dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

brief estimation and detection of unknown inputs using optimal fir filter the optimal finite impulse response fir filter is applied to systems with both unknown inputs and noise in order to obtain noise suppressed estimates of the unknown inputs the unknown inputs are modeled as random walk processes and estimated together with the state using the optimal fir filtering algorithm the estimate for each unknown input is shown to be independent of the window initial state and also of other unknown inputs a detection scheme for an unknown input is developed utilizing a test variable independently of other unknown inputs finally numerical examples are given to show the performance of the proposed estimation and detection method

robust feti dp methods for heterogeneous three dimensional elasticity problems iterative substructuring methods with lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses in particular results for algorithms belonging to the family of dual primal feti methods are presented the core issue of these algorithms is the construction of an appropriate global problem in order to obtain a robust method which converges independently of the material discontinuities in this article several necessary and sufficient conditions arising from the theory are numerically tested and confirmed furthermore results of numerical experiments are presented for situations which are not covered by the theory such as curved edges and material discontinuities not aligned with the interface and an attempt is made to develop rules for these cases

a hybrid discontinuous galerkin method for computing the ground state solution of bose einstein condensates a numerical method for computing the ground state solution of bose einstein condensates modeled by the gross pitaevskii equation is presented in this method the three dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions both polynomial and plane wave bases are considered for this purpose and lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution the ground state is computed by an iterative procedure for minimizing the energy the performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher order finite element method

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

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

a feti dp formulation of three dimensional elasticity problems with mortar discretization in this paper a feti dp formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered to resolve the nonconformity of the finite elements a mortar matching condition is imposed on the subdomain interfaces faces a feti dp algorithm is then built by enforcing the mortar matching condition in dual and primal ways in order to make the feti dp algorithm scalable a set of primal constraints which include average and momentum constraints over interfaces are selected from the mortar matching condition a condition number bound c 1 log h h 2 is then proved for the feti dp formulation for the elasticity problems with discontinuous material parameters only some faces need to be chosen as primal faces on which the average and momentum constraints are imposed numerical results are included

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

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

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

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

brief extension of minimum variance estimation for systems with unknown inputs in this paper we address the problem of minimum variance estimation for discrete time time varying stochastic systems with unknown inputs the objective is to construct an optimal filter in the general case where the unknown inputs affect both the stochastic model and the outputs it extends the results of darouach and zasadzinski automatica 33 1997 717 where the unknown inputs are only present in the model the main difficulty in treating this problem lies in the fact that the estimation error is correlated with the systems noises this fact leads generally to suboptimal filters necessary and sufficient conditions for the unbiasedness of this filter are established then conditions under which the estimation error and the system noises are uncorrelated are presented and an optimal estimator and a predictor filters are derived sufficient conditions for the existence of these filters are given and sufficient conditions for their stability are obtained for the time invariant case a numerical example is given in order to illustrate the proposed method

incorporation of linear multipoint constraints in domain decomposition based iterative solvers part ii blending feti dp and mortar methods and assembling floating substructures two approaches are presented for extending the feti dp domain decomposition based iterative method to the solution of problems with linear multipoint constraints lmpcs both cases of arbitrary lmpcs and lmpcs generated by the mortar method for enforcing a weak continuity of the solution across non matching finite element interfaces are considered in the first approach the lmpcs are addressed during the preconditioning step but enforced only at convergence in the second approach the lmpcs are enforced at each iteration through the solution of an auxiliary coarse problem both extended feti dp solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems the obtained results reveal that both solvers are numerically scalable and complement each other from the memory and cpu performance viewpoints

the schwarz alternating method in solid mechanics abstract we advance the schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics we prove that the schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well posed i e their corresponding energy density functions are quasi convex it is shown that the use of a newton type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems depending on the treatment of the dirichlet boundary conditions the first kind is a symmetric block diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain i e the off diagonal blocks are all zero and the coupling terms appear only on the right hand side the second kind is a nonsymmetric block system with off diagonal coupling terms several variants of the schwarz alternating method are proposed for the first kind of linear system including one in which the schwarz alternating iterations and the newton iterations are combined into a single scheme this version of the method is particularly attractive as it lends itself to a minimally intrusive implementation into existing finite element codes finally we demonstrate the performance of the proposed variants of the schwarz alternating method on several one dimensional and three dimensional examples

