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

lower bounds for three dimensional nonoverlapping domain decomposition algorithms lower bounds for the condition numbers of the preconditioned systems are obtained for the wire basket preconditioner and the neumann neumann preconditioner in three dimensions they show that the known upper bounds are sharp

a preconditioner with inexact element face solver for three dimensional p version finite element methods in this paper we propose to modify a preconditioner developed in pavarino and widlund comput math appl 33 1997 193 for three dimensional p version finite element or spectral element approximations by solving the sub problems on element faces with an inexact element face solver such a modification reduces the cost to evaluate the action of the preconditioner from o p4 to o p3 per element face where p is the polynomial degree used for approximation furthermore it is shown that the coefficient matrix of the inexact solver is spectrally equivalent to the original stiffness matrix on element faces therefore such a change will not affect the polylogarithmic estimate given in pavarino and widlund for the condition number of the preconditioned system numerical results are also given to confirm the theoretical analysis

nonoverlapping domain decomposition methods with a simple coarse space for elliptic problems we propose a substructuring preconditioner for solving three dimensional elliptic equations with strongly discontinuous coefficients the new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by bramble pasiack and schatz 1989 but with much simpler coarse solvers though the condition number of the preconditioned system may not have a good bound we are able to show that the convergence rate of the pcg method with such substructuring preconditioner is nearly optimal and also robust with respect to the possibly large jumps of the coefficient in the elliptic equation

critical dynamics of self organizing eulerian walkers the model of self organizing eulerian walkers is numerically investigated on the square lattice the critical exponents for the distribution of a number of steps tau l and visited sites tau s characterizing the process of transformation from one recurrent configuration to another are calculated using the finite size scaling analysis two different kinds of dynamical rules are considered the results of simulations show that both the versions of the model belong to the same class of universality with the critical exponents tau l tau s 1 75 pm 0 1

iterative substructuring methods for spectral element discretizations of elliptic systems ii mixed methods for linear elasticity and stokes flow iterative substructuring methods are introduced and analyzed for saddle point problems with a penalty term two examples of saddle point problems are considered the mixed formulation of the linear elasticity system and the generalized stokes system in three dimensions these problems are discretized with mixed spectral element methods the resulting stiffness matrices are symmetric and indefinite the interior unknowns of each element are first implicitly eliminated by using exact local solvers the resulting saddle point schur complement is solved with a krylov space method with block preconditioners the velocity block can be approximated by a domain decomposition method e g of wire basket type which is constructed from a local solver for each face of the elements and a coarse solver related to the wire basket of the elements the condition number of the preconditioned operator is independent of the number of spectral elements and is bounded from above by the product of the square of the logarithm of the spectral degree and the inverse of the discrete inf sup constant of the problem

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

incorporation of multipoint constraints into the balancing domain decomposition method and its parallel implementation the domain decomposition method ddm is a major solution algorithm that is used to parallelize the finite element method in the case of implicit structural analysis using the ddm the substructuring based iterative linear solver is a powerful tool when an effective preconditioner such as the balancing domain decomposition bdd method is used in the present study a method by which to incorporate a set of linear multipoint constraints mpc into the bdd method is proposed in this method when an mpc is enforced on the internal degrees of freedom dofs in some subdomains the dofs are converted into interface dofs that is all of the dofs constrained by mpcs become interface dofs then the interface problem with the set of mpcs for the interface dofs is solved by the conjugate projected gradient method in order to combine the above procedure with the preconditioner used in the bdd method the effect of the mpcs for the interface dofs is also imposed on the coarse grid problem of the bdd method using the penalty method a parallel implementation of the present method is also described some illustrative examples are solved and good convergence and parallel performance of the present method are demonstrated copyright 2006 john wiley sons ltd

dual primal feti methods for three dimensional elliptic problems with heterogeneous coefficients in this paper certain iterative substructuring methods with lagrange multipliers are considered for elliptic problems in three dimensions the algorithms belong to the family of dual primal finite element tearing and interconnecting feti methods which recently have been introduced and analyzed successfully for elliptic problems in the plane the family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual primal preconditioner it is shown that the condition numbers of several of the dual primal feti methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains the mesh size and jumps in the coefficients these results closely parallel those of other successful iterative substructuring methods of primal as well as dual type

