block partitioned solvers for coupled poromechanics a unified framework abstract coupled poromechanical problems appear in a variety of disciplines from reservoir engineering to biomedical applications this work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations these systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations recently several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution these schemes appear distinct but a unifying feature is that they exploit the block partitioned nature of the problem to accelerate convergence this paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness we begin by focusing on two specific methods a fully implicit and a sequential implicit scheme in the first approach the system matrix is treated monolithically and a krylov iteration is used to update pressure and displacement unknowns simultaneously to accelerate convergence a preconditioning operator is introduced based on an approximate block factorization of the linear system next we analyze a sequential implicit scheme based on the fixed stress split in this method one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached we re interpret this scheme as a block preconditioned richardson iteration and we show that the preconditioning operator is identical to that used within the fully implicit approach rapid convergence in both the richardson and krylov based methods results from a particular choice for a sparse schur complement approximation this analysis leads to a unified framework for developing solution schemes based on approximate block factorizations many classic fully implicit and sequential implicit schemes are simple sub cases the analysis also highlights several new approaches that have not been previously explored for illustration we directly compare the performance and robustness of several variants on a benchmark problem

diffusion in poro elastic media existence uniqueness and regularity theory is developed for a general initial boundary value problem for a system of partial differential equations which describes the biot consolidation model in poro elasticity as well as a coupled quasi static problem in thermoelasticity additional effects of secondary consolidation and pore fluid exposure on the boundary are included this quasi static system is resolved as an application of the theory of linear degenerate evolution equations in hilbert space and this leads to a precise description of the dynamics of the system

uniformly stable discontinuous galerkin discretization and robust iterative solution methods for the brinkman problem we consider robust iterative methods for discontinuous galerkin dg h div omega conforming discretizations of the brinkman equations we describe a simple uzawa iteration for the solution of this problem which requires the solution of a nearly incompressible linear elasticity type equation with mass term on every iteration we prove the uniform stability of the dg discretization for both problems then we analyze variable v cycle and w cycle multigrid methods with nonnested bilinear forms we prove that these algorithms are robust and their convergence rates are independent of the parameters in the brinkman problem and of the mesh size the theoretical analysis is confirmed by numerical results

a numerical study of a posteriori error estimators for convection diffusion equations this paper presents a numerical study of a posteriori error estimators for convection diausion equations the study involves the gradient indicator an a posteriori error estimator which is based on gradient recovery three residual based error estimators for diaerent norms and two error estimators which are defined by solutions of local neumann problems they are compared with respect to the reliable estimation of the global error and with respect to the accuracy of the computed solutions on adaptively refined grids the numerical study shows for both criteria of comparison that none of the considered error estimators works satisfactorily in all tests 2000 elsevier science s a all rights reserved msc 65n50 65n30

stress and flux reconstruction in biots poro elasticity problem with application to a posteriori error analysis we derive equilibrated reconstructions of the darcy velocity and of the total stress tensor for biots poro elasticity problem both reconstructions are obtained from mixed finite element solutions of local neumann problems posed over patches of elements around mesh vertices the darcy velocity is reconstructed using raviartthomas finite elements and the stress tensor using arnoldwinther finite elements so that the reconstructed stress tensor is symmetric both reconstructions have continuous normal component across mesh interfaces using these reconstructions we derive a posteriori error estimators for biots poro elasticity problem and we devise an adaptive spacetime algorithm driven by these estimators the algorithm is illustrated on test cases with analytical solution on the quarter five spot problem and on an industrial test case simulating the excavation of two galleries

guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the biot problem abstract the paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi static linear biot equations the system is decoupled by employing the fixed stress split scheme which leads to an iteratively solved semi discrete system the error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations the estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method they are fully computable do not contain mesh dependent constants and provide reliable global estimates of the error measured in the energy norm moreover they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures

the exact asymptotic behaviour of the unique solution to a singular nonlinear dirichlet problem abstract by karamata regular varying theory a perturbed argument and constructing comparison functions we show the exact asymptotic behaviour of the unique solution u c 2 c near the boundary to a singular dirichlet problem u b x g u f u u 0 x u 0 which is independent on f u and we also show the existence and uniqueness of solutions to the problem where is a bounded domain with smooth boundary in r n 0 g c 1 0 0 and there exists 1 such that lim t 0 g t g t 1 0 f c loc 0 0 the function f s s s 0 is decreasing on 0 for some s 0 0 and b is nonnegative nontrivial on which may be vanishing on the boundary

on ground state solutions for singular and semi linear problems including super linear terms at infinity abstract we establish a result concerning the existence of entire positive classical and bounded solutions which converge to zero at infinity for the semi linear equation u f x u x r n where f r n 0 0 is a suitable function and 0 is a real parameter this result completes the principal theorem of a mohammed a mohammed ground state solutions for singular semi linear elliptic equations nonlinear analysis 2008 doi 10 1016 j na 2008 11 080 mainly because his result does not address the super linear terms at infinity penalty arguments lower upper solutions and an approximation procedure will be used

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

analysis of a discontinuous galerkin method for the biot s consolidation problem in this paper a fully discrete stabilized discontinuous galerkin method is proposed to solve the biot s consolidation problem the existence and uniqueness of the finite element solution are obtained the stability of the fully discrete solution is discussed the corresponding error estimates for the approximation of displacement and pressure in a mesh dependent norm are obtained the error estimate for the approximation of pressure in l 2 norm is also obtained

the second expansion of the unique vanishing at infinity solution to a singular elliptic equation abstract this paper is considered with the second expansion of the unique vanishing at infinity solution to u b x g u x r n n 3 where b c r n is nonnegative and may be supercritical attenuation or critical attenuation at infinity g c 1 0 0 is non increasing on 0 with lim s 0 g s and g is normalized regularly varying at zero with index 1

asymptotic behavior of semidiscrete finite element approximations of biot s consolidation problem error estimates for spatially discrete galerkin finite element approximations of biot s model for consolidation of saturated porous media are presented the short and long time behaviors of such approximations based on both stable and unstable combinations of finite element spaces of displacement and pore pressure fields are discussed

ground state solutions for a semilinear elliptic problem with a convection term abstract by a sub supersolution method and a perturbed argument we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem u p x u q f x u u 0 x r n lim x u x 0 where q 1 2 p x c loc r n for some 0 1 p x 0 x r n and f r n 0 0 is a locally holder continuous function which may be singular at zero

theory of the dynamic biot allard equations and their link to the quasi static biot system we undertake establishing well posedness of the dynamic biot allard equations it is obtained using the precise properties of the dynamic permeability matrix following the homogenization derivation of the model by taking the singular limit of the contrast coefficient the quasi static biot system can be obtained from the dynamic biot equations these results can be used to formulate an efficient computational algorithm for solving dynamic biot allard equations for subsurface flows with the characteristic reservoir time scales larger than the intrinsical characteristic time this result appears to be completely new in the literature on biot s theory we conclude by showing that in the case of periodic deformable porous media the dynamic permeability has the required properties

a least squares mixed finite element method for biot s consolidation problem in porous media a least squares mixed finite element method for the coupled problem of flow and deformation is presented and analyzed in this paper for the analysis we restrict ourselves to fully saturated conditions for the flow process and to a linearly elastic material law for the deformation process this is known in the literature as biot s consolidation problem for simplicity the analysis is presented for the problem in two space dimensions our least squares approach is motivated by the fact that all process variables i e fluid pressure and flux as well as displacement field and stress tensor are approximated directly by suitable finite element spaces ellipticity of the corresponding variational formulation is proven for the stationary case as well as for the subproblems arising at each step of an implicit time discretization in the general time dependent case standard h1 conforming piecewise linear and quadratic finite elements are used for the fluid pressure and for each component of the displacement respectively for the flux and stress components the h div conforming raviart thomas spaces of lowest order are used computational results are presented for some two dimensional test problems

a posteriori error estimation for variational problems with uniformly convex functionals the objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations we consider variational problems of the form inf f v g v where f v r is a convex lower semicontinuous functional g y r is a uniformly convex functional v and y are reflexive banach spaces and v y is a bounded linear operator we show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and thus can be justified via the duality theory

bounded positive entire solutions of singular quasilinear elliptic equations abstract suppose that 0 is a constant and that f r n r r n r is a continuous function with r 0 this paper investigates n dimensional singular quasilinear elliptic equations of the form u f x u u u 0 x r n n 3 and gives some sufficient conditions for the equations to have i a decaying positive entire solution or ii infinitely many positive entire solutions each of which is bounded with a positive lower bound

existence and optimal estimates of solutions for singular nonlinear dirichlet problems abstract by applying the properties of the unique classical solution to the singular boundary value problem on half line p s g p s p s 0 s 0 p 0 0 lims p s b 0 and constructing the new comparison functions they show the existence and the optimal global estimates of solutions to singular nonlinear dirichlet problems u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in rn g s is nonincreasing and positive in 0 1 g t dt and lim s 0 g s k c is positive in and may be singular or zero on the boundary

a singular boundary value problem for a degenerate elliptic pde abstract we consider a singular boundary value problem associated with the laplacian and having nonseparable inhomogeneous term we establish the existence of ground state solutions to such a problem in bounded domains as well as in the whole euclidean space the investigation is based on subsolution and supersolution methods and existence of principal eigenfunctions to the eigenvalue problem of dirichlet type corresponding to the laplacian

