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

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

a local refinement finite element method for one dimensional parabolic systems vector systems of parabolic partial differential equations in one space dimension are solved by an adaptive local mesh refinement galerkin finite element procedure piecewise linear polynomials are used for the spatial representation of the solution and the backward euler method is used for temporal integration a local error indicator based on an estimate of the local discretization error is used to control an adaptive feedback mesh refinement strategy where finer space time meshes are recursively added to coarser ones in regions where greater solution resolution is needed a posteriori estimates of the local discretization error are obtained by a p refinement procedure that uses piecewise quadratic hierarchic finite element approximations in space and trapezoidal rule integration in time superconvergence properties of the finite element method are used to neglect errors at nodes and thereby increase the computational efficiency of the error estimation procedure further improvements in computational e

galerkin runge kutta discretizations for parabolic equations with time dependent coefficients a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients unlike any classical counterpart this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction in support of this claim error estimates are proved and computational results are presented additionally since the time stepping equations involve coefficient matrices changing at each time step a preconditioned iterative technique is used to solve the linear systems only approximately nevertheless the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

parameterized approximate block lu preconditioners for generalized saddle point problems abstract in this paper we are concerned with the iteration solution of generalized saddle point problems based on the exact block lu factorization of the coefficient matrix we construct a class of parameterized approximate block lu factorization preconditioners which rely on suitable approximations of the schur complement of the 1 1 block of the coefficient matrix convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced algebraic characteristics of the related preconditioned matrices are discussed including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial the established results extend those of the approximate factorization and variants of the hermitian and skew hermitian splitting and positive and skew hermitian splitting preconditioners for saddle point problems numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners

temporal error control for convection dominated equations in two space dimensions a new time integration strategy for the solution of convection dominated partial differential equations in two space dimensions by the method of lines is presented the strategy aims to ensure that the time integration error is less than the spatial discretisation error this is achieved by making use of the individual contributions of the local spatial discretisation error and the local time integration error to the global error in the numerical solution numerical results are used to illustrate the performance of this strategy

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

block preconditioners for symmetric indefinite linear systems this paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a krylov subspace method for solving symmetric indefinite linear systems the focus is on large scale real world problems where block approximations are a practical necessity the main illustration is the performance of the block diagonal constrained and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large scale finite element discretization of biot s consolidation equations this system of equations is of fundamental importance to geomechanics numerical studies show that simple diagonal approximations to the 1 1 block k and inexpensive approximations to the schur complement matrix s may not always produce the most spectacular time savings when k is explicitly available but is able to deliver reasonably good results on a consistent basis in addition the block diagonal preconditioner with a negative 2 2 block appears to be reasonably competitive when compared to the more complicated ones these observation are expected to remain valid for coefficient matrices whereby the 1 1 block is sparse diagonally significant a notion weaker than diagonal dominance moderately well conditioned and has a much larger block size than the 2 2 block copyright 2004 john wiley sons ltd

multi level hp adaptivity for cohesive fracture modeling summary discretization induced oscillations in the load displacement curve are a well known problem for simulations of cohesive crack growth with finite elements the problem results from an insufficient resolution of the complex stress state within the cohesive zone ahead of the crack tip this work demonstrates that the hp version of the finite element method is ideally suited to resolve this complex and localized solution characteristic with high accuracy and low computational effort to this end we formulate a local and hierarchic mesh refinement scheme that follows dynamically the propagating crack tip in this way the usually applied static a priori mesh refinement along the complete potential crack path is avoided which significantly reduces the size of the numerical problem studying systematically the influence of h refinement p refinement and hp refinement we demonstrate why the suggested hp formulation allows to capture accurately the complex stress state at the crack front preventing artificial snap through and snap back effects this allows to decrease significantly the number of degrees of freedom and the simulation runtime furthermore we show that by combining this idea with the finite cell method the crack propagation within complex domains can be simulated efficiently without resolving the geometry by the mesh copyright 2016 john wiley sons ltd

a moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

moving finite element methods for evolutionary problems i theory abstract in this paper the first of two on the subject we present a unified approach to moving and fixed finite element methods for evolutionary problems in terms of projections the central theoretical results are concerned with moving finite elements for one dimensional scalar problems particularly hyperbolic equations with shocks but the viewpoint extends to general systems in any number of dimensions

energy stable numerical methods for hyperbolic partial differential equations using overlapping domain decomposition overlapping domain decomposition methods otherwise known as overset grid or chimera methods are useful for simplifying the discretization of partial differential equations in or around complex geometries though in wide use such methods are prone to numerical instability unless numerical diffusion or some other form of regularization is used especially for higher order methods to address this shortcoming high order provably energy stable overlapping domain decomposition methods are derived for hyperbolic initial boundary value problems the overlap is treated by splitting the domain into pieces and using generalized summation by parts derivative operators and polynomial interpolation new implicit and explicit operators are derived that do not require regularization for stability in the linear limit applications to linear and nonlinear problems in one and two dimensions are presented where it is found the explicit operators are preferred to the implicit ones

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

nonexcellence of multiquadratic field extensions abstract let k0 be a field chark0 2 n 2 a b 1 b n k 0 such that the elements a b 1 b n k 0 k 0 2 are linearly independent we show that there exists a field extension f k0 and an anisotropic 4 dimensional form over f such that d a the form f b 1 b n is isotropic and the form f b 1 b n an is not defined over f