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

a fractional step solution method for the unsteady incompressible navier stokes equations in generalized coordinate systems abstract a fractional step method is developed for solving the time dependent three dimensional incompressible navier stokes equations in generalized coordinate systems the primitive variable formulation uses the pressure defined at the center of the computational cell and the volume fluxes across the faces of the cells as the dependent variables instead of the cartesian components of the velocity this choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell the governing equations are discretized by finite volumes using a staggered mesh system the solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a poisson equation this procedure combined with a consistent approximation of the geometric quantities is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain the favorable convergence properties of the poisson solver the momentum equations are solved by an approximate factorization method and a novel zebra scheme with four color ordering is devised for the efficient solution of the poisson equation several two and three dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method good agreement is obtained in all cases

schwarz methods with local refinement for the p version finite element method in some applications the accuracy of the numerical solution of an elliptic problem needs to be increased only in certain parts of the domain in this paper local refinement is introduced for an overlapping additive schwarz algorithm for the p version finite element method both uniform and variable degree refinements are considered the resulting algorithm is highly parallel and scalable

deflation and projection methods applied to symmetric positive semi definite systems linear systems with a singular symmetric positive semi definite matrix appear frequently in practice this usually does not lead to difficulties for cg methods as long as these systems are consistent however the construction of a preconditioner especially the construction of two level and multilevel methods becomes more complicated since singular coarse grid matrices or galerkin matrices may occur here we continue the work started in 21 22 where deflation is used for some special singular coefficient matrices here we show that deflation and other projection type preconditioners can be applied to arbitrary singular problems without any difficulties in each of these methods a two level preconditioner is involved where coarse grid systems based on a singular galerkin matrix should be solved we prove that each projection operator consisting of a singular galerkin matrix can be written as an operator with a nonsingular galerkin matrix therefore many results that hold for nonsingular galerkin matrices are also valid for problems with singular galerkin matrices

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

an additive schwarz preconditioner for the feti method a new additive schwarz preconditioner for the finite element tearing and interconnecting feti method is analyzed in this paper this preconditioner has the unique feature that the coefficient matrix of its coarse grid problem is mesh independent for a model second order heterogeneous elliptic boundary value problem in two dimensions the condition number of the preconditioned system is shown to be bounded by c 1 ln h h 2 where h is the mesh size h is the typical diameter of the subdomains and the constant c is independent of h h the number of subdomains and the coefficients of the boundary value problem

a robin type domain decomposition method with red black partition in this paper we present a new robin type nonoverlapping domain decomposition dd preconditioner the unknown variables to be solved in this preconditioned algebraic system are the robin transmission data on the interface which are different from the well known dd methods like substructuring nonoverlapping dd method and feti method through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate for the second order elliptic problem we prove that the condition number of the preconditioned system is c 1 log frac h h 2 where h is the coarse mesh size and h is the fine mesh size moreover in this paper we always assume that there is a red black partition for the whole domain omega numerical results are given to illustrate the efficiency of our dd preconditioner

balancing domain decomposition for nonconforming plate elements in this paper the balancing domain decomposition method is extended to nonconforming plate elements the condition number of the preconditioned system is shown to be bounded by c big 1 ln h h big 2 where h measures the diameters of the subdomains h is the mesh size of the triangulation and the constant c is independent of h h and the number of subdomains

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

explicit runge kutta method for three dimensional internal incompressible flows a computer code has been developed to obtain the steady state solutions for three dimensional laminar incompressible flow governed by the navier stokes equations a central difference finite volume formulation with an explicit one step multistage runge kutta time stepping scheme for the primitive variables i e velocity components and pressure is used the preconditioning matrix method which is a generalized version of the artificial compressibility method is used to construct a pressure equation from the continuity equation fourth order artificial dissipation is used to suppress high frequency oscillations

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

adaptive selection of face coarse degrees of freedom in the bddc and the feti dp iterative substructuring methods we propose a class of method for the adaptive selection of the coarse space of the bddc and feti dp iterative substructuring methods the methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures it is assumed that the starting coarse degrees of freedom are already sufficientto prevent relative rigid body motions in any selected pair of adjacent substructures a heuristic indicator of the the condition number is developed and a minimal number of coarse degrees of freedom is added to decrease the indicator under a given threshold it is shown numerically on 2d elasticity problems that the indicator based on pairs of substructures with common edges predicts the actual condition number reasonably well and that the method can select adaptively the hard part of the problem and concentrate computational work there to achieve good convergence of the iterations at a modest cost