stability and convergence of sequential methods for coupled flow and geomechanics fixed stress and fixed strain splits abstract we analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics in which the flow problem is solved first we employ the von neumann and energy methods for linear and nonlinear problems respectively we consider two sequential methods with the generalized midpoint rule for tn where is the parameter of time discretization namely the fixed strain and fixed stress splits the von neumann method indicates that the fixed strain split is only conditionally stable and that its stability limit is a coupling strength less than unity if 0 5 on the other hand the fixed stress split is unconditionally stable when 0 5 the amplification factors of the fixed stress split are different from those of the undrained split and are identical to the fully coupled method unconditional stability of the fixed stress split is also obtained from the energy method for poroelastoplasticity we show that the fixed stress split is contractive and b stable when 0 5 we also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward euler method in time from the estimates the fixed strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable the fixed stress split however is convergent for a fixed number of iterations showing better accuracy than the undrained split even when we cannot obtain the exact local bulk modulus or exact rock compressibility at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials the fixed stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus such as the dimension based estimation by which the employed local bulk modulus is less stiff than the exact local bulk modulus we provide numerical examples supporting all the estimates of stability and convergence for the fixed strain and fixed stress splits

an a posteriori based fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media in this work we develop an a posteriori based adaptive algorithm for thermal multiphase compositional flows in porous media the key ingredients are fully computable a posteriori error estimates bounding the dual norm of the residual supplemented by a nonconformity evaluation term the theory hinges on assumptions that allow the application to variety of discretization methods the estimators are then elaborated to estimate separately the space time linearization and algebraic errors this additional information is used to formulate a fully adaptive algorithm including adaptive stopping criteria for iterative solvers as well as refinement derefinement criteria for both the time step and the mesh size numerical validation is provided on an industrial case study in the context of oil recovery based on the steam assisted gravity drainage procedure implicit cell centered finite volumes with phase upwind and two point discretization of the diffusive fluxes are considered it is shown that significant gains in computational cost can be achieved in this example without hindering the quality of the results as measured by quantities of engineering interest

an efficient diagonal preconditioner for finite element solution of biot s consolidation equations finite element simulations of very large scale soil structure interaction problems e g excavations tunnelling pile rafts etc typically involve the solution of a very large ill conditioned and indefinite biot system of equations the traditional preconditioned conjugate gradient solver coupled with the standard jacobi sj preconditioner can be very inefficient for this class of problems this paper presents a robust generalized jacobi gj preconditioner that is extremely effective for solving very large scale biot s finite element equations using the symmetric quasi minimal residual method the gj preconditioner can be formed inverted and implemented within an element by element framework as readily as the sj preconditioner it was derived as a diagonal approximation to a theoretical form which can be proven mathematically to possess an attractive eigenvalue clustering property the effectiveness of the gj preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three dimensional footing problem this paper casts a new perspective on the potentialities of the simple diagonal preconditioner which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix copyright 2002 john wiley sons ltd

comment on nonlocal third order shear deformation plate theory with application to bending and vibration of plates by r aghababaei and j n reddy journal of sound and vibration 326 2009 277 289 abstract this paper aims to prove the inaccuracy of the navier solution presented by aghababaei and reddy 1 for the bending analysis of nanoplates based on the nonlocal theory of eringen according to the derived relations for bending of the nonlocal plate model the main cause of this inaccuracy is attributed to an incorrect approximation of the navier solution for a uniform transverse load of course this problem does not inherently occur for the navier solution in cases such as free vibration or the buckling of a nonlocal plate model in which the amount of transverse load is zero in order to obtain further verification the results reported based on the navier solution by aghababaei and reddy 2009 1 for the bending analysis of a nanoplate are compared with those computed by the differential quadrature dq and finite difference fd methods as shown the results obtained by both the fd and dq methods are consistently alike and unlike the solutions reported by aghababaei and reddy 2009 1 they are independent from small scale effect

on singular semilinear elliptic equations abstract for the semilinear elliptic equation u p x u 0 x r n n 3 0 we show via the barrier method the existence of a positive entire solution behaving like x 2 n near

stability and convergence of sequential methods for coupled flow and geomechanics drained and undrained splits abstract we perform a stability and convergence analysis of sequential methods for coupled flow and geomechanics in which the mechanics sub problem is solved first we consider slow deformations so that inertia is negligible and the mechanical problem is governed by an elliptic equation we use biot s self consistent theory to obtain the classical parabolic type flow problem we use a generalized midpoint rule parameter between 0 and 1 time discretization and consider two classical sequential methods the drained and undrained splits the von neumann method provides sharp stability estimates for the linear poroelasticity problem the drained split with backward euler time discretization 1 is conditionally stable and its stability depends only on the coupling strength and it is independent of time step size the drained split with the midpoint rule 0 5 is unconditionally unstable the mixed time discretization with 1 0 for mechanics and 0 5 for flow has the same stability properties as the backward euler scheme the von neumann method indicates that the undrained split is unconditionally stable when 0 5 we extend the stability analysis to the nonlinear regime poro elastoplasticity via the energy method it is well known that the drained split does not inherit the contractivity property of the continuum problem thereby precluding unconditional stability for the undrained split we show that it is b stable therefore unconditionally stable at the algorithmic level when 0 5 we also analyze convergence of the drained and undrained splits and derive the a priori error estimates from matrix algebra and spectral analysis we show that the drained split with a fixed number of iterations is not convergent even when it is stable the undrained split with a fixed number of iterations is convergent for a compressible system i e finite biot modulus for a nearly incompressible system i e very large biot modulus the undrained split loses first order accuracy and becomes non convergent in time we also study the rate of convergence of both splits when they are used in a fully iterated sequential scheme when the medium permeability is high or the time step size is large which corresponds to a high diffusion of pressure the error amplification of the drained split is lower and therefore converges faster than the undrained split the situation is reversed in the case of low permeability and small time step size we provide numerical experiments supporting all the stability and convergence estimates of the drained and undrained splits in the linear and nonlinear regimes we also show that our spatial discretization finite volumes for flow and finite elements for mechanics removes the well documented spurious instability in consolidation problems at early times

the existence and non existence of positive solutions to a singular quasilinear elliptic problem in rn abstract this paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation p u a x g u b x f u in r n 1 p n n 3 with u x 0 as x here either g or f or both of them are singular at 0 in the sense that g t f t as t 0 by using a perturbation method which eliminates the singularity on a ball with radius r and then let r tends to infinity with help of the bounded super solution of the original problem we obtain the existence of a weak solution of the problem the main results of this paper improve the corresponding results of dragos patru covei the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in r n nonlinear anal 69 2008 2615 2622 j v goncalves c a santos singular elliptic problems existence non existence and boundary behavior nonlinear anal 66 2007 2078 2090

convergence analysis of multirate fixed stress split iterative schemes for coupling flow with geomechanics abstract we consider multirate iterative schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium the multirate iterative coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we adapt the fixed stress split algorithm that decouples the flow and mechanics equations for the multirate case and perform an iteration between the two problems until convergence we provide a fully discrete scheme that uses backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics our analysis is based on studying the equations satisfied by the difference of iterates and using banach contraction argument to prove that the corresponding scheme is a fixed point contraction the analysis provides the value of an adjustable coefficient used in the proposed iterative coupling algorithms furthermore we show that the converged quantities satisfy the variational weak form for the coupled discrete system

a boundary value problem for normalized finsler infinity laplacian equations with singular nonhomogeneous terms abstract we study boundary value problems of finsler infinity laplacian equations with nonhomogeneous terms that may exhibit singularity when solutions vanish on the boundary

on stability and convergence of finite element approximations of biot s consolidation problem stability and convergence analysis of finite element approximations of biot s equations governing quasistatic consolidation of saturated porous media are discussed a family of decay functions parametrized by the number of time steps is derived for the fully discrete backward euler galerkin formulation showing that the pore pressure oscillations arising from an unstable approximation of the incompressibility constraint on the initial condition decay in time error estimates holding over the unbounded time domain for both semidiscrete and fully discrete formulations are presented and a post processing technique is employed to improve the pore pressure accuracy

a remark on the existence of entire solutions of a singular semilinear elliptic problem it is proved that the singular semilinear elliptic equation u p x g u 0 p x x rn 2 n lims 0 g s andg c1 0 0 which is strictly decreasing in 0 has a unique positivec2 loc rn solution that decays to zero near provided 0 t t dt where t imax i x tg x

a sequential adaptive strategy in space time with application to consolidation of porous media issues related to space time adaptivity for a class of nonlinear and time dependent problems are discussed the dg k methods are adopted for the time integration and the a posteriori error control is based on the appropriate dual problem in space time one key ingredient is to decouple the error generation in space and time with a hierarchical decomposition of the discrete space of dual solutions the main idea put forward in the paper is to increase the computational efficiency of the adaptive scheme by avoiding recursive adaptations of the whole time mesh rather the space mesh and the time step defining each finite space time slab are defined in a truly sequential fashion the proposed adaptive strategy is applied to the coupled consolidation problem in geomechanics involving large deformations its performance is investigated with the aid of a numerical example in 2d