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

the arbitrary local mesh replacement method an alternative to remeshing for crack propagation analysis a finite element based method is proposed to accommodate the arbitrary motion of a crack in the context of general two dimensional configurations of loading and body geometry the primary goal is to render consistently accurate simulations of crack propagation while maintaining a high degree of computational efficiency and without an undue burden on the analyst the essence of the method here called the arbitrary local mesh replacement method is the replacement of the finite element interpolant in the vicinity of the crack tip with one that is derived from a moving mesh patch the boundary of the patch is not required to be coincident with background mesh element edges accordingly a weak statement of compatibility is enforced on the interface between the two meshes thereby introducing a small set of additional force type unknowns also partial elements are manifest near the patch boundary a special numerical integration rule is devised for their treatment

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

moving finite element methods for evolutionary problems ii applications abstract in this the second of two papers on the subject we present applications of the moving finite element method to a number of test problems key features are linear elements a direct approach to parallelism and node overtaking avoiding penalty functions rapid inversion of the mass matrix by preconditioned conjugate gradients and explicit euler time stepping the resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy the paper includes a substantial section on changes of dependent variable and front tracking techniques for non linear diffusion problems test problems include non linear hyperbolic conservation laws and non linear parabolic equations in one and two dimensions

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

a contact finite element formulation for biological soft hydrated tissues as part of the highly complex numerical simulation of human joints a finite element formulation for contact between layers of articular cartilage has been developed a biphasic description of the cartilage is used where the material is considered to be a porous permeable solid matrix with fluid flowing through it governing equations and contact boundary conditions are summarized for this material before developing the finite element formulation from a galerkin weighted residual statement lagrange multipliers are introduced to enforce the two kinetic continuity equations across the contact surface and an iterative scheme is employed to determine the current contact area numerical examples are presented which demonstrate both that the appropriate continuity across the contact surface is enforced and that the contact surface definition is accurate

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

second order finite element approximations and a posteriori error estimation for two dimensional parabolic systems we discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two dimensional rectangular regions the partial differential system is discretized in space using a galerkin approach with piecewise eight node serendipity functions an a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems the error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented

a local space time adaptive scheme in solving two dimensional parabolic problems based on domain decomposition methods we describe a local space time adaptive refinement and derefinement method for solving systems of parabolic partial differential equations solutions are calculated on local space time meshes using a domain decomposition finite element algorithm in space and an extrapolation algorithm in time an a posteriori spatial error estimate controls a local spatial refinement strategy and an a posteriori temporal error estimate monitors temporal refinement and order of the integration a multiplicative schwarz algorithm is used to coordinate solutions between overlapping grids we use nearly minimal fine meshes which follow the dynamics of the problems in an efficient manner

combined hermite spectral finite difference method for the fokker planck equation the convergence of a class of combined spectral finite difference methods using hermite basis applied to the fokker planck equation is studied it is shown that the hermite based spectral methods are convergent with spectral accuracy in weighted sobolev space numerical results indicating the spectral convergence rate are presented a velocity scaling factor is used in the hermite basis and is shown to improve the accuracy and effectiveness of the hermite spectral approximation with no increase in workload some basic analysis for the selection of the scaling factors is also presented

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

stability analysis of interface temporal discretization in grid overlapping methods we investigate the stability of a temporal discretization of interface terms in grid overlapping methods a matrix stability analysis is performed on a model problem of the one dimensional diffusion equation on overlapping grids the scheme stability is first analyzed theoretically and a proof of the unconditional stability of the first order interface extrapolation scheme with the first and second order time integration for any overlap size is presented for the higher order schemes we obtain explicit estimates of the spectral radius of the corresponding discrete matrix operator and document the values of the stability threshold depending on the number of grid points and the size of overlap the influence of iterations on stability properties is also investigated numerical experiments are then presented relating the obtained stability bounds to the observed numerical values semidiscrete analysis confirms the derived scaling for the stability bounds

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

optimal numerical parameterization of discontinuous galerkin method applied to wave propagation problems this paper deals with the high order discontinuous galerkin dg method for solving wave propagation problems first we develop a one dimensional dg scheme and numerically compute dissipation and dispersion errors for various polynomial orders an optimal combination of time stepping scheme together with the high order dg spatial scheme is presented it is shown that using a time stepping scheme with the same formal accuracy as the dg scheme is too expensive for the range of wave numbers that is relevant for practical applications an efficient implementation of a high order dg method in three dimensions is presented using 1d convergence results we further show how to adequately choose elementary polynomial orders in order to equi distribute a priori the discretization error we also show a straightforward manner to allow variable polynomial orders in a dg scheme we finally propose some numerical examples in the field of aero acoustics

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

a numerical approach to study the kramers equation for finite geometries boundary conditions and potential fields the kramers equation for the phase space function which models the dynamics of an underdamped brownian particle is the subject of our study numerical solutions of this equation for natural boundaries unconfined geometries have been well reported in the literature but not much has been done on the kramers equation for finite confining geometries which require a set of additional constraints imposed on the phase space function at physical boundaries in this paper we present numerical solutions for the kramers equation with a variety of potential fields namely constant linear harmonic and periodic in the presence of fully absorbing and fully reflecting boundary conditions bcs the choice of the numerical method and its implementation take into consideration the type of bcs in order to avoid the use of ghost points or artificial conditions we study and assess the conditions under which the numerical method converges various aspects of the solutions for the phase space function are presented with figures and discussed in detail