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

a scalable substructuring method by lagrange multipliers for plate bending problems we present a new lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems the proposed method is essentially an extension of the finite element tearing and interconnecting substructuring algorithm to the biharmonic equation the main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations the resulting method is proved to have a condition number that does not grow with the number of subdomains but rather grows at most polylogarithmically with the number of elements per subdomain these optimal properties hold for numerous plate bending elements that are used in practice including the hsieh clough tocher element the discrete kirchhoff triangle and a class of nonlocking elements for the reissner mindlin plate models computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method computational efficiency is also demonstrated with the numerical solution in 45 iterations and 105 seconds on a 64 processor ibm sp2 of a plate bending problem with almost one million degrees of freedom

simultaneous feti and block feti robust domain decomposition with multiple search directions domain decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities in particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner recently a robust approach named feti geneo was proposed where troublesome modes are precomputed and deflated from the interface problem the cost of the feti geneo is however high we propose in this paper techniques that share similar ideas with feti geneo but where no pre processing is needed and that can be easily and efficiently implemented as an alternative to standard domain decomposition methods in the block iterative approaches presented in this paper the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition those search directions originate either from the domain wise preconditioner in the simultaneous feti method or from the block structure of the right hand side of the interface problem block feti we show on 2d structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities even when the interface is jagged or when the domains have a bad aspect ratio the simultaneous feti was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular feti method

brief paper unbiased minimum variance state estimation for linear systems with unknown input the problem of state estimation for a linear system with unknown input which affects both the system and the output is discussed in this paper a recursive optimal filter with global optimality in the sense of unbiased minimum variance over all linear unbiased estimators is provided the necessary and sufficient condition for the convergence and stability is also given which is milder than existing approaches

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

high scalable non overlapping domain decomposition method using a direct method for finite element analysis abstract in this paper a novel domain decomposition method is introduced to solve a large finite element model in this method the decomposed domains do not overlap and are connected using a simple connective finite element which influences the nodal point equilibrium between adjacent finite elements this approach has the advantage that it allows use of a direct method such as gauss elimination even in a singular problem the singular stiffness matrices from the floating domain without the dirichlet boundary conditions are changed into invertible stiffness matrices by assembling the connective elements another advantage is that computational time and storage can be reduced by using a banded matrix in the direct solver in order to describe this proposed method we first review the feti method which is the most popular domain decomposition method then the proposed method is introduced with a technical approach in a distributed computer system finally the high scalability and computational efficiency of the proposed method are verified by comparing with the traditional feti method for 2d and 3d finite element examples which have floating subdomains in the result we demonstrate high scalability for a large finite element model

the two level feti method part ii extension to shell problems parallel implementation and performance results the two level feti method presented in part i of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems in this paper we extend the two level feti method to shell problems describe a variant approach revisit the preconditioning problem address the efficient implementation of the corresponding iterative solvers on massively parallel processors highlight the computational price of mathematical optimality and discuss its consequences and report some impressive performance results on two paragon xp s and ibm sp2 massively parallel processors for several realistic plate and shell problems

optimal solution of the two stage kalman estimator the two stage kalman estimator was originally proposed to reduce the computational complexity of the augmented state kalman filter it was also applied to the tracking of maneuvering targets by treating the target acceleration as a bias term except in certain restrictive conditions the conventional two stage estimators are suboptimal in the sense that they are not equivalent to the augmented state filter in this paper the authors propose a new two stage kalman estimator i e new structure which is an extension of friedland s estimator and is optimal in general conditions in addition we provide some analytic results to demonstrate the computational advantages of two stage estimators over augmented ones

a feti method with a mesh independent condition number for the iteration matrix abstract we introduce a framework for feti methods using ideas from the decomposition via lagrange multipliers of h 0 1 derived by raviart thomas p a raviart j m thomas primal hybrid finite element methods for second order elliptic equations math comput 31 138 1977 391 413 and complemented with the detailed work on polygonal domains developed by grisvard p grisvard singularities in boundary value problems rech math appl 22 1992 we compute the action of the lagrange multipliers using the natural h 00 1 2 scalar product therefore no consistency error appears our analysis allows to deal with cross points and floating subdomains in a natural manner as a byproduct we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning this result improves the standard asymptotic bound for this condition number given by 1 log h h 2 where h and h are the characteristic subdomain size and element size respectively shown by mandel tezaur in j mandel r tezaur convergence of a substructuring method with lagrange multipliers numer math 73 1996 473 487 numerical results that confirm our theoretical analysis are presented

