restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

a chebyshev method for the solution of boundary value problems abstract an expansion procedure using the chebyshev polynomials as base functions is proposed the method yields more accurate results than either of the galerkin or tau methods as indicated from solving the orr sommerfeld equation for both the plane poiseuille flow and the blasius velocity profile the chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow

krylov methods for the incompressible navier stokes equations methods are presented for time evolution steady state solving and linear stability analysis for the incompressible navier stokes equations at low to moderate reynolds numbers the methods use krylov subspaces constructed by the arnoldi process from actions of the explicit navier stokes right hand side and of its jacobian without inversion of the viscous operator time evolution is performed by a nonlinear extension of the method of exponential propagation steady states are calculated by inexact krylov newton iteration using orthores and gmres linear stability analysis is carried out using an implicitly restarted arnoldi process with implicit polynomial filters a detailed implementation is described for a pseudospectral calculation of the stability of taylor vortices with respect to wavy vortices in the couette taylor problem 9 1994 academic press inc

bpx type preconditioners for second and fourth order elliptic problems on the sphere we develop two bramble pasciak xu type preconditioners for second resp fourth order elliptic problems on the surface of the two sphere to discretize the second order problem we construct c 0 linear elements on the sphere and for the fourth order problem we construct c 1 finite elements of powell sabin type on the sphere the main reason these bpx preconditioners work depends on this particular choice of basis we prove optimality and provide numerical examples furthermore we numerically compare the bpx preconditioners with the suboptimal hierarchical basis preconditioners

multilevel algorithms considered as iterative methods on semidefinite systems for the representation of piecewise d linear functions instead of the usual finite element basis a generating system is introduced that contains the nodal basis functions of the finest level and of all coarser levels of discretization this approach enables the author to work directly with multilevel decompositions of a function for a partial differential equation the galerkin scheme based on this generating system results in a semidefinite matrix equation that has in the one dimensional 1d case only about twice in the two dimensional 2d case about 4 3 times and in the three dimensional 3d case about 8 7 times as many unknowns as the usual system furthermore the semidefinite system possesses not just one but many solutions however the unique solution of the usual definite finite element problem can be easily computed from every solution of the semidefinite problem the author shows that modern multilevel algorithms can be considered as standard iterative methods over the semidefinite sy

c 1 hierarchical bases abstract in this note we derive estimates for the condition numbers of stiffness matrices relative to certain c1 conforming hierarchical bases for fourth order elliptic boundary value problems

resonant mode interactions in rayleigh benard convection department of physics university of california berkeley california 94720 received 5 february 1998 in systems with midplane re ection symmetry the dominant spatial resonance is the 1 3 resonance numeri cal continuation is used to study this and other 1 2k11 resonances in two dimensional convection betweenno slip perfectly conducting horizontal plates periodic boundary conditions are imposed in the horizontal these resonances in uence the process of wave number selection at moderate rayleigh numbers through thegeneration of hybrid solutions and thereby modify the eckhaus picture of wave number selection unlike thebetter known mixed modes the hybrid solutions have the same symmetry as a pair of primary rolls both hybridand symmetry breaking mixed modes are computed and their linear stability properties with respect to pertur bations preserving different spatial periods are determined a complete description of the effects of midplanere ection on wave number selection emerges only steady solutions are considered and the prandtl number is xed ats510 s1063 651x 98 05709 2 pacs number s 47 20 ky 05 45 1bi introduction

a hybrid block gmres method for nonsymmetric systems with multiple right hand sides the block gmres bgmres method is a natural generalization of the gmres algorithm for solving large non symmetric systems with multiple right hand sides unfortunately its cost increases significantly per iteration frequently rendering the method impractical in this paper we propose a hybrid block gmres method which offers significant performance improvements over bgmres this method uses the matrix polynomial obtained in the course of a bgmres step and combines the advantages of the block approach with those of successful hybrid methods we discuss the properties and several implementation variants of the method and report results from numerical experiments we also describe how to use these techniques in order to solve multiply shifted systems with multiple right hand sides