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

stable numerical method applying a total variation diminishing scheme for incompressible flow to solve incompressible navier stokes equations a stable finite difference method applying a total variation diminishing tvd upwind scheme originally designed for compressible flow is proposed the validity and the receptivity of this method employing such a tvd scheme for application to incompressible flow analysis are investigated by the evaluation of the accuracy stability and convergence rate using the generalized implicit simplified marker and cell smac solution method combined with the tvd scheme three dimensional developing entry flows through a square curved duct are calculated and compared with available experimental data and some computational results obtained by quadratic upstream interpolations for the convective kinematics quick schemes as well as by third order upwind schemes

two level domain decomposition preconditioning for the p version finite element method in three dimensions this paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain convergence factors can be bounded using properties of the local stiffness matrices only the method is applied to the p version finite element method for three dimensional elasticity we treat each element as a subdomain and compute explicit convergence bounds

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

additive schwarz methods for elliptic mortar finite element problems two variants of the additive schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed the methods are defined on subdomains without overlap and they use special coarse spaces resulting in algorithms that are well suited for parallel computation the condition number estimate for the preconditioned system in each method is proportional to the ratio h h where h and h are the mesh sizes and it is independent of discontinuous jumps of the coefficients for one of the methods presented the choice of the mortar nonmortar side is independent of the coefficients

a comparison of deflation and the balancing preconditioner in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare the well known balancing preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always i e for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the balancing preconditioner even more we establish that both preconditioners lead to almost the same spectra the zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used moreover we prove that the a norm of the errors of the iterates built by the deflation preconditioner is always below the a norm of the errors of the iterates built by the balancing preconditioner depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system if the amount of work is equal both preconditioners are sensitive with respect to inexact computations finally we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors numerical results for porous media flows emphasize the theoretical results

algebraic theory of multiplicative schwarz methods the convergence of multiplicative schwarz type methods for solving linear systems when the coefficient matrix is either a nonsingular m matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings the effect on convergence of algorithmic parameters such as the number of subdomains the amount of overlap the result of inexact local solves and of coarse grid corrections global coarse solves is analyzed in an algebraic setting results on algebraic additive schwarz are also included

unstructured additive schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients this paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes the effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a a1 ad and then considering certain positive coefficient sequences a m in which some of these values approach 0 or infty as m rightarrow infty we consider the performance of additive schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid we show that the preconditioned conjugate gradient method converges in both the energy and the euclidean norms with a number of iterations which grows only logarithmically in the size of the maximum jump cal j m max a k m a l m k l 1 ldots d as m rightarrow infty the result is obtained by a careful analysis of the preconditioned matrix it cannot be obtained by the usual procedure of estimating condition numbers a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in cal j m recent results of chan smith and zou have shown that by using this preconditioner together with the conjugate gradient method the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh our results now show that this method is also highly resilient to discontinuous coefficients even if no attention is paid to the coefficient discontinuity in the construction of the solver

dual primal feti methods for linear elasticity dual primal feti methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations as in primal iterative substructuring methods while most of the constraints are enforced by lagrange multipliers as in one level feti methods these methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems the purpose of this article is to develop strategies for selecting these constraints which are enforced throughout the iterations such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them the algorithms are described in terms of a change of basis that has proven to be quite robust in practice a theoretical analysis is provided for the case of linear elasticity and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the young s modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain the strategies have already proven quite successful in large scale implementations of these iterative methods 2006 wiley periodicals inc

avoiding singular coarse grid systems abstract here we consider the iterative solution of linear systems of equations with a symmetric positive semidefinite system matrix if multilevel methods in combination with krylov subspace methods are used to find the solution of these systems often singular subsystems or coarse grid systems have to be solved then the moore penrose inverse of the coarse grid systems can be used here we establish some theoretical techniques how to avoid the singularity of the coarse grid system while the resulting operator remains the same as we would have used the moore penrose inverse one option is to delete specific columns of the restriction and prolongation operator the other option is to perturb the system matrix