optimal filtering for systems with unknown inputs an optimal filtering formula is derived for linear time varying discrete systems with unknown inputs by making use of the well known innovations filtering technique the derivation is an extension of a new observer design method for time invariant deterministic systems with unknown inputs the systems under consideration have the most general form the derived optimal filter has a similar form to the standard kalman filter with some modified covariance and gain matrices

numerical methods based on rational variable substitution for wiener hopf equations of the second kind this paper considers numerical methods for wiener hopf equations of the second kind y t 0 k t s y s ds g t 0 t by applying rational variable substitution to integrals on the semi infinite interval 0 and using the well known clenshaw curtis quadrature to the resulted integral we get a clenshaw curtis rational ccr quadrature rule we then apply the ccr quadrature to wiener hopf equations the reduction of singularities in the transformed equation is considered numerical examples are given to illustrate the efficiency of the numerical methods proposed in this paper

operational quadrature methods for wiener hopf integral equations we study the numerical solution of wiener hopf integral equations by a class of quadrature methods which lead to discrete wiener hopf equations with quadrature weights constructed from the fourier transform of the kernel or rather from the laplace transforms of the kernel halves as the analytical theory of wiener hopf equations is likewise based on the fourier transform of the kernel this approach enables us to obtain results on solvability and stability and error estimates for the discretization the discrete wiener hopf equations are solved by using an approximate wiener hopf factorization obtained with fft numerical experiments with the milne equation of radiative transfer are included

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

deflated preconditioned conjugate gradient solvers for linear elasticity extensions of deflation techniques previously developed for the poisson equation to static elasticity are presented compared to the scalar poisson equation j comput phys 2008 227 24 10196 10208 int j numer meth engng 2010 doi 10 1002 nme 2932 int j numer meth biomed engng 2010 26 1 73 85 the elasticity equations represent a system of equations giving rise to more complex low frequency modes multigrid elsevier amsterdam 2000 in particular the straightforward extension from the scalar case does not provide generally satisfactory convergence however a simple modification allows to recover the remarkable acceleration in convergence and cpu time reached in the scalar case numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in cpu time for grids with moderate graph depths compared to the non deflated version furthermore a monotonic decrease of iterations with increasing subdomains as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included copyright 2011 john wiley sons ltd

a preconditioned dual primal finite element tearing and interconnecting method for solving three dimensional time harmonic maxwell s equations a new preconditioned dual primal nonoverlapping domain decomposition method is proposed for the finite element solution of three dimensional large scale electromagnetic problems with the aid of two lagrange multipliers the new method converts the original volumetric problem to a surface problem by using a higher order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation similar to the previous version a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration which further increases the rate of convergence in addition a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable perfectly matched layers pmls are considered for the boundary truncation when solving open region problems the influence of the pml truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the pml numerical examples including wave propagation and antenna radiation problems truncated with pmls are presented to demonstrate the validity and the capability of this method

incorporation of linear multipoint constraints in substructure based iterative solvers part 1 a numerically scalable algorithm we consider the iterative solution by a class of substructuring methods of the large scale systems of equations arising from the nite element discretization of structural models with an arbitrary set of linear multipoint constraints we present a methodology for generalizing to such problems numerically scalable substructure based iterative solvers without interfering with their formulations and their well established local and global preconditioners we apply this methodology to the feti method and show that the resulting algorithm is numerically scalable with respect to both the substructure and problem sizes 1998 john wiley sons ltd

extending substructure based iterative solvers to multiple load and repeated analyses direct solvers currently dominate commercial finite element structural software but do not scale well in the fine granularity regime targeted by emerging parallel processors substructure based iterative solvers often called also domain decomposition algorithms lend themselves better to parallel processing but must overcome several obstacles before earning their place in general purpose structural analysis programs one such obstacle is the solution of systems with many or repeated right hand sides such systems arise for example in multiple load static analyses and in implicit linear dynamics computations direct solvers are well suited for these problems because after the system matrix has been factored the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions on the other hand iterative solvers in general are ill suited for these problems because they often must restart from scratch for every different right hand side in this paper we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides basically we formulate the overall problem as a series of minimization problems over k orthogonal and supplementary subspaces and tailor the preconditioned conjugate gradient algorithm to solve them efficiently the resulting solution method is scalable whereas direct factorization schemes and forward and backward substitution algorithms are not we illustrate the proposed methodology with the solution of static and dynamic structural problems and highlight its potential to outperform forward and backward substitutions on parallel computers as an example we show that for a linear structural dynamics problem with 11640 degrees of freedom every time step beyond time step 15 is solved in a single iteration and consumes 1 0 second on a 32 processor ipsc 860 system for the same problem and the same parallel processor a pair of forward backward substitutions at each step consumes 15 0 seconds

integral equations on the half line a modified finite section approximation on considere la solution approchee des equations integrales de la forme y t 0 k t s y s ds f t comportant des noyaux du type de wiener hopf on etablit la convergence uniforme de la solution approchee de section finie vers la solution on traite un exemple numerique

brief paper unbiased minimum variance input and state estimation for linear discrete time systems this paper addresses the problem of simultaneously estimating the state and the input of a linear discrete time system a recursive filter optimal in the minimum variance unbiased sense is developed where the estimation of the state and the input are interconnected the input estimate is obtained from the innovation by least squares estimation and the state estimation problem is transformed into a standard kalman filtering problem necessary and sufficient conditions for the existence of the filter are given and relations to earlier results are discussed

framework for state and unknown input estimation of linear time varying systems the design of unknown input decoupled observers and filters requires the assumption of an existence condition in the literature this paper addresses an unknown input filtering problem where the existence condition is not satisfied instead of designing a traditional unknown input decoupled filter a double model adaptive estimation approach is extended to solve the unknown input filtering problem it is proved that the state and the unknown inputs can be estimated and decoupled using the extended double model adaptive estimation approach without satisfying the existence condition numerical examples are presented in which the performance of the proposed approach is compared to methods from literature

a deluxe feti dp algorithm for a hybrid staggered discontinuous galerkin method for h curl elliptic problems summary convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered dg finite element approximation of h curl elliptic problems in two dimensions in addition to the advantages of staggered dg methods the basis functions of the new hybrid staggered dg method are all locally supported in the triangular elements and a lagrange multiplier approach is applied to enforce the global connections of these basis functions the interface problem on the lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem our algorithm is shown to give a condition number bound ofc 1c log h h 2 independent of the two parameters whereh his the number of triangles across each subdomain numerical results are included to confirm our theoretical bounds copyright 2013 john wiley sons ltd received 12 september 2013 accepted 11 november 2013 in this paper a dual and primal finite element tearing and interconnecting feti dp algorithm is developed for a fast and stable solution of a new hybrid staggered dg hsdg method applied to h curl problems staggered dg sdg methods were first introduced in 1 3 for wave propagation problems the idea was subsequently applied to other problems such as convection diffusion equations 4 electromagnetic problems 5 9 stokes equations 10 and multiscale wave simulations 11 12 similar to 5 the advantages of using sdg for h curl problems are the preservation of the structures of the differential operators the local conservation property and the optimal convergence in particular the discretizations of the two curl operators by the sdg method are adjoint to each other and the null space of the discrete curl operator is exactly the gradients moreover the divergence condition is automatically satisfied in an appropriate weak sense despite the aforementioned advantages the implementation of sdg methods requires careful numbering of the supports of the basis functions to allow an easier implementation of the sdg method the hsdg method is introduced in this paper one distinctive advantage of the hsdg method is that the basis functions are all locally supported in the triangular elements hence the numbering can be easily performed the fact that the basis functions are totally discontinuous requires some global couplings in particular we will use the lagrange multiplier approach to enforce the global connection of the basis functions this idea is also related to the hybridized dg methods 13 16 in fact it is shown that the sdg method is the limit of a hybridized dg method

analysis of feti methods for multiscale pdes in this paper we study a variant of the finite element tearing and interconnecting feti method which is suitable for elliptic pdes with highly heterogeneous multiscale coefficients x in particular coefficients with strong variation within subdomains and or jumps that are not aligned with the subdomain interfaces using energy minimisation and cut off arguments we can show rigorously that for an arbitrary positive coefficient function alpha in l infty omega the condition number of the preconditioned feti system can be bounded by c 1 log h h 2 where h is the subdomain diameter and h is the mesh size and where the function c depends only on the coefficient variation in the vicinity of subdomain interfaces in particular if alpha omega i varies only mildly in a layer i of width near the boundary of each of the subdomains i then c alpha mathcal o h eta 2 independent of the variation of in the remainder i i of each subdomain and independent of any jumps of across subdomain interfaces the quadratic dependence of c on h can be relaxed to a linear dependence under stronger assumptions on the behaviour of in the interior of the subdomains our theoretical findings are confirmed in numerical tests

deadbeat unknown input state estimation and input reconstruction for linear discrete time systems abstract this paper considers discrete time input reconstruction and state estimation assuming that the system has no invariant zeros without assuming that the initial condition is known and without assuming that at least one markov parameter has full column rank algorithms based on the generalized inverse of a block toeplitz matrix are given for unknown input state estimation and simultaneous input reconstruction and state estimation in both cases the unknown input is an arbitrary signal both algorithms are deadbeat which means that exact input reconstruction and state estimation are achieved in a finite number of steps

newton krylov bddc solvers for nonlinear cardiac mechanics abstract the aim of this work is to design and study a balancing domain decomposition by constraints bddc solver for the nonlinear elasticity system modeling the mechanical deformation of cardiac tissue the contraction relaxation process in the myocardium is induced by the generation and spread of the bioelectrical excitation throughout the tissue and it is mathematically described by the coupling of cardiac electro mechanical models consisting of systems of partial and ordinary differential equations in this study the discretization of the electro mechanical models is performed by q1 finite elements in space and semi implicit finite difference schemes in time leading to the solution of a large scale linear system for the bioelectrical potentials and a nonlinear system for the mechanical deformation at each time step of the simulation the parallel mechanical solver proposed in this paper consists in solving the nonlinear system with a newton krylov bddc method based on the parallel solution of local mechanical problems and a coarse problem for the so called primal unknowns three dimensional parallel numerical tests on different machines show that the proposed parallel solver is scalable in the number of subdomains quasi optimal in the ratio of subdomain to mesh sizes and robust with respect to tissue anisotropy

the two level feti method for static and dynamic plate problems part i an optimal iterative solver for biharmonic systems abstract we present a lagrange multiplier based substructuring method for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems the proposed method is essentially an extension of the feti domain decomposition algorithm to fourth order problems the main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations this results in a two level feti substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures and grows at most polylogarithmically with the number of elements per substructure these theoretically proven optimal convergence properties of the new feti method are numerically demonstrated for several finite element plate bending static and transient problems the two level iterative solver presented in this paper is applicable to a large family of biharmonic time independent as well as time dependent systems it is also extendible to shell problems

a simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems several domain decomposition methods with lagrange multipliers have been recently designed for solving iteratively large scale systems of finite element equations while these methods differ typically by implementational details they share in most cases the same substructure based preconditioners that were originally developed for the feti method the success of these preconditioners is due to the fact that for homogeneous structural mechanics problems they ensure a computational performance that scales with the problem size in this paper we address the suboptimal behaviour of these preconditioners in the presence of material and or discretization heterogeneities we propose a simple and virtually no cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations we consider several intricate structural analysis problems and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities copyright 1999 john wiley sons ltd

technical communique extension of unbiased minimum variance input and state estimation for systems with unknown inputs this paper extends the existing results on joint input and state estimation to systems with arbitrary unknown inputs the objective is to derive an optimal filter in the general case where not only unknown inputs affect both the system state and the output but also the direct feedthrough matrix has arbitrary rank the paper extends both the results of gillijns and de moor gillijns s de moor b 2007b unbiased minimum variance input and state estimation for linear discrete time systems with direct feedthrough automatica 43 934 937 and darouach zasadzinski and boutayeb darouach m zasadzinski m boutayeb m 2003 extension of minimum variance estimation for systems with unknown inputs automatica 39 867 876 the resulting filter is an extension of the recursive three step filter ertsf and serves as a unified solution to the addressed unknown input filtering problem the relationship between the ertsf and the existing literature results is also addressed

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

the mosaic of high performance domain decomposition methods for structural mechanics formulation interrelation and numerical efficiency of primal and dual methods a multitude of domain decomposition methods ddm for structural mechanics is available in the literature today a unified framework for formulating primal and dual ddm is thus presented in this paper aiming at providing a mathematical platform for a uniform treatment of high performance ddm in structural mechanics a novel approach for developing new ddm from existing methods is also proposed and is applied to dual and primal methods in the field of the feti methods this approach leads to a new category of methods derived from existing feti variants furthermore two alternative formulations of the balancing domain decomposition method are described while interrelations between the introduced and existing methods are established finally comparative numerical tests demonstrate the differences in the computational performance of the methods in question

unbiased minimum variance linear state estimation abstract a method is developed for linear estimation in the presence of unknown or highly non gaussian system inputs the state update is determined so that it is unaffected by the unknown inputs the filter may not be globally optimum in the mean square error sense however it performs well when the unknown inputs take extreme or unexpected values in many geophysical and environmental applications it is performance during these periods which counts the most the application of the filter is illustrated in the real time estimation of mean areal precipitation

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

a scalable feti dp algorithm with non penetration mortar conditions on contact interface by combining feti algorithms of dual primal type with recent results for bound constrained quadratic programming problems we develop an optimal algorithm for the numerical solution of coercive variational inequalities the model problem is discretized using non penetration conditions of mortar type across the potential contact interface and a feti dp algorithm is formulated the resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem numerical experiments for non matching meshes across the contact interface confirm the theoretical scalability of the algorithm

the mosaic of high performance domain decomposition methods for structural mechanics part ii formulation enhancements multiple right hand sides and implicit dynamics domain decomposition methods ddm constitute a family of high performance solvers for large scale problems of computational mechanics and of structural mechanics in particular this paper compares popular ddm for structural analysis in both theory and computational performance while it investigates the impact of enhanced mapping operators in their convergence behavior multiple right hand side problems are considered while standard and new ddm solution techniques for implicit dynamics are tested the performed benchmark tests lead to useful conclusions regarding the comparative performance of the methods

a two level nonoverlapping schwarz algorithm for the stokes problem numerical study abstract a general framework of a two level nonoverlapping schwarz algorithm for the stokes problem is developed by relaxing average zero condition on pressure unknowns this framework allows both discontinuous and continuous pressure finite element spaces the coarse problem is built by algebraic manipulation after selecting appropriate primal unknowns just like in bddc algorithms performance of the suggested algorithm is presented depending on the selection of finite elements and primal unknowns under the same set of primal unknowns the algorithm for the case with discontinuous pressure functions outperforms one with continuous pressure functions for the two dimensional stokes problem the algorithm with a set of primal unknowns consisting of velocity unknowns at corners averages of velocity components over common edges and pressure unknowns at corners presents good scalability when continuous pressure test functions are used in both two and three dimensional stokes problems an improvement can be made for the case with continuous pressure test functions by applying the suggested algorithm to the interface problem which is obtained by eliminating velocity unknowns and pressure unknowns interior to each subdomains

a scalable lagrange multiplier based domain decomposition method for time dependent problems we present a new efficient and scalable domain decomposition method for solving implicitly linear and non linear time dependent problems in computational mechanics the method is derived by adding a coarse problem to the recently proposed transient feti substructuring algorithm in order to propagate the error globally and accelerate convergence it is proved that in the limit for large time steps the new method converges toward the feti algorithm for time independent problems computational results confirm that the optimal convergence properties of the time independent feti method are preserved in the time dependent case we employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors and demonstrate the effective scalability of the new transient feti method with the large scale finite element dynamic analysis on the paragon xp s and ibm sp2 systems of several diffraction grating finite element structural models we also show that this new domain decomposition method outperforms the popular direct skyline solver the coarse problem presented herein is applicable and beneficial to a large class of lagrange multiplier based substructuring algorithms for time dependent problems including the fictitious domain decomposition method

a study on the lumped preconditioner and memory requirements of feti and related primal domain decomposition methods in recent years domain decomposition methods ddms have emerged as advanced solvers in several areas of computational mechanics in particular during the last decade in the area of solid and structural mechanics they reached a considerable level of advancement and were shown to be more efficient than popular solvers like advanced sparse direct solvers the present contribution follows the lines of a series of recent publications on the relationship between primal and dual formulations of ddms in some of these papers the effort to unify primal and dual methods led to a family of ddms that was shown to be more efficient than the previous methods the present paper extends this work presenting a new family of related ddms thus enriching the theory of the relations between primal and dual methods with the primal methods which correspond to the dual ddm that uses the lumped preconditioner the paper also compares the numerical performance of the new methods with that of the previous ones and focuses particularly on memory requirement issues related to the use of the lumped preconditioner suggesting a particularly memory efficient formulation

bddc preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions balancing domain decomposition by constraints bddc algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with gauss lobatto legendre spectral elements in three dimensions initially mixed spectral elements are employed to discretize the almost incompressible elasticity system but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped appropriate sets of primal constraints can be associated with the subdomain vertices edges and faces so that the resulting bddc methods have a fast convergence rate independent of the almost incompressibility of the material in particular the condition number of the bddc preconditioned operator is shown to depend only weakly on the polynomial degree n the ratio h h of subdomain and element diameters and the inverse of the inf sup constants of the subdomains and the underlying mixed formulation while being scalable i e independent of the number of subdomains and robust i e independent of the poisson ratio and young s modulus of the material considered these results also apply to the related dual primal finite element tearing and interconnect feti dp algorithms defined by the same set of primal constraints numerical experiments carried out on parallel computing systems confirm these results

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

quadrature methods for integral equations of the second kind over infinite intervals convergence results are proved for a class of quadrature methods for integral equations of the form y t fit o k t s y s ds an important special case is the nystrom method in which the integral term is approximated by an ordinary quadrature rule for all of the methods considered here the rate of convergence is the same apart from a constant factor as that of the quadrature approximation to the integral term

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

a fast implementation of the feti dp method feti dp rbs lna and applications on large scale problems with localized non linearities as parallel and distributed computing gradually becomes the computing standard for large scale problems the domain decomposition method dd has received growing attention since it provides a natural basis for splitting a large problem into many small problems which can be submitted to individual computing nodes and processed in a parallel fashion this approach not only provides a method to solve large scale problems that are not solvable on a single computer by using direct sparse solvers but also gives a flexible solution to deal with large scale problems with localized non linearities when some parts of the structure are modified only the corresponding subdomains and the interface equation that connects all the subdomains need to be recomputed in this paper the dual primal finite element tearing and interconnecting method feti dp is carefully investigated and a reduced back substitution rbs algorithm is proposed to accelerate the time consuming preconditioned conjugate gradient pcg iterations involved in the interface problems linear non linear analysis lna is also adopted for large scale problems with localized non linearities based on subdomain linear non linear identification criteria this combined approach is named as the feti dp rbs lna algorithm and demonstrated on the mechanical analyses of a welding problem serial cpu costs of this algorithm are measured at each solution stage and compared with that from the ibm watson direct sparse solver and the feti dp method the results demonstrate the effectiveness of the proposed computational approach for simulating welding problems which is representative of a large class of three dimensional large scale problems with localized non linearities

on the general solution by a direct method of a large scale singular system of linear equations application to the analysis of floating structures finding the general solution of a singular system of linear equations requires computing a particular solution and a basis of the null space of the corresponding singular matrix in this paper we consider the case where the singular matrix is large and sparse and the application calls for a direct solution method we highlight the dependence of straightforward factorization algorithms on an arbitrary constant that can influence the correctness of the computed solution and describe a family of improved direct solution methods that alleviate this problem for structural mechanics applications we propose a hybrid geometric algebraic method that is more robust than the purely algebraic direct methods that are currently used for solving singular sparse systems of equations we illustrate the potential of our proposed solution algorithms with examples from structural mechanics and domain decomposition based iterative solvers 1998 john wiley sons ltd

technical communique unbiased minimum variance input and state estimation for linear discrete time systems with direct feedthrough this paper extends previous work on joint input and state estimation to systems with direct feedthrough of the unknown input to the output using linear minimum variance unbiased estimation a recursive filter is derived where the estimation of the state and the input are interconnected the derivation is based on the assumption that no prior knowledge about the dynamical evolution of the unknown input is available the resulting filter has the structure of the kalman filter except that the true value of the input is replaced by an optimal estimate

unbiased minimum variance estimation for systems with unknown exogenous inputs a new method is developed for the state estimation of linear discrete time stochastic systems in the presence of an unknown disturbance the filter obtained is optimal in the unbiased minimum variance sense the necessary and sufficient conditions for the existence and the stability of the filter are given

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

mesh partitioning for implicit computations via iterative domain decomposition impact and optimization of the subdomain aspect ratio optimal domain decomposition methods have emerged as powerful iterative algorithms for parallel implicit computations their key preprocessing step is mesh partitioning where research has focused so far on the automatic generation of load balanced subdomains with minimum interface nodes in this paper we emphasize the importance of the subdomain aspect ratio as a mesh partitioning factor and highlight its impact on the convergence rate of an optimal domain decomposition based iterative method we also present a fast optimization algorithm for improving the aspect ratio of existing mesh partitions and illustrate it with several examples from fluid dynamics and structural mechanics applications for a stiffened shell problem decomposed by the optimal recursive spectral bisection scheme and solved by the feti method this optimization algorithm is shown to improve the solution time by a factor equal to 1 54 and to restore numerical scalability

a dual primal feti method for solving a class of fluid structure interaction problems in the frequency domain summary the dual primal finite element tearing and interconnecting method feti dp is extended to systems of linear equations arising from a finite element discretization for a class of fluid structure interaction problems in the frequency domain a preconditioned generalized minimal residual method is used to solve the linear equations for the lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution the coupling between the fluid and the structure on the fluid structure interface requires an appropriate choice of coarse level degrees of freedom in the feti dp algorithm to achieve fast convergence several choices are proposed and tested by numerical experiments on three dimensional fluid structure interaction problems in the mid frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard feti dp method copyright 2011 john wiley sons ltd

bddc and feti dp preconditioners with adaptive coarse spaces for three dimensional elliptic problems with oscillatory and high contrast coefficients abstract bddc and feti dp algorithms are developed for three dimensional elliptic problems with adaptively enriched coarse components it is known that these enriched components are necessary in the development of robust preconditioners to form the adaptive coarse components carefully designed generalized eigenvalue problems are introduced for each faces and edges and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance t o l upper bounds for condition numbers of the preconditioned systems are shown to be c t o l with the constant c depending only on the maximum number of edges and faces per subdomain and the maximum number of subdomains sharing an edge numerical results are presented to test the robustness of the proposed approach

brief paper on the global optimality of unbiased minimum variance state estimation for systems with unknown inputs in this paper a globally optimal filtering framework is developed for unbiased minimum variance state estimation for systems with unknown inputs that affect both the system state and the output the resulting optimal filters are globally optimal within the unbiased minimum variance filtering over all linear unbiased estimators globally optimal state estimators with or without output and or input transformations are derived through the global optimality evaluation of this research the performance degradation of the filter proposed by darouach zasadzinski and boutayeb darouach m zasadzinski m boutayeb m 2003 extension of minimum variance estimation for systems with unknown inputs automatica 39 867 876 is clearly illustrated and the global optimality of the filter proposed by gillijns and de moor gillijns s de moor b 2007b unbiased minimum variance input and state estimation for linear discrete time systems with direct feedthrough automatica 43 934 937 is further verified the relationship with the existing literature results is addressed a unified approach to design a specific globally optimal state estimator that is based on the desired form of the distribution matrix from the unknown input to the output is also presented a simulation example is given to illustrate the proposed results

bridging multiple structural scales with a generalized finite element method summary this paper presents a generalized fem based on the solution of interdependent coarse scale global and fine scale local problems in order to resolve multiscale effects due to fine scale heterogeneities overall structural behavior is captured by the global problem while local problems focus on the resolution of fine scale solution features in regions where material heterogeneities may govern the structural response fine scale problems are accurately solved in parallel and to address the intrinsic coupling of scales these solutions are embedded into the global solution space using a partition of unity approach this method is demonstrated on representative heat transfer examples in order to examine its accuracy efficiency and flexibility copyright 2014 john wiley sons ltd

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