analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

extension operators on tensor product structures in two and three dimensions in this paper a uniformly elliptic second order boundary value problem in two dimensions is discretized by the p version of the finite element method an inexact dirichlet dirichlet domain decomposition preconditioner for the system of linear algebraic equations is investigated the ingredients of such a preconditioner are a preconditioner for the schur complement a preconditioner for the subdomains and an extension operator operating from the edges of the elements into their interior using methods of multiresolution analysis we propose a new method to compute the extension efficiently we prove that this type of extension is optimal i e the h 1 omega norm of the extended function is bounded by the h 0 5 partial omega norm of the given function numerical experiments show the optimal performance of the described extension

parallel domain oriented multilevel methods instead of the usual nodal basis we use a generating system for the discretization of pdes that contains not only the basis functions of the finest level of discretization but additionally the basis functions of all coarser levels of discretization the galerkin approach now results in a semidefinite system of linear equations to be solved standard iterative gs methods for this system turn out to be equivalent to elaborated multigrid methods for the fine grid system

a weakly overlapping domain decomposition preconditioner for the finite element solution of elliptic partial differential equations we present a new two level additive schwarz domain decomposition preconditioner which is appropriate for use in the parallel finite element solution of elliptic partial differential equations pdes as with most parallel domain decomposition methods each processor may be assigned one or more subdomains and the preconditioner is such that the processors are able to solve their own subproblem s concurrently the novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement and it is shown that this amount of overlap is sufficient to yield an optimal preconditioner some numerical experiments posed in both two and three space dimensions are included to confirm that the condition number when using the new preconditioner is indeed independent of the level of mesh refinement on the test problems considered

asynchronous fast adaptive composite grid methods for elliptic problems theoretical foundations accurate numerical modeling of complex physical chemical and biological systems requires numerical simulation capability over a large range of length scales with the ability to capture rapidly varying phenomena localized in space and or time adaptive mesh refinement amr is a numerical process for dynamically introducing local fine resolution on computational grids during the solution process in response to unresolved error in a computation fast adaptive composite grid fac methods are a class of algorithms that exploit the multilevel structure of amr grids to solve elliptic problems efficiently this paper develops a theoretical foundation for afacx an asynchronous fac method a new multilevel condition number estimate establishes that the convergence rate of the afacx algorithm does not degrade as the number of refinement levels in the amr hierarchy increases

mathematical analysis of robustness of two level domain decomposition methods with respect to inexact coarse solves convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level the geneo coarse space has been shown to lead to a fully robust two level schwarz preconditioner which scales well over multiple cores 27 19 as has been proved rigorously in 27 the robustness is due to its good approximation properties for problems with highly heterogeneous material parameters it is available in the finite element packages freefem feel and recently in dune and is implemented as a standalone library in hpddm but the coarse component of the preconditioner can ultimately become a bottleneck if the number of subdomains is very large and exact solves are used it is therefore interesting to consider the effect of approximate coarse solves in this paper robustness of geneo methods is analyzed with respect to approximate coarse solves interestingly the geneo 2 method has to be modified in order to be able to prove its robustness in this context

balancing survival and extinction in nonautonomous competitive lotka volterra systems abstract we generalise and unify some recent results about extinction in n th order nonautonomous competitive lotka volterra systems for each r n we show that if the coefficients are continuous bounded by strictly positive constants and satisfy certain inequalities then any solution with strictly positive initial values has the property that n r of its components vanish whilst the remaining r components asymptotically approach a canonical solution of an r dimensional restricted system in other words r of the species being modeled survive whilst the remaining n r are driven to extinction

some multilevel methods on graded meshes we consider yserentant s hierarchical basis method and multilevel diagonal scaling method on a class of refined meshes used in the numerical approximation of boundary value problems on polygonal domains in the presence of singularities we show as in the uniform case that the stiffness matrix of the first method has a condition number bounded by ln 1 h 2 where h is the meshsize of the triangulation for the second method we show that the condition number of the iteration operator is bounded by ln 1 h which is worse than in the uniform case but better than the hierarchical basis method as usual we deduce that the condition number of the bpx iteration operator is bounded by ln 1 h finally graded meshes fulfilling the general conditions are presented and numerical tests are given which confirm the theoretical bounds

