a multiscale flux basis for mortar mixed discretizations of stokes darcy flows abstract a multiscale flux basis algorithm is developed for the stokes darcy flow problem the method is based on a non overlapping domain decomposition algorithm where the global problem is reduced to a coarse scale mortar interface problem that is solved by an iterative solver subdomain solves are required at each interface iteration so the cost for the method without a multiscale basis can be high when the number of subdomains or the condition number of the interface problem is large the proposed algorithm involves precomputing a multiscale flux basis which consists of the flux or velocity trace response from each mortar degree of freedom it is computed by each subdomain independently before the interface iteration begins the subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis this may lead to a significant reduction in computational cost since the number of subdomain solves is fixed depending only on the number of mortar degrees of freedom associated with a subdomain several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation for large scale stokes darcy problems

stable cell centered finite volume discretization for biot equations in this paper we discuss a new discretization for the biot equations the discretization treats the coupled system of deformation and flow directly as opposed to combining discretizations for the two separate subproblems the coupled discretization has the following key properties the combination of which is novel 1 the variables for the pressure and displacement are co located and are as sparse as possible e g one displacement vector and one scalar pressure per cell center 2 with locally computable restrictions on grid types the discretization is stable with respect to the limits of incompressible fluid and small time steps 3 no artificial stabilization term has been introduced furthermore due to the finite volume structure embedded in the discretization explicit local expressions for both momentum balancing forces and mass conservative fluid fluxes are available we prove stability of the proposed method with respect to all relevant limits together with consistency this proves conve

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

a global jacobian method for mortar discretizations of nonlinear porous media flows we describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method there are two main ideas 1 linearize the global system in both subdomain and interface variables simultaneously to yield a single newton iteration and 2 algebraically eliminate subdomain velocities and optionally subdomain pressures to solve linear systems for the 1st or the 2nd schur complements solving the 1st schur complement system gives the multiscale solution without the need to solve an interface iteration solving the 2nd schur complement system gives a linear interface problem for a nonlinear model the methods are less complex than a previously developed nonlinear mortar algorithm which requires two nested newton iterations and a forward difference approximation furthermore efficient linear preconditioners can be applied to speed up the iteration the methods are implemented in parallel and a numerical study is performed

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

ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

regularity conditions characterizing fenchel lagrange duality and farkas type results in dc infinite programming abstract in this paper we consider a dc infinite programming problem p with inequality constraints by using the properties of the epigraph of the conjugate functions we introduce some new notions of regularity conditions for p under these new regularity conditions we completely characterize the fenchel lagrange duality and the stable fenchel lagrange duality for p similarly we also completely characterize the farkas type results and the stable farkas type results for p as applications we obtain the corresponding results for conic programming problems

strong duality and kkt conditions in nonconvex optimization with a single equality constraint and geometric constraint some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established in particular a hidden convexity of the conic hull of joint range of the pair of functions associated to the original problem is obtained applications to derive a characterization of the validity of kkt conditions without standard constraints qualification are also discussed it goes beyond the exact penalization technique several examples showing our results provide much more information than those appearing elsewhere are given finally the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail

a weaker regularity condition for subdifferential calculus and fenchel duality in infinite dimensional spaces abstract in this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping this condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior point regularity conditions given so far in the literature moreover it provides a weak sufficient condition for fenchel duality regarding convex optimization problems in infinite dimensional spaces as an application we discuss the strong conical hull intersection property chip for a finite family of closed convex sets

asymptotic closure condition and fenchel duality for dc optimization problems in locally convex spaces a b s t r a c t we consider the dc optimization problem p inf x f1 x f2 x g1 ax g2 ax where f1 f2 g1 and g2 are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear continuous operator from x to y adopting the standard convexification technique a fenchel dual problem of p is given by using properties of the epigraph of conjugate functions some sufficient and necessary conditions for the fenchel duality and for the stable fenchel duality of p are provided

implementation of a mortar mixed finite element method using a multiscale flux basis this paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems the algorithm uses non overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem which is then solved using an iterative method the original implementation by arbogast pencheva wheeler and yotov multiscale model simul 2007 required solving one local dirichlet problem on each subdomain per interface iteration we alter this implementation by forming a multiscale flux basis this basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom on each subdomain independently the computation of these basis functions requires solving a fixed number of dirichlet subdomain problems taking linear combinations of the multiscale flux basis functions replaces the need to solve any dirichlet subdomain problems during the interface iteration this new implementation yields the same solution as the original implementation and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain the gain in computational efficiency increases with the number of subdomains

