the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

blow ups and infinitesimal automorphisms of cr manifolds for a real analytic connected cr hypersurface m of cr dimension n geqslant 1 having a point of levi nondegeneracy the following alternative is demonstrated for its symmetry algebra mathfrak s mathfrak s m i either dim mathfrak s n 2 4n 3 and m is spherical everywhere ii or dim mathfrak s leqslant n 2 2n 2 delta 2 n and in the case of equality m is spherical and has fixed signature of the levi form in the complement to its levi degeneracy locus a version of this result is proved for the lie group of global automorphisms of m explicit examples of cr hypersurfaces and their infinitesimal and global automorphisms realizing the bound in ii are constructed we provide many other models with large symmetry using the technique of blow up in particular we realize all maximal parabolic subalgebras of the pseudo unitary algebras as a symmetry

a residual based a posteriori error estimator for the stokes darcy coupled problem in this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow the flows are governed by the stokes and darcy equations respectively and the transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the finite element subspaces consider bernardi raugel and raviart thomas elements for the velocities piecewise constants for the pressures and continuous piecewise linear elements for a lagrange multiplier defined on the interface we derive a reliable and efficient residual based a posteriori error estimator for this coupled problem the proof of reliability makes use of suitable auxiliary problems diverse continuous inf sup conditions satisfied by the bilinear forms involved and local approximation properties of the clement interpolant and raviart thomas operator on the other hand helmholtz decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are the main tools for proving the efficiency of the estimator up to minor modifications our analysis can be extended to other finite element subspaces yielding a stable galerkin scheme

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

on a higher order accurate fully discrete galerkin approximation to the navier stokes equations we consider approximating the solution of the initial and boundary value problem for the navier stokes equations in bounded twoand three dimensional domains using a nonstandard galerkin finite element method for the space discretization and the third order accurate three step backward differentiation method coupled with extrapolation for the nonlinear terms for the time stepping the resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions all these linear systems have the same matrix the resulting approximations of the velocity are shown to have optimal rate of convergence in l2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space

on the symmetry algebras of 5 dimensional cr manifolds abstract we show that for a real analytic connected holomorphically nondegenerate 5 dimensional cr hypersurface m and its symmetry algebra s one has either i dim s 15 and m is spherical with levi form of signature either 2 0 or 1 1 everywhere or ii dim s 11 where dim s 11 can only occur if on a dense open subset m is spherical with levi form of signature 1 1 furthermore we construct a series of examples of pairwise nonequivalent cr hypersurfaces with dim s 11

an alternating explicit implicit domain decomposition method for the parallel solution of parabolic equations explicit implicit domain decomposition eidd is a class of globally non iterative non overlapping domain decomposition methods for the numerical solution of parabolic problems on parallel computers which are highly efficient both computationally and communicationally for each time step in this paper an alternating eidd method is proposed which is algorithmically simple efficient for each time step highly parallel and satisfies a stability condition that imposes no additional restriction to the time step restriction imposed by the consistency condition which guarantees a convergence of order o dth 1n b n o h 2 in an h 1 type norm where n b and n respectively denote the number of gridpoints on the interface boundaries b and the number of gridpoints on the entire discrete domain

optimized schwarz methods with robin transmission conditions for parabolic problems in this paper optimized schwarz methods with robin transmission conditions are considered for solving the time dependent linear algebraic systems resulting from finite element approximations for parabolic problems we use both analysis and numerical experiments to investigate the convergence behavior particularly we check the influence of the time step size on the convergence rate we get the estimate of the upper bound of convergence rate 1 o min h 1 2 h 1 2 h 1 2 h 3 2 tau 1 where h is the mesh size h is the size of subdomains and tau is the time step size our numerical results show that the convergence rate of this method is fast we also search for the optimal parameter lambda by experiments and find the rule of its distribution

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

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

h div conforming finite element methods for the coupled stokes and darcy problem this paper proposes and analyzes a numerical method for solving the coupled stokes and darcy problem an interface problem between a fluid governed by stokes equations and a flow in a porous medium governed by darcy equations the method employs h div conforming finite elements for the velocity field in both stokes and darcy subdomains optimal order error estimates are established

coupled generalized nonlinear stokes flow with flow through a porous medium in this article we analyze the flow of a fluid through a coupled stokes darcy domain the fluid in each domain is non newtonian modeled by the generalized nonlinear stokes equation in the free flow region and the generalized nonlinear darcy equation in the porous medium a flow rate is specified along the inflow portion of the free flow boundary we show existence and uniqueness of a variational solution to the problem we propose and analyze an approximation algorithm and establish a priori error estimates for the approximation

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 unified stabilized method for stokes and darcy s equations we use the lowest possible approximation order piecewise linear continuous velocities and piecewise constant pressures to compute solutions to stokes equation and darcy s equation applying an edge stabilization term to avoid locking we prove that the formulation satisfies the discrete inf sup condition we prove an optimal a priori error estimate for both problems the formulation is then extended to the coupled case using a nitsche type weak formulation allowing for different meshes in the two subdomains finally we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach

infinitesimal cr automorphisms of real hypersurfaces let m be an analytic real hypersurface through the origin in c n 1 and let hol m denote the space of vector fields x re z m tangent to m in a neighborhood of the origin where z is a holomorphic vector field defined in a neighborhood of the origin the hypersurface m is holomorphically nondegenerate at the origin if there is no holomorphic vector field tangent to m in a neighborhood of the origin the main result of the paper is that hol m is finite dimensional if and only if m is holomorphically nondegenerate at the origin

stabilized explicit implicit domain decomposition methods for the numerical solution of parabolic equations we report a class of stabilized explicit implicit domain decomposition seidd methods for the numerical solution of parabolic equations explicit implicit domain decomposition eidd methods are globally noniterative nonoverlapping domain decomposition methods which when compared with schwarz algorithm based parabolic solvers are computationally and communicationally efficient for each simulation time step but suffer from small time step size restrictions by adding a stabilization step to eidd the seidd methods retain the time stepwise efficiency in computation and communication of the eidd methods but exhibit much better numerical stability three seidd algorithms are presented in this paper which are experimentally tested to show excellent stability computation and communication efficiencies and high parallel speedup and scalability

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

explicit implicit conservative domain decomposition procedures for parabolic problems based on block centered finite differences domain decomposition procedures for solving parabolic equations are considered the underlying discretization consists of block centered finite differences in these procedures fluxes at subdomain interfaces are calculated from the solution at the previous time level these fluxes serve as neumann boundary data for implicit block centered discretizations in the subdomains a priori error estimates are presented and numerical results examining the stability accuracy and parallelism of the scheme are presented

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

coupling of darcy forchheimer and compressible navier stokes equations with heat transfer this paper is devoted to the coupling of a 2d reservoir model with a 1 5d vertical wellbore model both written in axisymmetric form the physical problems are respectively described by the darcy forchheimer and the compressible navier stokes equations together with an exhaustive energy equation each model was previously studied and its finite element discretization was validated the two weak problems are bound together by means of transmission conditions at the perforations yielding a nonstandard mixed formulation a technical analysis is then carried out and the well posedness of the time discretized coupled problem in both the continuous and the discrete cases is established numerical tests including physical cases are presented validating the coupled code

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

first order system least squares for coupled stokes darcy flow the coupled problem with stokes flow in one subdomain and a darcy flow model in a second subdomain is studied in this paper both flow problems are treated as first order systems involving pseudostress and velocity in the stokes case and using a flux pressure formulation in the darcy subdomain as process variables respectively the beavers joseph saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems a combination of h mathrm div conforming raviart thomas and standard h 1 conforming elements is used for the stokes as well as for the darcy subsystem the homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates it also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies

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

a new least squares finite element model for combined navier stokes and darcy flows in geometrically complicated domains with solid and porous boundaries in this paper we present a novel method for linking navier stokes and darcy equations along a porous inner boundary in a flow regime which is governed by both types of these equations the method is based on a least squares finite element technique and uses isoparametric c1 continuous hermite elements for domain discretization we show that our technique is superior to previously developed models for the combined navier stokes darcy flows the previous works use weighted residual finite element procedures in conjunction with c0 elements which are inherently incapable of linking navier stokes and darcy equations the paper includes the application of our model to a geometrically complicated axisymmetric slurry filtration system

a unified stabilized mixed finite element method for coupling stokes and darcy flows abstract a stabilized mixed finite element method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach utilizes the same nonconforming crouzeix raviart element discretization on the entire domain a discrete inf sup condition and an optimal a priori error estimate are derived finally some numerical examples verifying the theoretical predictions are presented

a domain decomposition discretization of parabolic problems in recent years domain decomposition methods have attracted much attention due to their successful application to many elliptic and parabolic problems domain decomposition methods treat problems based on a domain substructuring which is attractive for parallel computation due to the independence among the subdomains in principle domain decomposition methods may be applied to the system resulting from a standard discretization of the parabolic problems or directly be carried out through a discretization of parabolic problems in this paper a direct domain decomposition method is introduced to discretize the parabolic problems the stability and convergence of this algorithm are analyzed

chern moser operators and polynomial models in cr geometry we consider the fundamental invariant of a real hypersurface in c n its holomorphic symmetry group and analyze its structure at a point of degenerate levi form generalizing the chern moser operator to hypersurfaces of finite multitype we compute the lie algebra of infinitesimal symmetries of the model and provide explicit description for each graded component compared with a hyperquadric it may contain additional components consisting of nonlinear vector fields defined in terms of complex tangential variables as a consequence we obtain exact results on jet determination for hypersurfaces with such models the results generalize directly the fundamental result of chern and moser from quadratic models to polynomials of higher degree c 2014 elsevier inc all rights reserved

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

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

ipic domain decomposition algorithm for parabolic problems a domain decomposition algorithm which is based on a semi implicit prediction and then a fully implicit correction at the interface for solving parabolic partial differential equations is presented the numerical results show that the algorithm is unconditionally stable and very efficient in this paper high speed up is analyzed with respect to the spectral radius and operation counts

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

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

boundary integral solutions of coupled stokes and darcy flows an accurate computational method based on the boundary integral formulation is presented for solving boundary value problems for stokes and darcy flows the method also applies to problems where the equations are coupled across an interface through appropriate boundary conditions the adopted technique consists of first reformulating the singular integrals for the fluid quantities as single and double layer potentials then the layer potentials are regularized and discretized using standard quadratures as a final step the leading term in the regularization error is eliminated in order to gain one more order of accuracy the numerical examples demonstrate the increase of the convergence rate from first to second order and show a decrease in magnitude of the error the coupled problems require the computation of the gradient of the stokes velocity at the common interface this boundary condition is also written as a combination of single and double layer potentials so that the same approach can be used to compute it accurately extensive numerical examples show the increased accuracy gained by the correction terms

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

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

a strongly conservative finite element method for the coupling of stokes and darcy flow we consider a model of coupled free and porous media flow governed by stokes and darcy equations with the beavers joseph saffman interface condition this model is discretized using divergence conforming finite elements for the velocities in the whole domain discontinuous galerkin techniques and mixed methods are used in the stokes and darcy subdomains respectively this discretization is strongly conservative in h d i v w and we show convergence numerical results validate our findings and indicate optimal convergence orders

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

long time stability of four methods for splitting the evolutionary stokes darcy problem into stokes and darcy subproblems this report analyzes the long time stability of four methods for non iterative sub physics uncoupling for the evolutionary stokes darcy problem the four methods uncouple each timestep into separate stokes and darcy solves using ideas from splitting methods three methods uncouple sequentially while one is a parallel uncoupling method we prove long time stability of four splitting based partitioned methods under timestep restrictions depending on the problem parameters the methods include those that are stable uniformly in s 0 the storativity coefficient for moderate k m i n the minimum hydraulic conductivity uniformly in k m i n for moderate s 0 and with no coupling between the timestep and the spacial meshwidth

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

multiplicative schwarz methods for parabolic problems the class of multiplicative schwarz methods originated from the classical schwarz alternating method it has been shown to be one of the most powerful methods for solving finite element or finite d

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

a finite difference domain decomposition algorithm for numerical solution of the heat equation a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented in this procedure interface values between subdomains are found by an explicit finite difference formula once these values are calculated interior values are determined by backward differencing in time a natural extension of this method allows for the use of different time steps in different subdomains maximum norm error estimates for these procedures are derived which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters

analysis of discontinuous finite element methods for ground water surface water coupling we derive and analyze new numerical approaches for modeling coupled ground water surface water flow in this coupled model surface water flow is described by the depth averaged shallow water equations while ground water is modeled by saturated darcy flow the coupling between the two models assumes continuity of pressure and water flux across the ground water surface water interface the coupled model is approximated by a local discontinuous galerkin method for ground water flow and a discontinuous galerkin method for surface water flow a priori error estimates are derived for this approach a closely related approach where the well known mixed finite element method is applied to the ground water flow equations is also described and analyzed one advantage of these approaches is that they allow for the ground water and surface water domains to be meshed independently under some mild restrictions