polynomial preconditioned gmres and gmres dr we look at solving large nonsymmetric systems of linear equations using polynomial preconditioned krylov methods we give a simple way to find the polynomial it is shown that polynomial preconditioning can significantly improve restarted gmres for difficult problems and the reasons for this are examined stability is discussed and algorithms are given for increased stability next we apply polynomial preconditioning to gmres with deflated restarting it is shown that this is worthwhile for sparse matrices and for problems with many small eigenvalues multiple right hand sides are also considered

temporal spatial and thermal features of 3 d rayleigh benard convection by a least squares finite element method numerical solutions of 3 d time dependent rayleigh benard convection are presented in this work the temporal spatial and thermal features of convective patterns are studied for four different geometric aspect ratios 2 1 2 4 1 4 5 1 5 and 3 5 1 2 1 at supercritical rayleigh numbers ra 8 103 2 4 104 and prandtl numbers pr 0 71 2 5 several physical phenomena such as multicellular flow pattern oscillatory transient solution t shaped rolls at the ends of a rectangular box and roll alignment are observed in our simulations the numerical technique is based on an implicit fully coupled and time accurate method which consists of the crank nicolson scheme for time integration newton s method for the convective terms with extensive linearization steps and a least squares finite element method a matrix free algorithm of the jacobi conjugate gradient method is implemented to solve the symmetric positive definite linear system of equations

from potential theory to matrix iterations in six steps the theory of the convergence of krylov subspace iterations for linear systems of equations conjugate gradients biconjugate gradients gmres qmr bi cgstab and so on is reviewed for a computation of this kind an estimated asymptotic convergence factor rho le 1 can be derived by solving a problem of potential theory or conformal mapping six approximations are involved in relating the actual computation to this scalar estimate these six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming

generalizations of the field of values useful in the study of polynomial functions of a matrix for a given square matrix a and positive integer k we consider sets in the complex plane satisfying p a maxz p z for all polynomials p of degree k or less the largest such set referred to as the polynomial numerical hull of degree k was introduced by o nevanlinna convergence of iterations for linear equations birkhauser basel 1993 and a number of properties of this set were derived for both matrices and linear operators we give several equivalent characterizations of the polynomial numerical hull of degree k and we actually compute these sets for several matrices for k 1 this set is just the field of values of a and for k m where m is the degree of the minimal polynomial of a it is the spectrum of a for 1 k m these sets are intermediate between the field of values and the spectrum and sometimes resemble pseudospectra

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

a robust gmres based adaptive polynomial preconditioning algorithm for nonsymmetric linear systems in this study a hybrid generalized minimal residual gmres polynomial preconditioning algorithm for solving nonsymmetric systems of linear equations is defined the algorithm uses the results from cycles of restarted gmres to form an effective polynomial preconditioner typically resulting in decreased work requirements the algorithm has the advantage over other hybrid algorithms in that its convergence behavior is well understood the new algorithm converges for all starting vectors if and only if restarted gmres converges the results of numerical experiments with the algorithm are presented

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

hierarchical conforming finite element methods for the biharmonic equation the paper deals with hierarchical bases in spaces of conforming c 1 elements in connection with the approximate solution of the biharmonic equation delta 2 u f quad text in omega qquad u frac partial u partial n 0 quad text on partial omega on a plane polygonal domain omega two different composite finite elements are studied piecewise quadratic powell sabin elements and piecewise cubic elements of clough tocher type the main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization the so called hierarchical multilevel method will yield suboptimal convergence rates in comparison with standard multigrid schemes

convergence analysis of variational and non variational multigrid algorithms for the laplace beltrami operator we design and analyze variational and non variational multigrid algorithms for the laplace beltrami operator on a smooth and closed surface in both cases a uniform convergence for the v cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid the main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface in addition the vanishing mean value constraint is imposed on each level thereby avoiding singular quadratic forms without adding additional computational cost numerical results supporting our analysis are reported in particular the algorithms perform well even when applied to surfaces with a large aspect ratio