uniform convergence of the schwarz alternating method for solving singularly perturbed advection diffusion equations our purpose in this paper is to study the convergence rate of some variants of the schwarz alternating method for solving two types of singularly perturbed advection diffusion equations one in which the advection term first order term dominates and the other in which the lowest order term zeroth order term dominates for such problems boundary or other layer regions may be present for singular perturbation problems in which the advection dominates we choose two overlapping subregions one corresponding to an outer or elliptic region and the other corresponding to an inner or hyperbolic region our main result for this problem shows that if the subdomains can be chosen to follow the flow i e if the boundary interface of one of the subdomains corresponds to an outflow boundary for the streamlines of the flow then the schwarz iterates converge in the maximum norm with an error reduction factor per iteration that exponentially decays with increasing overlap or decreasing diffusion for the singular perturbation problem in which the lowest order term dominates we consider many overlapping subregions with no special requirement on outflow or inflow boundaries our main result here which is valid for two variants of the classical schwarz algorithm shows that the error reduction factor per iteration decays exponentially in the maximum norm with increasing overlap and increasing dominance of the zeroth order term for both problems our proofs rely on the maximum principle and the construction of barrier functions

balancing domain decomposition for problems with large jumps in coefficients the balancing domain decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent the abstract theory introduced recently by the first named author is used to develop condition number bounds for conforming linear elements in two and three dimensions the bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains and grow only as the squared logarithm of the mesh size h computational experiments for two and three dimensional problems confirm the theory

additive schwarz methods for the crouzeix raviart mortar finite element for elliptic problems with discontinuous coefficients in this paper we propose two variants of the additive schwarz method for the approximation of second order elliptic boundary value problems with discontinuous coefficients on nonmatching grids using the lowest order crouzeix raviart element for the discretization in each subdomain the overall discretization is based on the mortar technique for coupling nonmatching grids the convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements the condition number bound for the preconditioned systems is independent of the jumps of the coefficient and depend linearly on the ratio between the subdomain size and the mesh size the performance of the methods is illustrated by some numerical results

self organizing height arrow model numerical and analytical results the recently introduced self organizing height arrow ha model is studied numerically on the square lattice and analytically on the bethe lattice the concentration of occupied sites and critical exponents of distributions of avalanches are evaluated for two slightly different versions of the model the exponents found for distributions of avalanches by mass area duration and appropriate fractal dimensions are close to those for the original sandpile model which suggests that the ha model belongs to the same universality class for comparison the concentration of occupied sites in the ha model is calculated exactly on the bethe lattice with coordination number q 4

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

nonstandard coarse spaces and schwarz methods for elliptic problems with discontinuous coefficients using non conforming elements two level domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems a bound is established for the condition number of these iterative methods that grows only logarithmically with the number of degrees of freedom in each subregion this bound holds for two and three dimensions and is independent of jumps in the value of the coefficients and number of subregions we introduce face coarse spaces and isomorphisms to map between conforming and nonconforming spaces

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

new substructuring domain decomposition methods for advection diffusion equations abstract in this paper we consider a nonsymmetric elliptic problem and we use the techniques related to the steklov poincare operators to propose new substructuring iterative procedures in particular we propose two methods that generalize the well known neumann neumann and dirichlet neumann iterative procedures we prove that our methods that use symmetric and positive definite preconditioners lead to the construction of iterative schemes with optimal convergence properties numerical results for the finite element discretization are given

general three dimensional viscous primary secondary flow analysis generalized primary secondary flow equations are derived as an approximation to the navier stokes equations for three dimensional viscous flows with a dominant flow direction the primary secondary flow equations are well posed for solution by spatial marching and can be solved one to two orders of magnitude faster than the navier stokes equations a key element in the approximations which is a distinguishing feature of the present approach is that accuracy is related to curvature terms obtained from a local primary flow direction rather than the coordinate system used to describe geometry of the flowfield potential flow streamlines for the flow geometry under consideration are a suitable choice for the primary flow direction a sequentially decoupled implicit algorithm has been developed that exploits the form of the primary secondary flow equations to obtain decoupled subsets of equations through choices for dependent variables the sequence of equations and the linearization scheme each of the equation subsets is solved by efficient and appropriate implicit numerical procedures computed solutions for flow in 90 deg bends agree very well with experimental data and navierstokes solutions the combined efficiency and accuracy of the approximate equations and solution algorithm make this approach attractive for problems in which a suitable primary flow direction can be identified

a domain decomposition algorithm for elliptic problems in three dimensions most domain decomposition algorithms have been developed for problems in two dimensions one reason for this is the difficulty in devising a satisfactory easy to implement robust method of providing global communication of information for problems in three dimensions several methods that work well in two dimension do not perform satisfactorily in three dimensions a new iterative substructuring algorithm for three dimensions is proposed it is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain the condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains the new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions

restricted additive schwarz preconditioners with harmonic overlap for symmetric positive definite linear systems a restricted additive schwarz ras preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems in this paper we provide one level and two level extensions of ras for symmetric positive definite problems using the so called harmonic overlaps rasho both ras and rasho outperform their counterparts of the classical additive schwarz variants as the design of rasho is based on a much deeper understanding of the behavior of schwarz type methods in overlapping subregions and in the construction of the overlap in rasho the overlap is obtained by extending the nonoverlapping subdomains only in the directions that do not cut the boundaries of other subdomains and all functions are made harmonic in the overlapping regions as a result the subdomain problems in rasho are smaller than those of as and the communication cost is also smaller when implemented on distributed memory computers since the right hand sides of discrete harmonic systems are always zero and therefore do not need to be communicated we also show numerically that rasho preconditioned cg takes fewer iterations than the corresponding as preconditioned cg a nearly optimal theory is included for the convergence of rasho preconditioned cg for solving elliptic problems discretized with a finite element method

schwarz methods of neumann neumann type for three dimensional elliptic finite element problems several domain decomposition methods of neumann neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements we will only consider problems in three dimensions several new variants of the basic algorithm are introduced in a schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi level methods the neumann neumann algorithms have several advantages over other domain decomposition methods the subregions which define the subproblems only share the boundary degrees of freedom with their neighbors the subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries in its original form however the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases in the new variants of the algorithms considered in this paper the preconditioners include global components of low rank to overcome this difficulty bounds are established for the condition number of the iteration operator which are independent of the number of subregions and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions 1995 john wiley sons inc

explicit synchronous partitioned algorithms for interface problems based on lagrange multipliers abstract traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem as a result these schemes may require multiple subdomain solves acceleration techniques or optimized transmission conditions to achieve sufficient accuracy and or stability we present a new synchronous partitioned method derived from a well posed mixed finite element formulation of the coupled problem we transform the resulting differential algebraic equation dae to a hessenberg index 1 form in which the algebraic equation defines the lagrange multiplier as an implicit function of the states using this fact we eliminate the multiplier and reduce the dae to a system of explicit odes for the states explicit time integration both discretizes this system in time and decouples its equations as a result the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes we establish sufficient conditions for the formulation to be well posed and prove that classical mortar finite elements on the interface are a stable choice for the lagrange multiplier we show that in this case the condition number of the schur complement involved in the elimination of the multiplier is bounded by a constant the paper concludes with numerical examples illustrating the approach for two different interface problems

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

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

domain decomposition for variational optical flow computation we present an approach to parallel variational optical flow computation by using an arbitrary partition of the image plane and iteratively solving related local variational problems associated with each subdomain the approach is particularly suited for implementations on pc clusters because interprocess communication is minimized by restricting the exchange of data to a lower dimensional interface our mathematical formulation supports various generalizations to linear nonlinear convex variational approaches three dimensional image sequences spatiotemporal regularization and unstructured geometries and triangulations results concerning the effects of interface preconditioning as well as runtime and communication volume measurements on a pc cluster are presented our approach provides a major step toward real time two dimensional image processing using off the shelf pc hardware and facilitates the efficient application of variational approaches to large scale image processing problems

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

positive definite balancing neumann neumann preconditioners for nearly incompressible elasticity in this paper a positive definite balancing neumann neumann bnn solver for the linear elasticity system is constructed and analyzed the solver implicitly eliminates the interior degrees of freedom in each subdomain and solves iteratively the resulting schur complement involving only interface displacements using a bnn preconditioner based on the solution of a coarse elasticity problem and local elasticity problems with natural and essential boundary conditions while the schur complement becomes increasingly ill conditioned as the materials becomes almost incompressible the bnn preconditioned operator remains well conditioned the main theoretical result of the paper shows that the proposed bnn method is scalable and quasi optimal in the constant coefficient case this bound holds for material parameters arbitrarily close to the incompressible limit while this result is due to an underlying mixed formulation of the problem both the interface problem and the preconditioner are positive definite numerical results in two and three dimensions confirm these good convergence properties and the robustness of the methods with respect to the almost incompressibility of the material

multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions multilevel schwarz methods are developed for a conforming finite element approximation of second order elliptic problems we focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions we establish a condition number estimate for the iterative operator which is independent of the coefficients and grows at most as the square of the number of levels we also characterize a class of distributions of the coefficients called quasi monotone for which the weighted l 2 projection is stable and for which we can use the standard piecewise linear functions as a coarse space in this case we obtain optimal methods i e bounds which are independent of the number of levels and subregions we also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas we consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods

a numerical study on neumann neumann and feti methods for hp approximations on geometrically refined boundary layer meshes in two dimensions abstract we present extensive numerical tests showing the performance and robustness of certain balancing neumann neumann and one level feti methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions the numerical results confirm the theoretical results derived in a toselli x vasseur technical report 02 15 seminar fur angewandte mathematik eth zurich september 2002 the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients in addition they only grow polylogarithmically with the polynomial degree as in the case of p approximations on shape regular meshes

a neumann neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients neumann neumann algorithm have been well developed for standard finite element discretization of elliptic problems with discontinuous coefficients in this paper an algorithm of this kind is designed and analyzed for a mortar finite element discretization of problems in three dimensions it is established that its rate of convergence is independent of the discretization parameters and jumps of coefficients between subregions the algorithm is well suited for parallel computations

parallel adaptive subspace correction schemes with applications to elasticity in this paper we give a survey on the three main aspects of the efficient treatment of pdes i e adaptive discretization multilevel solution and parallelization we emphasize the abstract approach of subspace correction schemes and summarize its convergence theory then we give the main features of each of the three distinct topics and treat the historical background and modern developments furthermore we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic pdes and especially of linear elasticity problems we report on numerical experiments for the adaptive parallel multilevel solution of some test problems namely the poisson equation and lame s equation here we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute which is achieved through hash storage techniques and space filling curves

balancing neumann neumann methods for the cardiac bidomain model balancing neumann neumann preconditioners are constructed analyzed and numerically studied for the cardiac bidomain model in three dimensions this reaction diffusion system is discretized by low order finite elements in space and implicit explicit methods in time yielding very ill conditioned linear systems that must be solved at each time step the proposed algorithm is based on decomposing the domain into nonoverlapping subdomains and on solving iteratively the bidomain schur complement obtained by implicitly eliminating the degrees of freedom interior to each subdomain the iteration is preconditioned by a balancing neumann neumann method employing local neumann solves on each subdomain and a coarse bidomain solve a novel approach for the estimation of the average operator of the nonoverlapping decomposition provides a framework for designing coarse spaces for balancing neumann neumann methods the theoretical estimates obtained show that the proposed method is scalable quasi optimal and robust with respect to possible coefficient discontinuities of the bidomain operator the results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory

iterative substructuring methods for spectral element discretizations of elliptic systems i compressible linear elasticity an iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed the pure displacement formulation for compressible materials is discretized with the spectral element method the resulting stiffness matrix is symmetric and positive definite the proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse global solver related to the wire basket of the elements as in the scalar case the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

iterative substructuring methods for spectral elements problems in three dimensions based on numerical quadrature abstract iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems two preconditioners for p version finite element methods based on continuous piecewise qp functions are considered for second order elliptic problems in three dimensions these special methods can also be viewed as spectral element methods the first iterative method is designed for the galerkin formulation of the problem the second applies to linear systems for a discrete model derived by using gauss lobatto legendre quadrature for both methods it is established that the condition number of the relevant operator grows only in proportion to 1 log p 2 these bounds are independent of the number of elements into which the given region has been divided their diameters as well as the jumps in the coefficients of the elliptic equation between elements results of numerical computations are also given which provide upper bounds on the condition numbers as functions of p and which confirms the correctness of our theory

lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions lower bounds for the condition numbers of the preconditioned systems are obtained for the bramble pasciak schatz substructuring preconditioner and the neumann neumann preconditioner in two dimensions they show that the known upper bounds are sharp

convergence of a substructuring method with lagrange multipliers we analyze the convergence of a substructing iterative method with lagrange multipliers propose recently by farhat and roux the method decomposes finite element discretization of an elliptic boundary value problem into neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components for linear conforming elements and preconditioning by the dirichlet problems on the subdomains we prove the asymptotic bound on the condition number c 1 log h h alpha alpha 2 or 3 where h is the characteristic element size h subdomain size and c is independent of the number of subdomains

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd