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

a multi grid technique for coupling fluid flow with porous media flow abstract in this paper we consider the coupling of fluid flow with porous media flow a multi grid finite element method for the coupled stokes darcy problem with the beavers joseph interface condition is proposed and discussed the optimal error estimates are obtained numerical experiment is given to verify the theoretical analysis and indicate the accuracy and efficiency of the multi grid method

the efficient rotational pressure correction schemes for the coupling stokes darcy problem abstract in this paper the rotational pressure correction methods for the stokes darcy system are developed and analyzed the central advantage of these methods is a time dependent version of domain decomposition these methods have first order second order accuracy without the incompressibility constraint of the stokes darcy system their main feature is the implementation efficiency in that we only solve one vector valued elliptic equation and one scalar valued poisson equation for the stokes equations per time step the unconditional stability and long time stability are established and numerical experiments are also presented to show their performance

finite element approximations for stokes darcy flow with beavers joseph interface conditions numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits such situations arise for example for groundwater flows in karst aquifers the coupled flow is modeled by the darcy equation in a porous medium and the stokes equations in the conduit domain on the interface between the matrix and conduit beavers joseph interface conditions instead of the simplified beavers joseph saffman conditions are imposed convergence and error estimates for finite element approximations are obtained numerical experiments illustrate the validity of the theoretical results

coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

spectral methods based on new formulations for coupled stokes and darcy equations in this paper we consider the numerical solution of the stokes and darcy coupled equations which frequently appears in porous media modeling the main contribution of this work is as follows first we introduce a new formulation for the stokes darcy coupled equations subject respectively to the beavers joseph saffman interface condition and an alternative matching interface condition secondly we prove the well posedness of these weak problems by using the classical saddle point theory thirdly some spectral approximations to the weak problems are proposed and analyzed and some error estimates are provided it is found that the new formulations significantly simplify the error analysis and numerical implementation finally some two dimensional spectral and spectral element numerical examples are provided to demonstrate the efficiency of our methods

decision trees old and new results abstract in this paper we prove two general lower bounds for algebraic decision trees which test membership in a set s r n which is defined by linear inequalities let rank s be the maximal dimension of a linear sub space contained in the closure of s in euclidean topology first we show that any decision tree for s which uses products of linear functions we call such functions mlf functions must have depth at least n rank s this solves an open question raised by a c yao and can be used to show that mlf functions are not really more powerful than simple comparisons between the input variables when computing the largest k out of n elements yao proved this result in the special case when products of at most two linear functions are allowed our proof also shows that any decision tree for this problem must have exponential size using the same methods we can give an alternative proof of rabin s theorem namely that the depth of any decision tree for s using arbitrary analytic functions is at least n rank s

least squares approach for the time dependent nonlinear stokes darcy flow abstract a time dependent nonlinear model of stokes darcy flows is considered for numerical approximations by an optimization based domain decomposition method the coupled system is formulated as a constrained optimal control problem where a neumann type control is introduced to enforce boundary conditions on the interface of subdomains a computational algorithm is discussed for the least squares functional which measures the jump of flow balance across the interface numerical results are presented to validate convergence and accuracy of the method

a crank nicolson discontinuous finite volume element method for a coupled non stationary stokes darcy problem abstract in this paper we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a crank nicolson discretization in time and we obtain the solution at the first time step by applying a first order backward euler method to solve the coupling of time dependent stokes darcy problem the proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head whereas the pressure is approximated by the piecewise constant elements then the unique solvability of the approximate solution for the discrete problem is derived in addition the optimal error estimates of the full discretization in standard l 2 norm and broken h 1 norm are obtained finally a series of numerical experiments are provided to illustrate the features of the proposed method such as the optimal accuracy orders adaptive mesh refinement mass conservation capability to conveniently deal with complicated geometries with different types of boundary conditions and the applicability to the problems with realistic parameters in particular the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain it encompasses both fractures with high permeability which act as the prior flow conduit and fractures with low permeability as a flow barrier

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

a domain decomposition method for the steady state navier stokes darcy model with beavers joseph interface condition this paper proposes and analyzes a robin type multiphysics domain decomposition method ddm for the steady state navier stokes darcy model with three interface conditions in addition to the two regular interface conditions for the mass conservation and the force balance the beavers joseph condition is used as the interface condition in the tangential direction the major mathematical difficulty in adopting the beavers joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system y cao et al comm math sci 8 2010 pp 1 25 y cao et al siam j numer anal 47 2010 pp 4239 4256 in this paper the well posedness of the navier stokes darcy model with beavers joseph condition is analyzed by using a branch of nonsingular solutions by following the idea in y cao et al numer math 117 2011 pp 601 629 the three physical interface conditions are utilized together to construct the robin type boundary conditions on th

existence of a weak solution for the fully coupled navier stokes darcy transport problem abstract this paper analyzes the surface subsurface flow coupled with transport the flow is modeled by the coupling of navier stokes and darcy equations the transport of a species is modeled by a convection dominated parabolic equation the two way coupling between flow and transport is nonlinear and it is done via the velocity field and the viscosity this problem arises from a variety of natural phenomena such as the contamination of the groundwater through rivers the main result is existence and stability bounds of a weak solution

decoupled linear and energy stable finite element method for the cahn hilliard navier stokes darcy phase field model in this paper we consider the numerical approximation for a phase field model of the coupled two phase free flow and two phase porous media flow this model consists of cahn hilliard navier stokes equations in the free flow region and cahn hilliard darcy equations in the porous media region that are coupled by seven interface conditions the coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step a fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system in order to deal with the difficulties arising from the interface conditions the decoupled scheme needs to be constructed appropriately for the interface terms and a modified discrete energy is introduced with an interface component furthermore the scheme is linearized and energy stable hence at each time step one need only solve a linear elliptic system for each of the two decoupled equations stability o

an efficient and long time accurate third order algorithm for the stokes darcy system a third order in time numerical imex type algorithm for the stokes darcy system for flows in fluid saturated karst aquifers is proposed and analyzed a novel third order adams moulton scheme is used for the discretization of the dissipative term whereas a third order explicit adams bashforth scheme is used for the time discretization of the interface term that couples the stokes and darcy components the scheme is efficient in the sense that one needs to solve at each time step decoupled stokes and darcy problems therefore legacy stokes and darcy solvers can be applied in parallel the scheme is also unconditionally stable and with a mild time step restriction long time accurate in the sense that the error is bounded uniformly in time numerical experiments are used to illustrate the theoretical results to the authors knowledge the novel algorithm is the first third order accurate numerical scheme for the stokes darcy system possessing its favorable efficiency stability and accuracy properties

bounds and policies for dynamic routing in loss networks we consider the problem of maximizing a weighted sum of expected rewards in steady state in multiclass loss networks under dynamic routing and admission control with poisson arrivals and exponentially distributed holding times by aggregating the underlying markov decision process we derive linear programming relaxations that contain the achievable performance region under all admissible policies and lead to a series of progressively tighter upper bounds these relaxations allow stronger bounds at the expense of higher computational times we further propose a series of routing and admission control policies from the relaxations that outperform in computational experiments other heuristic policies such as the dynamic alternative routing with trunk reservation and the least busy alternative routing variations of which are used in practice the suboptimality guarantees defined as best bound best policy range from 0 to 2 5 under symmetry conditions symmetric network topology arrival rates link capacities rewards and from 4 to 10 under asymmetry conditions we discuss the qualitative behavior of these policies and obtain insights about their performance

a parallel robin robin domain decomposition method for the stokes darcy system we propose a new parallel robin robin domain decomposition method for the coupled stokes darcy system with beavers joseph saffman jones interface boundary condition in particular we prove that with an appropriate choice of parameters the scheme converges geometrically independent of the mesh size

local and parallel finite element post processing scheme for the stokes problem abstract a local and parallel finite element post processing scheme based on partition of unity method is proposed and analyzed in this paper for the stokes problem firstly a standard galerkin finite element method on a relatively coarse grid is used to obtain the approximation of the lower frequency components secondly the higher frequency components are computed on fine grids by some local and parallel procedure to post process the standard galerkin approximation the motivation of the proposed local and parallel finite element post processing scheme is based on the superposition principle finally to eliminate the effect of the dirichlet boundary conditions which are imposed on the internal artificial boundaries a global coarse grid correction is done to improve the l 2 accuracy of the approximation

routing and capacity allocation in networks with trunk reservation in this paper we extend recent work on routing and capacity allocation in circuit switched networks to include the possible use of trunk reservation trunk reservation is an easily implemented control mechanism which allows priority to be given to chosen traffic streams it has the potential to be an extremely efficient control mechanism but its use on any single link of a network has ramifications which may extend throughout the network and it is important to take this into account using a simplified analytical model of a circuit switched network incorporating trunk reservation we show that there exist implied costs associated with the priority and nonpriority calls routed through a link and shadow prices associated with the capacity for priority and nonpriority traffic through a link mathematically this involves calculating derivatives of an implicitly defined function and expressing these derivatives as simple functions of certain variables these variables the implied costs and shadow prices are determined as solutions to a set of linear equations possessing a local or decentralized character and can be used as a basis for routing and capacity allocation strategies

a parallel domain decomposition method for coupling of surface and groundwater flows abstract in this paper we construct a robust parallel method based on a recently developed non overlapping domain decomposition methodology to accurately model natural coupling of surface and groundwater flows stokes and darcy equations are formulated and solved within the surface and subsurface regions respectively a new type of robin robin boundary condition is proposed on the common boundary for the coupling of those systems the formulation provides great flexibility for multi physics coupling and is suitable for efficient parallel implementation meanwhile it is stable with inherent system parameter variation a numerical example is provided to verify the theory

a modified two grid decoupling method for the mixed navier stokes darcy model in this paper a modified two grid decoupling method for the mixed navier stokes darcy model is proposed and analyzed the solution on interface condition is approximated by the solution obtained on the fine grid stability and convergence of the method are proved the convergence rate for the l 2 norm of u and p is improved from o h 3 2 to o h 2 the numerical results are consistent with the theoretical results and illustrate that the proposed method is quite efficient

on the solution of coupled stokes darcy model with beavers joseph interface condition abstract in this paper we first get the uniqueness result of the possible solution to the steady state coupled stokes darcy model with beavers joseph interface condition for any physical parameters especially for any 0 then we show the existence of solutions for any 0 by using galerkin method furthermore we analyze the error of the corresponding coupled finite element scheme and derive the optimal error estimates

asymptotically optimal loss network control we consider a loss network which employs alternative routing and derive the asymptotically optimal call acceptance and routing policy the limit being as the number of links becomes large we show that a well known control policy least busy alternative routing with trunk reservation is asymptotically optimal the approach of the paper is not specific to the loss network control problem that we consider but is applicable more generally the results are obtained by combining weak convergence with linear programming techniques

stokes darcy boundary integral solutions using preconditioners in a system where a free fluid flow is coupled to flow in a porous medium different pdes are solved simultaneously in two subdomains we consider steady stokes equations in the free region coupled across a fixed interface to darcy equations in the porous substrate recently the numerical solution of this system was obtained using the boundary integral formulation combined with a regularization correction procedure the correction process also results in the improvement of the condition number of the linear system in this paper an appropriate preconditioner based on the singular part of corrections is introduced to improve the convergence of a krylov subspace method applied to solve the integral formulation

preconditioning techniques for a mixed stokes darcy model in porous media applications we study numerical methods for a mixed stokes darcy model in porous media applications the global model is composed of two different submodels in a fluid region and a porous media region coupled through a set of interface conditions the weak formulation of the coupled model is of a saddle point type the mixed finite element discretization applied to the saddle point problem leads to a coupled indefinite and nonsymmetric linear system of algebraic equations we apply the preconditioned gmres method to solve the discrete system and are particularly interested in efficient and effective decoupled preconditioning techniques several decoupled preconditioners are proposed theoretical analysis and numerical experiments show the effectiveness and efficiency of the preconditioners effects of physical parameters on the convergence performance are also investigated

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

an asymptotically compatible formulation for local to nonlocal coupling problems without overlapping regions abstract in this paper we design and analyze an explicit partitioned procedure for a 2d dynamic local to nonlocal ltn coupling problem based on a new nonlocal robin type transmission condition the nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter characterizing the range of nonlocal interactions and the local subproblem is described by the classical heat equation we consider a heterogeneous system where the local and nonlocal subproblems present different physical properties and employ no overlapping region between the two subdomains we first propose a new generalization of classical local neumann type condition by converting the local flux to a correction term in the nonlocal model and show that the proposed neumann type boundary formulation recovers the local case as o 2 in the l norm we then extend the nonlocal neumann type boundary condition to a robin type boundary condition and develop a local to nonlocal coupling formulation with robin dirichlet transmission conditions to stabilize the explicit coupling procedure and to achieve asymptotic compatibility the choice of the coefficient in the robin condition is obtained via amplification factor analysis for the discretized system with coarse grids employing a high order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver the selection of optimal robin coefficients are verified with numerical experiments on heterogeneous and complicated domains with the developed optimal coupling strategy we numerically demonstrate the coupling framework s asymptotic convergence to the local limit with an o o h rate when there is a fixed ratio between the horizon size and the spatial discretization size h

a multilevel mesh independence principle for the navier stokes equations multilevel finite element discretization methods for the navier stokes equations are considered in contrast to usual multilevel methods a superlinear scaling of the consecutive meshwidths h j 1 mathcal o h j alpha j is used on the coarsest mesh the discretized system is solved which is small and nonlinear on each subsequent mesh only one or two newton correction steps are performed so that only one or two larger linear systems are solved the scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the h 1 and l 2 norm are investigated an error analysis of independent interest is also presented for the basic finite element method using the conservation form of the nonlinear term this formulation leads to a nonlinear system whose nonlinearity is easier to resolve than the systems arising when either the convective or explicitly skew symmetric corms are used for the nonlinear term

robin robin domain decomposition methods for the stokes darcy coupling in this paper we consider a coupled system made of the stokes and darcy equations and we propose some iteration by subdomain methods based on robin conditions on the interface we prove the convergence of these algorithms and for suitable finite element approximations we show that the rate of convergence is independent of the mesh size h special attention is paid to the optimization of the performance of the methods when both the kinematic viscosity nu of the fluid and the hydraulic conductivity tensor k of the porous medium are very small

an iterative substructuring method for coupled fluid solid acoustic problems a fast parallel iterative method is proposed for the solution of linear equations arising from finite element discretization of the time harmonic coupled fluid solid systems in fluid pressure and solid displacement formulation the fluid and the solid domains are decomposed into nonoverlapping subdomains continuity of the solution is enforced by lagrange multipliers the system is augmented by duplicating the degrees of freedom on the wet interface the original degrees of freedom are then eliminated and the resulting system is solved by iterations preconditioned by a coarse space correction in each iteration the method requires the solution of one independent local acoustic problem per subdomain and the solution of a global problem with several degrees of freedom per subdomain computational results show that the method is scalable with the problem size frequency and the number of subdomains the method generalizes the feti h method for the helmholtz equation to coupled fluid elastic scattering the number of iterations is about same as for the feti h method for the related helmholtz problem with neumann boundary condition instead of an elastic scatterer if enough coarse space functions are used convergence behavior is explained from the spectrum of the iteration operator and from numerical near decoupling of the equations in the fluid and in the solid regions

domain decomposition methods for solving stokes darcy problems with boundary integrals we consider a coupled problem of stokes and darcy equations this involves solving pdes of different orders simultaneously to overcome this difficulty we apply a nonoverlapping domain decomposition method based on a robin boundary condition obtained by combining the velocity and force interface conditions the coupled system is then reduced to solving each problem separately by an iterative procedure using a krylov subspace method the numerical solution in each subdomain is based on the boundary integral formulation where the kernels are regularized and correction terms are added to reduce the regularization error

bounds on the performance of dynamic routing schemes for highly connected networks we describe a procedure for bounding the performance of dynamic routing schemes for loss or queueing networks the bound is developed from a network flow synthesis of a collection of markov decision processes one for each resource of the network the bound emerges as a solution to a dual pair of convex programming problems where the primal problem describes mean flows through the network and the dual problem describes implied costs and surplus values at resources of the network the bound is particularly appropriate for large highly connected networks where it may be approached by simple trunk reservation or threshold routing schemes

on the approximation property of excellent rings where d is a smooth finite type a algebra a theorem which would imply the existence of an embedding of the above type yields the positive solution of the bass quillen conjecture so far this theorem is only known in special cases there is the following result by artin 3 4

a stabilized finite element method based on two local gauss integrations for a coupled stokes darcy problem in this paper a stabilized mixed finite element method for a coupled steady stokes darcy problem is proposed and investigated this method is based on two local gauss integrals for the stokes equations its originality is to use a difference between a consistent mass matrix and an under integrated mass matrix for the pressure variable of the coupled stokes darcy problem by using the lowest equal order finite element triples this new method has several attractive computational features parameter free flexible and altering the difficulties inherited in the original equations stability and error estimates of optimal order are obtained by using the lowest equal order finite element triples p 1 p 1 p 1 and q 1 q 1 q 1 for approximations of the velocity pressure and hydraulic head finally a series of numerical experiments are given to show that this method has good stability and accuracy for the coupled problem with the beavers joseph saffman jones and beavers joseph interface conditions

on globally defined semianalytic sets in this work we present the concept of c semianalytic subset of a real analytic manifold and more generally of a real analytic space c semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by cartan c analytic sets for short more precisely s is a c semianalytic subset of a real analytic space x ox if each point of x has a neighborhood u such that s u is a finite boolean combinations of global analytic equalities and strict inequalities on x by means of paracompactness c emianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on x the family of c semianalytic sets is closed

mixed stabilized finite element method for the stationary stokes dual permeability fluid flow model abstract this article proposes a stabilized mixed finite element method to investigate the stationary stokes dual permeability fluid flow model which couples the free flow region modeled by the stokes equations and dual porous medium modeled by the dual permeability equations the robust mixed finite element method developed here does not require any lagrange multiplier but a mesh dependent stabilization term with a relaxation parameter is introduced essentially to ensure the numerical stability of the algorithm the well posedness and the optimal error estimate of the finite element scheme are derived furthermore an iterative scheme is developed based on the traditional coupled algorithm we prove that the iterative scheme converges to the coupled scheme and the rate of convergence is estimated the feature of the model is illustrated by considering five numerical examples which reveal the applicability of the fluid flow model as well as the accuracy of the numerical method

computational complexity of loss networks abstract in this paper we examine the theoretical limits on developing algorithms to find blocking probabilities in a general loss network we demonstrate that exactly computing the blocking probabilities of a loss network is a p complete problem we also show that a general algorithm for approximating the blocking probabilities is also intractable unless rp np which seems unlikely according to current common notions in complexity theory given these results we examine implications for designing practical algorithms for finding blocking probabilities in special cases

local and parallel finite element algorithms for the stokes problem based on two grid discretizations some local and parallel finite element algorithms for the stokes problem are proposed and analyzed in this paper these algorithms are motivated by the observation that for a solution to the stokes problem low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure one technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape regular grids

on the solution of the coupled navier stokes and darcy equations abstract this paper introduces and analyzes two mathematical models for coupling the incompressible navier stokes equations with the porous media flow equations a numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium is proposed existence and uniqueness results under small data condition of the numerical solution are proved optimal a priori error estimates are derived numerical examples comparing the two models under varying physical parameters are provided

on stokes ritz projection and multistep backward differentiation schemes in decoupling the stokes darcy model we analyze a parallel noniterative multiphysics domain decomposition method for decoupling the stokes darcy model with multistep backward differentiation schemes for the time discretization and

monolithic multigrid method for the coupled stokes flow and deformable porous medium system abstract the interaction between fluid flow and a deformable porous medium is a complicated multi physics problem which can be described by a coupled model based on the stokes and poroelastic equations a monolithic multigrid method together with either a coupled vanka smoother or a decoupled uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids a specialty in our modeling approach is that at the interface of the fluid and poroelastic medium two unknowns from the different subsystems are defined at the same grid point we propose a special discretization at and near the points on the interface which combines the approximation of the governing equations and the considered interface conditions in the decoupled uzawa smoother local fourier analysis lfa helps us to select optimal values of the relaxation parameter appearing to implement the monolithic multigrid method grid partitioning is used to deal with the interface updates when communication is required between two subdomains numerical experiments show that the proposed numerical method has an excellent convergence rate the efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients

a divergence free weak virtual element method for the stokes darcy problem on general meshes abstract this paper presents a weak virtual element method on general meshes for the stokes darcy problem with the beavers joseph saffman interface condition the velocity is discretized by the h div virtual element the pressure is approximated by discontinuous piecewise polynomials besides a polynomial space on the element faces is introduced to approximate the tangential trace of the velocity in the stokes equations the velocity on the discrete level is exactly divergence free and thus the exact mass conservation is preserved in the discretization the well posedness of the discrete problem is proved and an a priori error estimate is derived that implies the error for the velocity in a suitable norm does not depend on the pressure a series of numerical experiments are reported to illustrate the performance of the method

space time domain decomposition for two phase flow between different rock types in previous work e ahmed s ali hassan c japhet m kern and m vohralik a posteriori error estimates and stopping criteria for space time domain decomposition for two phase flow between different rock types the smai journal of computational mathematics 5 2019 pp 195 227 https doi org 10 5802 smai jcm 47 a space time domain decomposition method was proposed for two phase ow in a porous medium composed of two different rock types so that the capillary pressure field is discontinuous at the interface between the rocks for this nonlinear and degenerate parabolic problem with nonlinear and discontinuous transmission conditions on the interface the optimized schwarz waveform relaxation method oswr with robin or ventcell transmission conditions was considered a guaranteed and fully computable a posteriori error estimate was derived which in particular took into account the domain decomposition error in this paper we provide a mathematical and numerical analysis of this space time domain decomposition method in the robin case complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous galerkin method in time we prove the existence of a weak solution of the two phase ow subdomain problem with robin boundary conditions by analyzing the convergence of the finite volume scheme the domain decomposition algorithm is based on the solution of space time nonlinear subdomain problems over the whole time interval allowing for different time steps in dfferent parts of the domain adapted to the physical properties of each subdomain and we show that such an algorithm is well defined numerical experiments on three dimensional problems with different rock types illustrate the performance of the domain decomposition method

numerical solution to a mixed navier stokes darcy model by the two grid approach we study numerical methods for a coupled navier stokes darcy model which describes a fluid flow filtrating through porous media a decoupled and linearized two grid algorithm is proposed numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm

a two grid decoupling method for the mixed stokes darcy model in this paper we consider the mixed stokes darcy problem which describes a fluid flow coupled with a porous media we present a modified two grid method for decoupling this mixed model stability is proved and optimal error estimates are derived the numerical results show that the modified two grid method is effective and has the same accuracy as the coupling scheme when we choose h h 2

unified stabilized finite element formulations for the stokes and the darcy problems in this paper we propose stabilized finite element methods for both stokes and darcy s problems that accommodate any interpolation of velocities and pressures apart from the interest of this fact the important issue is that we are able to deal with both problems at the same time in a completely unified manner in spite of the fact that the functional setting is different concerning the stabilization formulation we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters both in what refers to stability and to accuracy this choice is shown to be crucial in the case of darcy s problem as an additional feature of this work we treat two types of stabilized formulations showing that they have a very similar behavior

a stabilized finite volume method for the evolutionary stokes darcy system abstract in this paper we propose a stabilized fully discrete finite volume method based on two local gauss integrals for a non stationary stokes darcy problem this stabilized method is free of stabilized parameters and uses the lowest equal order finite element triples p 1 p 1 p 1 for approximating the velocity pressure and hydraulic head of the stokes darcy model under a modest time step restriction in relation to physical parameters we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements finally a series of numerical experiments are provided to demonstrate the validity of the theoretical results

analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater surface water flows the most effective simulations of the multiphysics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes partitioned methods which solve the coupled problem by successively solving the subphysics problems have recently been studied for the stokes darcy coupling with convergence established over bounded time intervals with constants growing exponentially in t this report analyzes and tests two such partitioned noniterative domain decomposition methods for the fully evolutionary stokes darcy problem under a modest time step restriction of the form delta t leq c where c c physical parameters we prove unconditional long time over 0 leq t infty stability of both partitioned methods from this we derive an optimal error estimate that is uniform in time over 0 leq t infty

robin robin domain decomposition methods for the steady state stokes darcy system with the beavers joseph interface condition domain decomposition methods for solving the coupled stokes darcy system with the beavers joseph interface condition are proposed and analyzed robin boundary conditions are used to decouple the stokes and darcy parts of the system then parallel and serial domain decomposition methods are constructed based on the two decoupled sub problems convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence

uzawa smoother in multigrid for the coupled porous medium and stokes flow system the multigrid solution of coupled porous media and stokes flow problems is considered the darcy equation as the saturated porous medium model is coupled to the stokes equations by means of appropriate interface conditions we focus on an efficient multigrid solution technique for the coupled problem which is discretized by finite volumes on staggered grids giving rise to a saddle point linear system special treatment is required regarding the discretization at the interface an uzawa smoother is employed in multigrid which is a decoupled procedure based on symmetric gauss seidel smoothing for velocity components and a simple richardson iteration for the pressure field since a relaxation parameter is part of a richardson iteration local fourier analysis is applied to determine the optimal parameters highly satisfactory multigrid convergence is reported and moreover the algorithm performs very well for small values of the hydraulic conductivity and fluid viscosity which are relevant for applications

decoupled schemes for a non stationary mixed stokes darcy model we study numerical methods for solving a non stationary mixed stokes darcy problem that models coupled fluid flow and porous media flow a decoupling approach based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model error estimates are derived and numerical experiments are conducted to demonstrate the computational effectiveness of the decoupling approach

efficient and long time accurate second order methods for the stokes darcy system we propose and study two second order in time implicit explicit methods for the coupled stokes darcy system that governs flows in karst aquifers and other subsurface flow systems the first method is a combination of a second order backward differentiation formula and the second order gear s extrapolation approach the second is a combination of the second order adams moulton and second order adams bashforth methods both algorithms only require the solution of decoupled stokes and darcy problems at each time step hence these schemes are very efficient and can be easily implemented using legacy codes we establish the unconditional and uniform in time stability for both schemes the uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes e g contaminant sequestration and release that occur over very long time scales error estimates for fully discretized schemes using finite element spatial discretizations are derived nume

partitioned time stepping method for fully evolutionary stokes darcy flow with beavers joseph interface conditions in this report a partitioned time stepping algorithm for transient flow in a matrix domain coupled to a free flow in embedded conduits is analyzed the coupled flow is modeled by the fully evolutionary stokes darcy problem this method requires solving only one uncoupled stokes and darcy subphysics and subdomain per time step on the interface between the matrix and conduit beavers joseph interface conditions instead of the simplified beavers joseph saffman condition are imposed under a modest time step restriction of the form triangle t leq c where c c physical parameters we prove stability of the method we also derive error estimates numerical tests illustrate the validity of the theoretical results

parallel iterative stabilized finite element algorithms based on the lowest equal order elements for the stationary navier stokes equations abstract based on a fully overlapping domain decomposition technique and the lowest equal order finite elements three parallel iterative stabilized finite element algorithms for the stationary navier stokes equations are proposed and studied where the stabilization term is based on two local gauss integrations at element level in these parallel algorithms each processor independently computes a local stabilized solution in its own subdomain making the algorithms have low communication cost and easy to implement based on a sequential solver the algorithms can yield an approximate solution with an accuracy comparable to that of the standard stabilized finite element solution with a substantial reduction in computational time theoretical and numerical results demonstrated the effectiveness and efficiency of the algorithms

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

solving composite problems with interface relaxation this paper deals with a solution approach suitable for composite partial differential equations pdes with interface conditions we present a general framework based on interface relaxation which provides a uniform platform for building problem solving environments through efficient software integration and for implementing various relaxation schemes mathematically this framework contains many existing domain decomposition methods and also allows the extension to a variety of new relaxers in particular we describe a class of relaxers which are suitable for general and complicated interface conditions interface relaxation is more general than the traditional domain decomposition methods in that it allows unrelated pde problems on different subdomains convergence analysis error estimates and preconditioning strategies are presented which show that these relaxers are competitive with existing domain decomposition methods for model problems involving a single pde we present experimental results which demonstrate the wide applicability of this approach differences between this approach and other domain decomposition methods are also discussed

a modified local and parallel finite element method for the mixed stokes darcy model abstract based on two grid discretizations a modified local and parallel finite element algorithm for the mixed stokes darcy problem is proposed and analyzed in this paper the idea is that on the coarse grid we solve a coupled stokes darcy problem on the fine grid we first consider the discrete darcy problem with a coarse grid approximation u h to the interface coupling conditions then solve a series of independent local stokes subproblems with the numerical solution of darcy problem on fine grid to the interface coupling conditions in parallel optimal error estimates are obtained and numerical experiments demonstrate the accuracy and efficiency of the modified local and parallel algorithm

dg approximation of coupled navier stokes and darcy equations by beaver joseph saffman interface condition in this work we couple the incompressible steady navier stokes equations with the darcy equations by means of the beaver joseph saffman s condition on the interface under suitable smallness conditions on the data we prove existence of a weak solution as well as some a priori estimates we establish local uniqueness when the data satisfy additional smallness restrictions then we propose a discontinuous galerkin scheme for discretizing the equations and do its numerical analysis

a finite element method based on weighted interior penalties for heterogeneous incompressible flows we propose a finite element scheme for the approximation of multidomain heterogeneous problems arising in the general framework of linear incompressible flows e g stokes and darcy s equations we exploit stabilized mixed finite elements together with nitsche type matching conditions that automatically adapt to the coupling of different subproblem combinations optimal error estimates are derived for the coupled problem then we propose and analyze an iterative splitting strategy for the approximation of the multidomain solution by means of a sequence of independent and local subproblems thanks to the introduction of a suitable relaxation strategy the iterative method turns out to be convergent for any possible coupling between subproblems

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

discontinuous galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems abstract we formulate a coupled surface subsurface flow model that relies on hydrostatic equations with free surface in the free flow domain and on darcy s law in the subsurface the model is discretized using the local discontinuous galerkin method and a statement of discrete energy stability is proved for the fully non linear coupled system the performance of the proposed scheme is illustrated by a numerical convergence study and a realistic example

a multilevel decoupled method for a mixed stokes darcy model this paper studies decoupled numerical methods for a mixed stokes darcy model for coupling fluid and porous media flows a two level algorithm is proposed and analyzed in mu and xu 2007 10 we generalize the two level algorithm to a multilevel algorithm in this paper and present numerical analysis on the error estimates for the multilevel algorithm the multilevel algorithm solves the mixed stokes darcy system by applying efficient legacy code for single model solvers to solve two decoupled stokes and darcy subproblems on all the subsequently refined meshes except for a much smaller global problem only on a very coarse initial mesh numerical experiments are conducted for both the two level and multilevel algorithms to illustrate their effectiveness and efficiency and validate the related theoretical analysis

a multilevel decoupling method for the navier stokes darcy model abstract this paper considers a multilevel decoupling method for the coupled navier stokes darcy model describing a free flowing fluid over a porous medium the method utilizes a sequence of meshes on which a low dimensional fully coupled nonlinear problem is solved only on a very coarse initial mesh on subsequent finer meshes the approximate solution in each flow region is obtained by solving a linear decoupled problem and performing a correction step the correction step in each domain is achieved by solving a linear system that differs from the original decoupled system only in the right hand side we prove optimal error estimates and demonstrate that for a sequence of meshes with spacing h j h j 1 2 the decoupling method is computationally efficient and achieves the same order of approximation as the fully coupled method

robin robin preconditioned krylov methods for fluid structure interaction problems in this work we propose a robin robin preconditioner combined with krylov iterations for the solution of the interface system arising in fluid structure interaction fsi problems it can be seen as a partitioned fsi procedure and in this respect it generalizes the ideas introduced in s badia f nobile c vergara j comput phys 227 2008 7027 7051 we analyze the convergence of gmres iterations with the robin robin preconditioner on a model problem and compare its efficiency with some existing algorithms the method is shown to be very efficient for many challenging fluid structure interaction problems such as those characterized by a large added mass effect or by enclosed fluids in particular the possibility to solve balloon type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches c 2009 elsevier b v all rights reserved