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

bpx preconditioning for isogeometric analysis we consider elliptic pdes partial differential equations in the framework of isogeometric analysis i e we treat the physical domain by means of a b spline or nurbs mapping which we assume to be regular the numerical solution of the pde is computed by means of tensor product b splines mapped onto the physical domain we construct additive multilevel preconditioners and show that they are asymptotically optimal i e the spectral condition number of the resulting stiffness matrix is independent of h together with a nested iteration scheme this enables an iterative solution scheme of optimal linear complexity the theoretical results are substantiated by numerical examples in two and three space dimensions

icosahedral discretization of the two sphere we present an almost uniform triangulation of the two sphere derived from the icosahedron and describe a procedure for discretization of a partial differential equation using this triangular grid the accuracy of our procedure is described by a strong theoretical estimate and verified by large scale numerical experiments we also describe a data structure for this spherical discretization that allows fast computation on either a vector computer or an asynchronous parallel computer

different modes of rayleigh benard instability in two and three dimensional rectangular enclosures the article describes a complete numerical solution of a recently formulated benchmark problem devoted to the parametric study of rayleigh benard instability in rectangular two and three dimensional boxes the solution is carried out by the spectral galerkin method with globally defined three dimensional divergent free basis functions which satisfy all boundary conditions the general description of these three dimensional basis functions which can be used for a rather wide spectrum of problems is presented the results of the parametric calculations are presented as neutral curves showing the dependence of the critical rayleigh number on the aspect ratio of the cavity the neutral curves consist of several continuous branches which belong to different modes of the most dangerous perturbation the patterns of different perturbations are also reported the results obtained lead to some new conclusions about the patterns of the most dangerous perturbations and about the similarities between two and three dimensional models some extensions of the considered benchmark problem are discussed

bpx preconditioners for the bidomain model of electrocardiology the aim of this work is to develop a bpx preconditioner for the bidomain model of electrocardiology this model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non linear parabolic reaction diffusion partial differential equation pde and an elliptic linear pde modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue the evolution equation is coupled through the non linear reaction term with a stiff system of ordinary differential equations the so called membrane model describing the ionic currents through the cellular membrane the discretization of the coupled system by finite elements in space and semi implicit finite differences in time yields at each time step the solution of an ill conditioned linear system the goal of the present study is to construct analyze and numerically test a bpx preconditioner for the linear system arising from the discretization of the bidomain model optimal convergence rate estimates are established and verified by two and three dimensional numerical tests on both structured and unstructured meshes moreover in a full heartbeat simulation on a three dimensional wedge of ventricular tissue the bpx preconditioner is about 35 faster in terms of cpu times than ilu 0 and an algebraic multigrid preconditioner

the onset of convection in a finite container due to surface tension and buoyancy abstract a method for predicting instabilities which combines recent techniques from bifurcation theory with the finite element method is described it is applied to the prediction of the onset of convection driven by both surface tension and buoyancy in rectangular containers for zero buoyancy the critical values of the marangoni number for the first two bifurcations from the trivial solution are found for a two dimensional cavity of aspect ratio 2 the variation of these critical values with aspect ratio is obtained by continuation methods and this reveals an interlacing of modes as the container size increases it is established that the bifurcation to an even number of cells is transcritical rather than pitchfork and the turning point on the subcritical branch is located as a function of aspect ratio the hysteresis associated with the transcritical bifurcation is small as the surface tension forces decrease to zero so that the convection is driven by buoyancy alone the amount of hysteresis associated with the transcritical bifurcation becomes vanishingly small the reason for this is not fully understood

wavelet least squares methods for boundary value problems this paper is concerned with least squares methods for the numerical solution of operator equations our primary focus is the discussion of the following conceptual issues the selection of appropriate least squares functionals their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations we describe first a general format of variational problems that are well posed in a certain natural topology in order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems their formulation as a first order system transmission problems the system of stokes equations or more general saddle point problems particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions we propose a unified treatment based on wavelet expansions in particular we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context this provides access to difficult norms efficient preconditioners and in the case of first order systems optimal l2 error estimates

multilevel solvers for unstructured surface meshes parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing such parameterization approaches give rise to large and exceedingly ill conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies since the underlying meshes are very fine to begin with such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy in this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature we introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner