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

gtm the generative topographic mapping latent variable models represent the probability density of data in a space of several dimensions in terms of a smaller number of latent or hidden variables a familiar example is factor analysis which is based on a linear transformations between the latent space and the data space in this paper we introduce a form of non linear latent variable model called the generative topographic mapping for which the parameters of the model can be determined using the em algorithm gtm provides a principled alternative to the widely used self organizing map som of kohonen 1982 and overcomes most of the significant limitations of the som we demonstrate the performance of the gtm algorithm on a toy problem and on simulated data from flow diagnostics for a multi phase oil pipeline

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

coupling biot and navier stokes equations for modelling fluid poroelastic media interaction the interaction between a fluid and a poroelastic structure is a complex problem that couples the navier stokes equations with the biot system the finite element approximation of this problem is involved due to the fact that both subproblems are indefinite in this work we first design residual based stabilization techniques for the biot system motivated by the variational multiscale approach then we state the monolithic navier stokes biot system with the appropriate transmission conditions at the interface for the solution of the coupled system we adopt both monolithic solvers and heterogeneous domain decomposition strategies different domain decomposition methods are considered and their convergence is analyzed for a simplified problem we compare the efficiency of all the methods on a test problem that exhibits a large added mass effect as it happens in hemodynamics applications

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

the exchange property for row and column finite matrix rings abstract the ring b r of all row and column finite matrices over any unital von neumann regular ring r is shown to be an exchange ring more generally the same is true of the multiplier ring of a non unital regular ring with a countable unit

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

robust preconditioning for coupled stokes darcy problems with the darcy problem in primal form abstract the coupled darcy stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part in this work we consider preconditioners for monolithic solution algorithms of the coupled darcy stokes problem where the darcy problem is in primal form we employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters i e the permeability viscosity and beavers joseph saffman condition our approach is similar to that of holter et al 2020 but since the darcy problem is in primal form expressing mass conservation at the interface involves the normal derivative which introduces some mathematical challenges these challenges will be specifically addressed in this paper in particular we will employ fractional laplacians at the interface numerical experiments illustrating the performance are provided the preconditioner is posed in non standard sobolev spaces which may be perceived as an obstacle for its use in applications however we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice

probabilistic non linear principal component analysis with gaussian process latent variable models summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure in this paper we provide an overview of some existing techniques for discovering such embeddings we then introduce a novel probabilistic interpretation of principal component analysis pca that we term dual probabilistic pca dppca the dppca model has the additional advantage that the linear mappings from the embedded space can easily be non linearised through gaussian processes we refer to this model as a gaussian process latent variable model gp lvm through analysis of the gp lvm objective function we relate the model to popular spectral techniques such as kernel pca and multidimensional scaling we then review a practical algorithm for gp lvms in the context of large data sets and develop it to also handle discrete valued data and missing attributes we demonstrate the model on a range of real world and artificially generated data sets

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

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

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

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

domain decomposition for coupled stokes and darcy flows abstract a non overlapping domain decomposition method is presented to solve a coupled stokes darcy flow problem in parallel by partitioning the computational domain into multiple subdomains upon which families of coupled local problems of lower complexity are formulated the coupling is based on appropriate interface matching conditions the global problem is reduced to an interface problem by eliminating the interior subdomain variables the interface problem is solved by an iterative procedure which requires solving subdomain problems at each iteration finite element techniques appropriate for the type of each subdomain problem are used to discretize it the condition number of the resulting algebraic system is analyzed and numerical tests verifying the theoretical estimates are provided

long time stability and convergence rate of maccormack rapid solver method for nonstationary stokes darcy problem abstract we propose and study a combination of two second order implicit explicit imex methods for the coupled stokes darcy system that governs flows in karst aquifers the first is a second order explicit two step maccormack scheme and the second is a second order implicit crank nicolson method both algorithms only require the solution of two decoupled problems at each time step one stokes and the other darcy this combination so called the maccormack rapid solver method is very efficient faster at least of second order accuracy in time and space and can be easily implemented using legacy codes under time step limitation of the form t c h where h t are mesh size and time step respectively and c is a physical parameter we prove both long time stability and the rate of convergence of the method some numerical experiments are presented and discussed

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

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

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

bayesian nonlinear principal component analysis using random fields we propose a novel model for nonlinear dimension reduction motivated by the probabilistic formulation of principal component analysis nonlinearity is achieved by specifying different transformation matrices at different locations of the latent space and smoothing the transformation using a markov random field type prior the computation is made feasible by the recent advances in sampling from von mises fisher distributions the computational properties of the algorithm are illustrated through simulations as well as an application to handwritten digits data

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

relating properties of a ring with properties of matrix rings coming from ornstein dual pairs abstract in this paper we study the ring theoretic classification of intermediate rings e r of infinite matrix rings rcfm r e r rfm r from ornstein s dual pairs see d ornstein dual vector spaces ann math 69 1959 520 534 a del rio j j simon intermediate rings between matrix rings and ornstein dual pairs arch math basel 75 2000 256 263 in case r is semisimple artinian semiprimary or left or right perfect to do this we develop a technique of decomposing an infinite matrix as infinite sum of submatrices of less size then we show that r is a semisimple artinian ring if and only if e r is a von neumann regular ring we then describe the lattice of finitely generated left ideals and the lattice of two sided ideals of e r by equivalence of idempotents that is in classical terms we also describe the jacobson radical of e r for an arbitrary ring r and then we show among other results that r is left perfect if and only if e r is a semiregular ring

model averaging and dimension selection for the singular value decomposition many multivariate data analysis techniques for an m n matrix y are related to the model y m e where y is an m n matrix of full rank and m is an unobserved mean matrix of rank k m n typically the rank of m is estimated in a heuristic way and then the least squares estimate of m is obtained via the singular value decomposition of y yielding an estimate that can have a very high variance in this article we suggest a model based alternative to the preceding approach by providing prior distributions and posterior estimation for the rank of m and the components of its singular value decomposition in addition to providing more accurate inference such an approach has the advantage of being extendable to more general data analysis situations such as inference in the presence of missing data and estimation in a generalized linear modeling framework

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

the structure of countably generated projective modules over regular rings abstract we prove that for every regular ring r there exists an isomorphism between the monoids of isomorphism classes of finitely generated projective right modules over the rings end r r r and rcfm r where the latter denotes the ring of countably infinite row and column finite matrices over r we use this result to give a precise description of the countably generated projective modules over simple regular rings and over regular rings satisfying s comparability

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

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

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

a diffuse interface method for the navier stokes darcy equations perfusion profile for a patient specific human liver based on mri scans abstract we present a diffuse interface method for coupling free and porous medium type flows modeled by the navier stokes and darcy equations its essential component is a diffuse geometry model generated from the phase field solution of a separate initial boundary value problem that is based on the allen cahn equation phase field approximations of the interface and its gradient are then employed to transfer all interface terms in the coupled variational flow formulation into volumetric terms this eliminates the need for an explicit interface parametrization between the two flow regimes we illustrate accuracy and convergence for a series of benchmark examples using standard low order stabilized finite element discretizations our diffuse interface method is particularly attractive for coupled flow analysis on imaging data with complex implicit interfaces where procedures for deriving explicit surface parametrizations constitute a significant bottleneck we demonstrate the potential of our method to establish seamless imaging through analysis workflows by computing a perfusion profile for a full scale 3d human liver based on mri scans

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

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

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 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

stability analysis of maccormack rapid solver method for evolutionary stokes darcy problem abstract a numerical method based on the explicit implicit scheme of maccormack finite difference scheme was applied to the solution of parabolized navier stokes equations partitioned methods which solve the coupled problem by successively solving the sub physics problems have recently been studied for the full evolutionary groundwater surface water flow with convergence established over both bounded and long time intervals a maccormack rapid solver method based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model under a time step limitation of the form c 1 h t c 2 h where h t are mesh size and time step respectively and c j j 1 2 are two physical parameters we prove both uniform and asymptotic stabilities over long time intervals some numerical experiments are presented and discussed

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

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

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