certain future and past energy function we extend this interpretation to the nonlinear case furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system we give the relation between these two methods of balancing which is similar to the relation in the linear case

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

multiscale stabilization for convection diffusion equations with heterogeneous velocity and diffusion coefficients abstract we present a new stabilization technique for multiscale convection diffusion problems stabilization for these problems has been a challenging task especially for the case with high peclet numbers our method is based on a constraint energy minimization idea and the discontinuous petrov galerkin formulation in particular the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions and are related to the trial space the resulting test functions have a localization property and can therefore be computed locally we will prove the stability and present several numerical results our numerical results confirm that our test space gives a good stability in the sense that the solution error is close to the best approximation error

the method of subspace corrections abstract this paper gives an overview for the method of subspace corrections the method is first motivated by a discussion on the local behavior of high frequency components in a solution to an elliptic problem a simple domain decomposition method is discussed as an illustrative example and multigrid methods are discussed in more detail brief discussions are also given to some non linear examples including eigenvalue problems obstacle problems and liquid crystal modelings the relationship between the method of subspace correction and the method of alternating projects is observed and discussed

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

a symmetric nodal conservative finite element method for the darcy equation this work introduces and analyzes novel stable petrov galerkin enriched methods pgem for the darcy problem based on the simplest but unstable continuous mathbb p 1 mathbb p 0 pair stability is recovered inside a petrov galerkin framework where elementwise dependent residual functions named multiscale functions enrich both velocity and pressure trial spaces unlike the velocity test space that is augmented with bubble like functions multiscale functions correct edge residuals as well the multiscale functions turn out to be the well known lowest order raviart thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure the enrichment strategy suggests the way to recover the local mass conservation property for nodal based interpolation spaces we prove that the method and its symmetric version are well posed and achieve optimal error estimates in natural norms numerical validations confirm claimed theoretical results