the hierarchical basis multigrid method we derive and analyze the hierarchical basis multigrid method for solving discretizations of self adjoint elliptic boundary value problems using piecewise linear triangular finite elements the method is analyzed as a block symmetric gau seidel iteration with inner iterations but it is strongly related to 2 level methods to the standard multigridv cycle and to earlier jacobi like hierarchical basis methods the method is very robust and has a nearly optimal convergence rate and work estimate it is especially well suited to difficult problems with rough solutions discretized using highly nonuniform adaptively refined meshes

average conditions for permanence and extinction in nonautonomous lotka volterra system abstract an n species nonautonomous competitive lotka volterra system is considered in this paper the average conditions on the coefficients are given to guarantee that all but one of the species are driven to extinction the generalization for the result is presented that is for each r n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n r are driven to extinction it is shown that these average conditions are improvement of those of ahmad and montes de oca appl math comput 90 1998 155 166 and montes de oca and zeeman proc amer math soc 124 1996 3677 3687 and j math anal appl 192 1995 360 370

nested iteration and first order system least squares for incompressible resistive magnetohydrodynamics this paper develops a nested iteration algorithm to solve time dependent nonlinear systems of partial differential equations for each time step newton s method is used to form approximate solutions from a sequence of nested spaces where the resolution of the approximations increases as the algorithm progresses nested iteration results in most of the iterations being performed on coarser grids where minimal work is needed to reduce error to the level of discretization error the approximate solution on a given coarse grid is interpolated to a refined grid and is used as an initial guess for the problem posed there the approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem due to the rapid convergence of newton s method near a solution the paper develops an algorithm that attempts to optimize accuracy per computational cost on each grid so that essentially no unnecessary work is done on any grid the nested iteration algorithm is then applied to a reduced two dimensional model of the incompressible resistive magnetohydrodynamic mhd equations using this algorithm on the mhd equations in the context of a first order system least squares finite element discretization and algebraic multigrid to solve the linearized systems instabilities in a model tokamak fusion reactor are simulated numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30 80 fine grid relaxation sweeps per time step

the algebra of weyl symmetrised polynomials and its quantum extension the algebra of weyl symmetrised polynomials in powers of hamiltonian operatorsp andq which satisfy canonical commutation relations is constructed this algebra is shown to encompass all recent infinite dimensional algebras acting on two dimensional phase space in particular the moyal bracket algebra and the poisson bracket algebra of which the moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra we propose the introduction of a second deformation by the replacement of the heisenberg algebra forp q with aq deformed commutator and construct algebras ofq symmetrised polynomials

two level schwarz algorithms with overlapping subregions for mortar finite elements preconditioned conjugate gradient methods based on two level overlapping schwarz methods often perform quite well such a preconditioner combines a coarse space solver with local components which are defined in terms of subregions that form an overlapping covering of the region on which the elliptic problem is defined precise bounds on the rate of convergence of such iterative methods have previously been obtained in the case of conforming lower order and spectral finite elements as well as in a number of other cases in this paper this domain decomposition algorithm and analysis are extended to mortar finite elements it is established that the condition number of the relevant iteration operator is independent of the number of subregions and varies with the relative overlap between neighboring subregions linearly as in the conforming cases previously considered

a stabilized explicit lagrange multiplier based domain decomposition method for parabolic problems a fully explicit stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations pdes is presented writing the semi discretized equations as a differential algebraic equation dae system where the interface continuity constraints between subdomains are enforced by lagrange multipliers the method uses the runge kutta chebyshev projection scheme to integrate the dae explicitly and to enforce the constraints by a projection with mass lumping techniques and node to node matching grids the method is fully explicit without solving any linear system a stability analysis is presented to show the extended stability property of the method the method is straightforward to implement and to parallelize numerical results demonstrate that it has excellent performance

on schwarz s domain decomposition methods for elliptic boundary value problems summary we study the additive and multiplicative schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness r 1 the finite element method reduces an elliptic boundary value problem to a linear system of equations it is well known that as the number of triangles in the underlying triangulation is increased which is indispensable for increasing the accuracy of the approximate solution the size and condition number of the linear system increases the schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size we shall show in this paper that the approximate solutions from the multiplicative schwarz domain decomposition method converge to the exact solution of the linear system geometrically we also show that the additive schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method we tested these methods for the biharmonic equation with dirichlet boundary condition over an arbitrary polygonal domain using c 1 cubic spline functions over a quadrangulation of the given domain the computer experiments agree with our theoretical results

an anisotropic superconvergent nonconforming plate finite element the classical finite element convergence analysis relies on the following regularity condition there exists a constant c independent of the element k and the mesh such that h k r k

a multilevel hybrid newton krylov schwarz method for the bidomain model of electrocardiology abstract a multilevel hybrid newton krylov schwarz nks method is constructed and studied numerically for implicit time discretizations of the bidomain reaction diffusion system in three dimensions this model describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with a stiff system of ordinary differential equations the nks bidomain solver employs an outer inexact newton iteration to solve the nonlinear finite element system originating at each time step of the implicit discretization the jacobian update during the newton iteration is solved by a krylov method employing a multilevel hybrid overlapping schwarz preconditioner additive within the levels and multiplicative among the levels several parallel tests on linux clusters are performed showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality

on the deformability of heisenberg algebras based on the vanishing of the second hochschild cohomology group of the weyl algebra it is shown that differential algebras coming from quantum groups do not provide a non trivial deformation of quantum mechanics for the case of aq oscillator there exists a deforming map to the classical algebra it is shown that the differential calculus on quantum planes with involution i e if one works in position momentum realization can be mapped on aq difference calculus on a commutative real space although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not possess a proper group of global transformations

new convergence estimates for multilevel algorithms for finite element approximations new convergence estimates are established for some multilevel algorithms for finite element methods applied to elliptic problems with jump coefficients a uniform rate of convergence is derived if the coefficient has only one jump interface if the coefficient has multi jump interfaces which meet at only one interior point in the domain the convergence rate is bounded by 1 cj 1 where j is the number of levels and c is a constant independent of the jump

a multigrid preconditioner for the mixed formulation of linear plane elasticity in this paper we develop a multigrid preconditioner for the discrete system of linear equations that results from the mixed formulation of the linear plane elasticity problem using the arnold winther elements this in turn can be reduced to the problem of finding a multigrid preconditioner for the form cdot cdot vdiv cdot vdiv cdot in the symmetric matrix space resulting from arnold winther elements since the form is not uniformly elliptic a helmholtz type decomposition is essential the arnold winther finite element space gives rise to nonnested multilevel spaces adding difficulty to the analysis we prove that for the variable v cycle multigrid preconditioner the condition number of the preconditioned system is independent of the number of levels the results of numerical experiments are also presented

a multiscale rectangular element method for elliptic problems with entirely small periodic coefficients the purpose of this paper is to solve elliptic problems with entirely small periodic configuration by so called multiscale finite element method a special multiscale rectangular element space is constructed whose base functions consist of a standard bilinear conforming finite element base functions defined on a relatively coarse partition compared with small configuration parameter plus special bubble like functions which include the small configuration information meanwhile the error of the multiscale finite element solution is analysed and the optimal error estimate is obtained finally a multilevel additive schwarz preconditioning method is presented for solving the discrete problem

necessary and sufficient conditions for permanence and extinction in a three dimensional competitive lotka volterra system abstract a three dimensional nonautonomous competitive lotka volterra system is considered in this paper it is shown that if the growth rates are positive bounded and continuous functions and the averages of the growth rates satisfy certain inequalities then any positive solution has the property that one of its components vanishes moreover if one of the above inequalities is changed then all components of any positive solution have positive infimum

optimal additive schwarz methods for the h p bem we propose and analyze an overlapping schwarz preconditioner for the p and h p boundary element method for the hypersingular integral equation in 3d we consider surface triangulations consisting of triangles the condition number is bounded uniformly in the mesh size h and the polynomial order p the preconditioner handles adaptively refined meshes and is based on a local multilevel preconditioner for the lowest order space numerical experiments on different geometries illustrate its robustness

multilevel preconditioning for the boundary concentrated hp fem abstract the boundary concentrated finite element method is a variant of the hp version of the finite element method that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and low regularity boundary conditions for this method we present two multilevel preconditioners that lead to preconditioned stiffness matrices with condition numbers that are bounded uniformly in the problem size n the cost of applying the preconditioners is o n numerical examples illustrate the efficiency of the algorithms

multilevel schwarz methods for elliptic partial differential equations we investigate multilevel schwarz domain decomposition preconditioners to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method in our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner we start from two level preconditioners with either aggregative or interpolative coarse level components then we focus on a strategy to increase the number of levels for all preconditioners we consider the additive residual update and its multiplicative variants within and between levels moreover we compare the preconditioners behaviour regarding scalability and rate of convergence numerical results are provided for elliptic boundary value problems including a convection diffusion problem when suitable stabilization becomes necessary c 2011 elsevier b v all rights reserved

efficiency based adaptive local refinement for first order system least squares formulations in this paper we propose new adaptive local refinement alr strategies for first order system least squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration the goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements to accomplish this the refinement decision at each refinement level is determined based on optimizing efficiency measures that take into account both error reduction and computational cost two efficiency measures are discussed predicted error reduction and predicted computational cost these methods are first applied to a two dimensional 2d poisson problem with steep gradients and the results are compared with the threshold based methods described in w dorfler siam j numer anal 33 1996 pp 1106 1124 next these methods are applied to a 2d reduced model of the incompressible resistive magnetohydrodynamic equations these equations are used to simulate instabilities in a large aspect ratio tokamak we show that by using the new alr strategies on this system we are able to resolve the physics using only 10 percent of the computational cost used to approximate the solutions on a uniformly refined mesh within the same error tolerance

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

competitive exclusion in a periodic lotka volterra system abstract we study a lotka volterra system for two competing species assuming continuous and periodic coefficients introducing a suitable average condition we extend the known principle of competitive exclusion to the periodic case

multilevel additive schwarz method for the h p version of the galerkin boundary element method we study a multilevel additive schwarz method for the h p version of the galerkin boundary element method with geometrically graded meshes both hypersingular and weakly singular integral equations of the first kind are considered as it is well known the h p version with geometric meshes converges exponentially fast in the energy norm however the condition number of the galerkin matrix in this case blows up exponentially in the number of unknowns m we prove that the condition number p of the multilevel additive schwarz operator behaves like o mlog 2 m as a direct consequence of this we also give the results for the 2 level preconditioner and also for the h p version with quasi uniform meshes numerical results supporting our theory are presented

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

the simple modules of certain generalized crossed products abstract we consider a dedekind domaindand az graded ring r i z r i withr0 dand eachri dvibeing a freed module of rank 1 the structure ofris described by an automorphism ofdand a generalized 2 cocyclec z z dnot necessarily taking its values in the units ofd the aim of this paper is to classify the simpler modules say in the case wherec i j 0 for alli j z we also deal with this problem in then graded case as a consequence we obtain a description of the simple modules of some classical algebras and of generalized weyl algebras

optimal additive schwarz preconditioning for adaptive 2d iga boundary element methods abstract we define and analyze local multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions hypersingular and weakly singular integral equations are considered we prove that the condition number of the preconditioned systems of linear equations is independent of the mesh size and the refinement level therefore the computational complexity when using appropriate iterative solvers is optimal our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results

saddle point least squares preconditioning of mixed methods abstract we present a simple way to discretize and precondition mixed variational formulations our theory connects with and takes advantage of the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric positive definite operators efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete spaces that are always compatible are provided for the proposed discrete spaces and solvers a basis is needed only for the test spaces and assembly of a global saddle point system is avoided we prove sharp approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete inf sup and sup sup constants of the pair of discrete spaces

on the robustness of the bpx preconditioner with respect to jumps in the coeffiencients we determine the worst case behavior of the standard bpx preconditioner for elliptic problems with arbitrary coefficient jumps along the boundaries of the coarsest partition the counterexamples are also useful for other problems

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

a hybrid multilevel schwarz method for the bidomain model abstract a hybrid multilevel schwarz method is studied numerically for the anisotropic bidomain model in both two and three dimensions this multiscale system models the electrical activity of the heart and it consists of two degenerate parabolic non linear reaction diffusion equations coupled with a stiff system of ordinary differential equations the numerical discretization of the whole system by finite elements in space and semi implicit methods in time generates ill conditioned linear systems that must be solved at each time step the multilevel algorithm studied employs a hierarchy of nested meshes with overlapping schwarz preconditioners on each level and is additive within the levels and multiplicative among the levels we perform several parallel tests on two linux clusters showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality moreover the comparison with the traditional block jacobi ilu parallel preconditioner and the v cycle multigrid parallel preconditioner shows that on a whole heart cycle simulation the proposed method attains the best performances in terms of cpu times

multilevel additive and multiplicative methods for orthogonal spline collocation problems multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise hermite bicubics is applied to the dirichlet boundary value problem for a self adjoint elliptic partial differential equation on a rectangle additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space a general theory of additive and multiplicative schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the laplacian with the spectral constants independent of the fine partition stepsize and the number of levels the preconditioned conjugate gradient and preconditioned orthomin methods are considered for the solution of collocation problems an implementation of the methods is discussed and the results of numerical experiments are presented

adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations by a well known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned cg iterations using a multilevel preconditioner in the spirit of bramble pasciak and xu designed for standard nonconforming approximations local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results the performance of the preconditioner and the error estimator is illustrated by several numerical examples

parallel point and domain oriented multilevel methods for elliptic pde s on workstation networks we present new multilevel methods for the solution of linear elliptic pdes they show the same convergence behavior as conventional multigrid methods but possess interesting properties with respect to parallelization regarding communication the number of setup steps is only dependent on the number of processors and not on the number of levels like for parallelized multigrid methods this makes our new methods perfectly suited for parallel computing systems with relatively dominant communication setup like for example clusters of workstations

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

p adic commutation relations representations of the canonical and deformed commutation relations by bounded operators on p adic banach spaces are constructed functions from the mahler basis of the space of p adic continuous functions and their multiplicative analogues are shown to be the p adic counterparts of the hermite and q hermite functions the analogue of the stone von neumann uniqueness theorem fails in the p adic case

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 methods for mixed finite elements in three dimensions in this paper we consider second order scalar elliptic boundary value problems posed over three dimensional domains and their discretization by means of mixed raviart thomas finite elements 18 this leads to saddle point problems featuring a discrete flux vector field as additional unknown following ewing and wang 26 the proposed solution procedure is based on splitting the flux into divergence free components and a remainder it leads to a variational problem involving solenoidal raviart thomas vector fields a fast iterative solution method for this problem is presented it exploits the representation of divergence free vector fields as bf curl s of the bf h bf curl conforming finite element functions introduced by nedelec 43 we show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem duality techniques in quotient spaces and modern algebraic multigrid theory 50 10 31 are the main tools for the proof

convergence of nonconforming v cycle and f cycle multigrid algorithms for second order elliptic boundary value problems the convergence of v cycle and f cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems

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

subspace correction methods for nonsymmetric parabolic problems abstract a convergence proof for general subspace correction techniques applied to an abstract nonsymmetric parabolic equation is given one of the main concerns is to give a unified convergence analysis for domain decomposition and multigrid methods for parabolic problems the analysis is also valid for nonsymmetric problems for second order parabolic problems the convection can dominate the diffusion the algorithms can be applied to domain decomposition methods with or without the coarse mesh for applications to multigrid methods the coarsest mesh does not need to be very coarse a relation between the coarse mesh size and the time step is needed to get a convergence rate independent of the mesh the number of iterations at each time step for the algorithms is also estimated some numerical experiments are presented for domain decomposition methods with minimum overlap which support the theoretical predictions the algorithms are able to capture the sharp traveling shocks for convection dominated problems

the unitary irreducible representations of the quantum heisenberg algebra the goal of this work is to describe the irreducible representations of the quantum heisenberg algebra and the unitary irreducible representation of one of its real forms the solution of this problem is obtained through the investigation of theleft spectrum of the quantum heisenberg algebra using the result about spectra of generic algebras of skew differential operators cf r

boundary treatments for multilevel methods on unstructured meshes in applying multilevel iterative methods on unstructured meshes the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid in this case the standard coarse to fine grid transfer operators with linear interpolants are not accurate enough at neumann boundaries so special care is needed to correctly handle different types of boundary conditions we propose two effective ways to adapt the standard coarse to fine interpolations to correctly implement boundary conditions for two dimensional polygonal domains and we provide some numerical examples of multilevel schwarz methods and multigrid methods which show that these methods are as efficient as in the structured case in addition we prove that the proposed interpolants possess the local optimal l2 approximation and h1 stability which are essential in the convergence analysis of the multilevel schwarz methods using these results we give a condition number bound for two level schwarz methods

multigrid algorithms for c 0 interior penalty methods multigrid algorithms for c 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper it is shown that v cycle f cycle and w cycle algorithms are contractions if the number of smoothing steps is sufficiently large the contraction numbers of these algorithms are bounded by cm alpha where m is the number of presmoothing and postsmoothing steps alpha is the index of elliptic regularity and the positive constant c is mesh independent these estimates are established for a smoothing scheme that uses a poisson solve as a preconditioner which can be easily implemented because the c 0 finite element spaces are standard spaces for second order problems furthermore the variable v cycle algorithm is also shown to be an optimal preconditioner

on the abstract theory of additive and multiplicative schwarz algorithms in recent years it has been shown that many modern iterative algorithms multigrid schemes multilevel preconditioners domain decomposition methods etc for solving problems resulting from the discretization of pdes can be interpreted as additive jacobi like or multiplicative gauss seidel like subspace correction methods the key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space v into a sum of subspaces v j and the splitting of the variational problem on v into auxiliary problems on these subspaces in this paper we propose a modification of the abstract convergence theory of the additive and multiplicative schwarz methods that makes the relation to traditional iteration methods more explicit the analysis of the additive and multiplicative schwarz iterations can be carried out in almost the same spirit as in the traditional block matrix situation making convergence proofs of multilevel and domain decomposition methods clearer or at least more classical in addition we present a new bound for the convergence rate of the appropriately scaled multiplicative schwarz method directly in terms of the condition number of the corresponding additive schwarz operator these results may be viewed as an appendix to the recent surveys x ys

the spectrum of the algebra of skew differential operators and the irreducible representations of the quantum heisenberg algebra the left spectrum of a wide class of the algebras of skew differential operators is described as a sequence we determine and classify all the algebraically irreducible representations of the quantum heisenberg algebra over an arbitrary field

the left spectrum the levitzki radical and noncommutative schemes this note contains a brief exposition of the basics of a noncommutative version of affine quasi affine and projective algebraic geometry in this version to any associative ring with unity a quasi affine resp affine left scheme is assigned the notion of the left spectrum of a ring plays the key role

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

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

an adaptive two level method for the coupling of nonlinear fem bem equations in this paper an a posteriori error estimate for nonlinear coupled fem bem equations is derived by using hierarchical basis techniques based on the properties of two level additive schwarz operators for linearized differential equations and weakly singular integral equations easily computable local error indicators are obtained an algorithm for adaptive error control which allows independent refinements of the finite elements and the boundary elements is formulated and numerical results for the linear and nonlinear case are included

multilevel preconditioners for non self adjoint or indefinite orthogonal spline collocation problems efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation osc discretization of a dirichlet boundary value problem with a non self adjoint or an indefinite operator the osc solution is sought in the hermite space of piecewise bicubic polynomials it is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal osc equation the preconditioners are used with the preconditioned conjugate gradient method and numerical results are presented that demonstrate their efficiency

a robust hierarchical basis preconditioner on general meshes in this paper we introduce a multi level direct sum space decomposition of general possibly locally refined linear or multi linear finite element spaces the resulting additive schwarz preconditioner is optimal for symmetric second order elliptic problems moreover it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh perturbations with positive zeroth order terms and after a further decomposition of the spaces also with respect to anisotropy along the grid lines important for an efficient implementation is that stable bases of the subspaces defining our decomposition consisting of functions having small supports can be easily constructed

average growth and extinction in a competitive lotka volterra system abstract in this paper we answer an earlier conjecture consider a class of nonautonomous competitive lotka volterra systems we show that certain inequalities involving the interaction coefficients and averages of the growth rates imply that the system is persistent however reversing one of these inequalities implies that the n th species goes to extinction

two level schwarz methods for wilson element approximation of elasticity problem in this paper a two level additive and multiplicative schwarz methods are developed for the wilson nonconforming element approximation of elasticity problem a new intergrid operator for the wilson element is constructed it is well known that the two level additive schwarz method can be seen as a preconditioning system so the condition number of the preconditioning system for elasticity problem is proved to be bounded by a constant independent of mesh size and the number of subdomains for the multiplicative method the optimal rate of convergence is also obtained

multiresolution weighted norm equivalences and applications we establish multiresolution norm equivalences in weighted spaces l2w 0 1 with possibly singular weight functions w x 0 in 0 1 our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions the discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function w x within each scale since norm equivalences for sobolev norms are by now well known our result can also be applied to weighted sobolev norms we apply our theory to the problem of preconditioning p version fem and wavelet discretizations of degenerate elliptic and parabolic problems from finance

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