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

new regularity conditions for strong and total fenchel lagrange duality in infinite dimensional spaces abstract we give new regularity conditions for convex optimization problems in separated locally convex spaces we completely characterize the stable strong and strong fenchel lagrange duality then we give similar statements for the case when a solution of the primal problem is assumed as known obtaining complete characterizations for the so called total and stable totalfenchel lagrange duality respectively for particular settings the conditions that we consider turn into some constraint qualifications already used by different authors like farkas minkowski cq locally farkas minkowski cq and basic cq and we rediscover and improve some recent results from the literature

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

variational multiscale enrichment method with mixed boundary conditions for modeling diffusion and deformation problems abstract this manuscript presents the formulation and implementation of the variational multiscale enrichment vme method using canopy shaped microscale enrichment functions obtained through the use of a new family of microscale boundary conditions the purpose of the new enrichment functions and the new boundary condition is to relax the overconstraint imposed by the homogeneous microscale boundary condition e g residual free bubbles commonly employed in the variational multiscale literature the formulation and implementation of the method are presented for diffusion and elasticity problems the performance of the proposed method is assessed by comparing with direct numerical simulations on diffusion and deformation problems a boundary parameter identification approach is proposed to obtain near optimal boundary conditions the identification approach is verified in the context of the deformation response of particle reinforced composites

a spectral multiscale hybridizable discontinuous galerkin method for second order elliptic problems abstract we design a multiscale model reduction framework within the hybridizable discontinuous galerkin finite element method our approach uses local snapshot spaces and local spectral decomposition following the concept of generalized multiscale finite element methods we propose several multiscale finite element spaces on the coarse edges that provide a reduced dimensional approximation for numerical traces within the hdg framework we provide a general framework for systematic construction of multiscale trace spaces using local snapshots we avoid high dimensional representation of trace spaces and use some local features of the solution space in constructing a low dimensional trace space we investigate the solvability and numerically study the performance of the proposed method on a representative number of numerical examples

mortar multiscale finite element methods for stokes darcy flows we investigate mortar multiscale numerical methods for coupled stokes and darcy flows with the beavers joseph saffman interface condition the domain is decomposed into a series of subdomains coarse grid of either stokes or darcy type the subdomains are discretized by appropriate stokes or darcy finite elements the solution is resolved locally in each coarse element on a fine scale allowing for non matching grids across subdomain interfaces coarse scale mortar finite elements are introduced on the interfaces to approximate the normal stress and impose weakly continuity of the normal velocity stability and a priori error estimates in terms of the fine subdomain scale h h and the coarse mortar scale h h are established for fairly general grid configurations assuming that the mortar space satisfies a certain inf sup condition several examples of such spaces in two and three dimensions are given numerical experiments are presented in confirmation of the theory

adaptive generalized multiscale finite element methods for h curl elliptic problems with heterogeneous coefficients abstract in this paper we construct an adaptive multiscale method for solving h curl elliptic problems in highly heterogeneous media our method is based on the generalized multiscale finite element method we will first construct a suitable snapshot space and a dimensional reduction procedure to identify important modes of the solution we next develop and analyze an a posteriori error indicator and the corresponding adaptive algorithm in addition we will construct a coupled offline online adaptive algorithm which provides an adaptive strategy to the selection of offline and online basis functions our theory shows that the convergence is robust with respect to the heterogeneities and contrast of the media we present several numerical results to illustrate the performance of our method

hp adaptive composite discontinuous galerkin methods for elliptic eigenvalue problems on complicated domains in this paper we develop the a posteriori error estimation of hp adaptive discontinuous galerkin composite finite element methods dgfems for the discretization of second order elliptic eigenvalue problems dgfems allow for the approximation of problems posed on computational domains which may contain local geometric features the dimension of the composite finite element space is independent of the number of geometric features this is in contrast with standard finite element methods as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space computable upper bounds on the error for both eigenvalues and eigenfunctions are derived numerical experiments highlighting the practical application of the proposed estimators within an automatic hp adaptive refinement procedure will be presented

global local nonlinear model reduction for flows in heterogeneous porous media abstract in this paper we combine discrete empirical interpolation techniques global mode decomposition methods and local multiscale methods such as the generalized multiscale finite element method gmsfem to reduce the computational complexity associated with nonlinear flows in highly heterogeneous porous media to solve the nonlinear governing equations we employ the gmsfem to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid computing the gmsfem solution involves calculating the residual and the jacobian on a fine grid as such we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid the resulting reduced order approach significantly reduces the flow problem size while accurately capturing the behavior of fully resolved solutions we consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high contrast porous media

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

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

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

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

analytical effective coefficient and a first order approximation for linear flow through block permeability inclusions we present a closed form solution for the upscaled diffusion coefficient and derive a first order homogenized approximation to linear flow equations with periodic and rapidly oscillating coefficients the coefficients are defined as step functions describing inclusions of various shapes in a main matrix this constitutes the n dimensional upscaled version of darcy s law for linear flow in such systems we consider the two scale asymptotic expansion of the solution of the flow equation and develop a corrector to an analytical approximation for the solution of the periodic cell problem we demonstrate that the proposed analytical form for the effective coefficient satisfies the generalized voigt reiss inequality and is in agreement with other known theoretical results including the geometric average for the checkerboard geometry and with some published numerical results the zeroth order approximation in h 1 w is readily obtained and the first order approximation in l 2 w is derived from the proposed analytical approximation to the basis functions the analytical basis functions are also used to define a correction function that incorporates the heterogeneous features into the zeroth order approximation to the gradient and flux which considerably improves the convergence results we illustrate the procedure with coefficients describing square inclusions with contrast ratios between the inclusion and the matrix as 10 1 100 1 1000 1 and 1 10 respectively we demonstrate numerically that the convergence properties of the proposed approximations agree with the classical theoretical results in homogenization theory

forecasting methods using fuzzy concepts in this paper fuzzy concepts are applied in forecasting product price and sales in the semiconductor industry which is often conceived as a highly dynamic environment first two fuzzy forecasting methods including fuzzy interpolation fi and fuzzy linear regression flr are developed and discussed forecasts generated by these methods are fuzzy valued next the subjective beliefs about whether the industry is booming or slumping and the speed at which this change in prosperity takes place during a given period are also considered two subjective functions are defined and used to adjust fuzzy forecasts practically fuzzy forecasts are incorporated with fuzzy programming like fuzzy linear programming flp or fuzzy nonlinear programming fnp for mid term or long term planning advantages over traditional methods are shown in our examples

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

upscaling for the laplace problem using a discontinuous galerkin method scientists and engineers generally tackle problems that include multiscale effects and that are thus difficult to solve numerically the main difficulty is to capture both the fine and the coarse scales to get an accurate numerical solution indeed the computations are generally performed by using numerical schemes based on grids but the stability and thus the accuracy of the numerical method depends on the size of the grid which must be refined drastically in the case of very fine scales that implies huge computational costs and in particular the limitations of the memory capacity are often reached it is thus necessary to use numerical methods that are able to capture the fine scale effects with computations on coarse meshes operator based upscaling is one of them and we present a first attempt to adapt that technique to a discontinuous galerkin method dgm we consider the laplace problem as a benchmark and we compare the performance of the resulting numerical scheme with the classical one using lagrange finite elements the comparison involves both an accuracy analysis and a complexity calculus this work shows that there is an interest of combining dgm with upscaling

a locally conservative enriched galerkin approximation and efficient solver for elliptic and parabolic problems we present and analyze an enriched galerkin finite element method eg to solve elliptic and parabolic equations with jump coefficients eg is formulated by enriching the conforming continuous galerkin finite element method cg with piecewise constant functions which can be considered as a penalty stabilization the method is shown to be locally and globally conservative while keeping fewer degrees of freedom in comparison with discontinuous galerkin finite element methods dg moreover we present and analyze a fast effective eg solver whose cost is roughly that of cg and which can handle an arbitrary order of approximations a number of numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of eg when coupled with transport

multiscale direction splitting algorithms for parabolic equations with highly heterogeneous coefficients in this paper we discuss two methods for upscaling of highly heterogeneous data for parabolic problems in the context of a direction splitting time approximation the first method is a direct application of the idea of jenny et al 2003 in the context of the direction splitting approach the second method devises the approximation from the schur complement corresponding to the interface unknowns of the coarse grid by applying a proper l 2 projection operator to it the spatial discretization employed in this paper is based on a mac finite volume stencil but the same approach can be implemented within a proper finite element discretization a key feature of the present approach is that it can extend to 3d problems with very little computational overhead the properties of the resulting approximations are demonstrated numerically on some benchmark coefficient data available in the literature

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

constraint qualifications for extended farkas s lemmas and lagrangian dualities in convex infinite programming for an inequality system defined by a possibly infinite family of proper functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions under the new constraint qualifications we obtain characterizations of those reverse convex inequalities which are a consequence of the constrained system and we provide necessary and or sufficient conditions for a stable farkas lemma to hold similarly we provide characterizations for constrained minimization problems to have the strong or strong stable lagrangian dualities several known results in the conic programming problem are extended and improved

a two grid discretization method for decoupling systems of partial differential equations in this paper we propose a two grid finite element method for solving coupled partial differential equations e g the schrodinger type equation with this method the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid it is shown both theoretically and numerically that the resulting solution still achieves asymptotically optimal accuracy

fuzzy weighted average an improved algorithm abstract this paper suggests a modification on the fuzzy weighted average fwa method developed by dong and wong 1987 a steepest descend ascend method is proposed to facilitate steps 3 and 4 of the fwa to find the minimum maximum of f x1 x2 xn w1 w2 wn i 1n wixi i 1n wi an example used by dong and wong is again employed to illustrate the new method the same results are obtained but the new method requires much less evaluations and computations the advantages and disadvantages of the two algorithms are discussed also several relevant theorems are stated and discussed in this paper and the proofs of these theorems are listed in the appendix

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

on strong and total lagrange duality for convex optimization problems abstract we give some necessary and sufficient conditions which completely characterize the strong and total lagrange duality respectively for convex optimization problems in separated locally convex spaces we also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals in the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so called farkas minkowski and locally farkas minkowski conditions for systems of convex inequalities recently used in the literature moreover we show that our new results extend some existing ones in the literature

cluster analysis based on fuzzy equivalence relation in this paper a cluster analysis method based on fuzzy equivalence relation is proposed at first the distance formula between two trapezoidal fuzzy numbers is used to aggregate subjects linguistic assessments about attributes ratings to obtain the compatibility relation then a fuzzy equivalence relation based on the fuzzy compatibility relation can be constructed finally using a cluster validity index to determine the best number of clusters and taking suitable cut value the clustering analysis can be effectively implemented by utilizing this clustering analysis the subjects fuzzy assessments with various rating attitudes can be taken into account in the aggregation process to assure more convincing and accurate cluster analysis

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

multiscale stabilization for convection dominated diffusion in heterogeneous media abstract we develop a petrov galerkin stabilization method for multiscale convection diffusion transport systems existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion which may not be sufficient to stabilize multiscale systems we seek a local reduced order model for this kind of multiscale transport problems and thus develop a systematic approach for finding reduced order approximations of the solution we start from a petrov galerkin framework using optimal weighting functions we introduce an auxiliary variable to a mixed formulation of the problem the auxiliary variable stands for the optimal weighting function the problem reduces to finding a test space a dimensionally reduced space for this auxiliary variable which guarantees that the error in the primal variable representing the solution is close to the projection error of the full solution on the dimensionally reduced space that approximates the solution to find the test space we reformulate some recent mixed generalized multiscale finite element methods we introduce snapshots and local spectral problems that appropriately define local weight and trial spaces in particular we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable the resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system we discuss the stability and its relation to the approximation property of the test space we design online basis functions which accelerate convergence in the test space and consequently improve stability we present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the peclet number

residual driven online mortar mixed finite element methods and applications abstract in this paper we develop an online basis enrichment method with the mortar mixed finite element method using the oversampling technique to solve for flow problems in highly heterogeneous media we first compute a coarse grid solution with a certain number of offline basis functions per edge which are chosen as standard polynomials basis functions we then iteratively enrich the multiscale solution space with online multiscale basis functions computed by using residuals the iterative solution converges to the fine scale solution rapidly we also propose an oversampling online method to achieve faster convergence speed the oversampling refers to using larger local regions in computing the online multiscale basis functions we present extensive numerical experiments including both 2d and 3d to demonstrate the performance of our methods for both steady state flow and two phase flow and transport problems in particular for the time dependent two phase flow and transport problems we apply the online method to the initial model without updating basis along the time evolution our numerical results demonstrate that by using a few number of online basis functions one can achieve a fast convergence

a stochastic mortar mixed finite element method for flow in porous media with multiple rock types this paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types the governing equations are based on darcy s law with nonstationary stochastic permeability represented as a sum of local karhunen loeve expansions the approximation uses stochastic collocation on either a tensor product or a sparse grid coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method the latter method requires solving a coarse scale mortar interface problem via an iterative procedure the traditional implementation requires the solution of local fine scale linear systems on each iteration we employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration in the stochastic setting the basis is further reused over multiple realizations leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude error analysis and numerical experiments are presented

reduced order variational multiscale enrichment method for elasto viscoplastic problems abstract this manuscript presents the formulation and implementation of a novel reduced order variational multiscale enrichment rovme method for elasto viscoplastic problems this method provides a hierarchical model order reduction technique based on the eigenstrain concept to approximate the fine scale response resolved at subdomains of interest by eliminating the requirement of direct fine scale discretization the computational effort associated with the variational multiscale enrichment vme method is significantly reduced the model order reduction is achieved in the scale coupled inelastic problem by automatically satisfying the microscale equilibrium state through the eigenstrain concept and coarse discretization of inelastic strain fields within the microscale domain the inelastic material behavior is idealized with coupled perzyna type viscoplasticity and flow stress evolution based on the johnson cook model numerical verifications are performed to assess the capabilities of the proposed methodology against the direct vme method with detailed fine scale resolutions the verification results demonstrate high accuracy and computational efficiency of the reduced order vme framework for elasto viscoplastic problems with material heterogeneity

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

flux formulation of parabolic equations with highly heterogeneous coefficients abstract in this paper we study the flux formulation of unsteady diffusion equations with highly heterogeneous permeability coefficients and their discretization in the proposed approach first an equation governing the flux of the unknown scalar quantity is solved and then the scalar is recovered from its flux the problem for the flux is further discretized by splitting schemes that yield locally one dimensional problems and therefore the resulting linear systems are tridiagonal if the spatial discretization uses cartesian grids a first and a formally second order time discretization splitting scheme have been implemented in both two and three dimensions and we present results for a few model problems using a challenging benchmark dataset

locally conservative stabilized finite element methods for variably saturated flow standard galerkin finite element methods for variably saturated groundwater flow have several deficiencies for instance local oscillations can appear around sharp infiltration fronts without the use of mass lumping and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries here we consider conforming finite element discretizations based on a multiscale formulation along with recently developed local postprocessing schemes the resulting approach maintains the basic flexibility and appeal of traditional finite element methods while controlling nonphysical oscillations and producing element wise mass conservative velocity fields accuracy and efficiency of the proposed schemes are evaluated through a series of steady state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains the schemes are formulated for a generic nonlinear advection diffusion equation and are thus applicable to many other flow models

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

generalized multiscale finite element method symmetric interior penalty coupling motivated by applications to numerical simulations of flows in highly heterogeneous porous media we develop multiscale finite element methods for second order elliptic equations we discuss a multiscale model reduction technique in the framework of the discontinuous galerkin finite element method we propose two different finite element spaces on the coarse mesh the first space is based on a local eigenvalue problem that uses an interior weighted l 2 norm and a boundary weighted l 2 norm for computing the mass matrix the second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension the approximation with these multiscale spaces is based on the discontinuous galerkin finite element method framework we investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples

constraint qualifications for optimality conditions and total lagrange dualities in convex infinite programming abstract for an inequality system defined by an infinite family of proper convex functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications under the new constraint qualifications we provide necessary and or sufficient conditions for the kkt rules to hold similarly we provide characterizations for constrained minimization problems to have total lagrangian dualities several known results in the conic programming problem are extended and improved

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

characterizations of e duality gap statements for composed optimization problems abstract in this paper we present different regularity conditions that equivalently characterize e duality gap statements for optimization problems consisting of minimizing the sum of a function with the precomposition of a cone increasing function to a vector function these regularity conditions are formulated by using epigraphs and e subdifferentials taking e 0 one can rediscover recent results on stable strong and total duality and zero duality gap from the literature moreover we deliver as byproducts e optimality conditions and e saddle point statements for the aforementioned kind of problems and e farkas statements involving the sum of a function with the precomposition of a cone increasing function to a vector function

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

stable and total fenchel duality for convex optimization problems in locally convex spaces we consider the optimization problem p a inf x in x f x g ax where f and g are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear operator from x to y by using the properties of the epigraph of the conjugated functions some sufficient and necessary conditions for the strong fenchel duality and the strong converse fenchel duality of p a are provided sufficient and necessary conditions for the stable fenchel duality and for the total fenchel duality are also derived

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

on a numerical subgrid upscaling algorithm for stokes brinkman equations this paper discusses a numerical subgrid resolution approach for solving the stokes brinkman system of equations which is describing coupled flow in plain and in highly porous media various scientific and industrial problems are described by this system and often the geometry and or the permeability vary on several scales a particular target is the process of oil filtration in many complicated filters the filter medium or the filter element geometry are too fine to be resolved by a feasible computational grid the subgrid approach presented in this paper is aimed at describing how these fine details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid this is done via a systematic and careful procedure of modifying and updating the coefficients of the stokes brinkman system in chosen cells this numerical subgrid approach is motivated from one side from homogenization theory from which we borrow the formulations for the so called cell problem and from the other side from the numerical upscaling approaches such as multiscale finite volume multiscale finite element etc results on the algorithm s efficiency both in terms of computational time and memory usage are presented comparison of the full fine grid solution when possible of the stokes brinkman system with the subgrid solution of the upscaled stokes brinkman system including effective permeabilities for the quasi porous cells are presented in order to evaluate the accuracy and the efficiency advantages and limitations of the considered subgrid approach are discussed

on the heterogeneous multiscale method with various macroscopic solvers abstract the heterogeneous multiscale method hmm is a general method for efficient numerical solution of problems with multiscales it consists of two components an overall macroscopic solver for macrovariables on a macrogrid and an estimation of the missing macroscopic data from the microscopic model in this paper we present a state of the art review of the hmm with various macroscopic solvers including finite differences finite elements discontinuous galerkin mixed finite elements control volume finite elements nonconforming finite elements and mixed covolumes the first four solvers have been studied in the hmm setting the others are not as example the hmm with the nonconforming finite element macroscopic solver for nonlinear and random homogenization problems is also studied here

necessary and sufficient conditions for some constraint qualifications in quasiconvex programming abstract in this paper we investigate relations between constraint qualifications in quasiconvex programming at first we show a necessary and sufficient condition for the closed cone constraint qualification for quasiconvex programming q cccq and investigate some sufficient conditions for the q cccq also we consider a relation between the q cccq and the basic constraint qualification for quasiconvex programming q bcq and we compare the q bcq with some constraint qualifications

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

an alternative formulation for a new closed cone constraint qualification abstract we give an alternative formulation for the so called closed cone constraint qualification cccq related to a convex optimization problem in banach spaces recently introduced in the literature this new formulation allows to prove in a simple way that cccq is weaker than some generalized interior point constraint qualifications given in the past by means of some insights from the theory of conjugate duality we also show that strong duality still holds under some weaker hypotheses than the ones considered so far in the literature

a two scale solution algorithm for the elastic wave equation operator based upscaling is a two scale algorithm that speeds up the solution of the wave equation by producing a coarse grid solution which incorporates much of the local fine scale solution information we present the first implementation of operator upscaling for the elastic wave equation by using the velocity displacement formulation of the three dimensional elastic wave equation basis functions that are linear in all three directions and applying mass lumping the subgrid solve first stage of the two step algorithm reduces to solving explicit difference equations at the second stage of the algorithm we upscale both velocity and displacement by using local subgrid information to formulate the coarse grid problem the coarse grid system matrix is independent of time sparse and banded this paper explores both serial and parallel implementations of the method the main simplifying assumption of the method special zero boundary conditions imposed on coarse blocks in the first stage of the algorithm leads to an easily parallelizable algorithm because very little communication is required between processors in fact for this upscaling implementation calculation of the load vector for the coarse solve dominates the cost of a time step we show that for a homogeneous medium convergence is second order in space and time so long as both the coarse and fine grids are simultaneously refined a series of heterogeneous medium numerical experiments demonstrate that the upscaled solution captures the fine scale fluctuations in the input parameters accurately most notable for use in a seismic inversion algorithm the upscaling algorithm accurately locates the depth of reflectors interface changes

necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities abstract in this paper we present constraint qualifications which completely characterize the farkas minkowski and the locally farkas minkowski convex possibly infinite inequality systems posed in topological vector spaces the number of constraints and the dimension of the linear space are arbitrary possibly infinite the constraint qualifications considered in this paper are expressed in terms of the solvability of certain parametric convex linear systems and the uniform strong duality or the uniform min max duality relative to the lagrange haar dual problems of suitable convex linear parametric optimization problems

convergence analysis for the numerical boundary corrector for elliptic equations with rapidly oscillating coefficients we develop the convergence analysis of a numerical scheme for approximating the solution of the elliptic problem l epsilon u epsilon frac partial partial x i a ij x epsilon frac partial partial x j u epsilon f mbox in omega u epsilon 0 mbox on partial omega where a y a ij y is a periodic symmetric positive definite matrix and omega 0 1 2 the major goal of the numerical scheme is to capture the epsilon scale of the oscillations of the solution u epsilon on a mesh size h epsilon mbox or h epsilon the numerical scheme is based on asymptotic expansions constructive boundary corrector and finite element approximations new a priori error estimates are established for the asymptotic expansions and for the constructive boundary correctors under weak assumptions on the regularity of the problem these estimates permit to establish sharp finite element error estimates and to consider composite materials applications depending on the regularity of the problem we establish for the numerical scheme a priori error estimates of o h 2 epsilon 3 2 epsilon h on the l 2 norm and o h epsilon 1 hat delta for the broken h 1 norm where hat delta in frac 1 4 0

a stochastic variational multiscale method for diffusion in heterogeneous random media a stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media the exact solution of the governing equations is split into two components a coarse scale solution that can be captured on a coarse mesh and a subgrid solution a localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse scale solution the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model the generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models finally various numerical examples are considered for evaluating the method

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

on constraint qualification for an infinite system of convex inequalities in a banach space for a general infinite system of convex inequalities in a banach space we study the basic constraint qualification and its relationship with other fundamental concepts including various versions of conditions of slater type the mangasarian fromovitz constraint qualification as well as the pshenichnyi levin valadier property introduced by li nahak and singer applications are given in the restricted range approximation problem constrained optimization problems as well as in the approximation problem with constraints by conditionally positive semidefinite functions

subjective assessment of mental workload a fuzzy linguistic multi criteria approach abstract in this paper a fuzzy linguistic multi criteria method for subjective assessment of mental workload is proposed the criteria ratings and the corresponding importance weights are assessed in linguistic terms which are described by fuzzy numbers with triangular and trapezoidal membership function a fuzzy weighted average algorithm based on the extension principle is employed to aggregate these fuzzy numbers the aggregated fuzzy number is then translated back to linguistics and the overall mental workload is expressed in linguistic terms

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

decision support for fuzzy comprehensive evaluation of urban development urban development often impacts the economic development pattern and the spacial distribution of resource allocation of a particular area urban development can be conceptualized as a multicriteria decision making problem therefore multicriteria evaluation has been used to trace the trend of urban development in this paper a fuzzy multicriteria evaluation model is developed to study the trend of urban development first a multilayer evaluation index system is presented it is used to conduct horizontal analysis on urban systems second a fuzzy multicriteria multilayer evaluation model is proposed third the detailed modeling aspects are discussed which include the fuzzy evaluation models under different evaluation rules the methods used to determine the weights of criteria and the determination of the membership grades of evaluation matrix the model has been implemented and the application results indicate that the proposed model is able to represent those important factors affecting urban development layer by layer and stage by stage

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

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