efficient block preconditioning for a c1 finite element discretisation of the dirichlet biharmonic problem we present an efficient block preconditioner for the two dimensional biharmonic dirichlet problem discretised by c1 bicubic hermite finite elements in this formulation each node in the mesh has four different degrees of freedom dofs grouping dofs of the same type together leads to a natural blocking of the galerkin coefficient matrix based on this block structure we develop two preconditioners a 2x2 block diagonal preconditioner bd and a block bordered diagonal bbd preconditioner we prove mesh independent bounds for the spectra of the bd preconditioned galerkin matrix under certain conditions the eigenvalue analysis is based on the fact that the proposed preconditioner like the coefficient matrix itself is symmetric positive definite and is assembled from element matrices we demonstrate the effectiveness of an inexact version of the bbd preconditioner which exhibits near optimal scaling in terms of computational cost with respect to the discrete problem size finally we study robustness of this preconditioner with respect to element stretching domain distortion and non convex domains

convergence of nonconforming multigrid methods without full elliptic regularity we consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity we prove that there is a bound 1 for the contraction number of the w cycle algorithm which is independent of mesh level provided that the number of smoothing steps is sufficiently large we also show that the symmetric variable v cycle algorithm is an optimal preconditioner

a v cycle multigrid method and additive multilevel preconditioners for the plate bending problem discretized by conforming finite elements it is well known that many conforming plate finite element spaces in the multilevel methods are nonnested even when the triangulations are nested so it is difficult to construct the effective v cycle multigrid method and multilevel preconditioners for the plate bending problem in this paper a new method dealing with multilevel methods for finite element discretisation equations of the plate problem is developed an optimal v cycle multigrid method and multilevel preconditioners for some famous conforming plate elements are constructed moreover our methods can be applied to the plate problem with complicated geometries

variable v cycle multigrid preconditioners for the discrete systems from combined hybrid quadrilateral elements abstract combined hybrid element method is an absolutely convergent and stable hybrid element method in which approximation spaces for field variables avoid the restriction of babuska brezzi condition so the spaces can be adopted in a larger range compared with other kinds of mixed hybrid element methods in this paper two variable v cycle multigrid preconditioners are proposed for the algebraic system resulting from combined hybrid quadrilateral element discretization of linear elasticity problem based on two kinds of simple intergrid transfer operators constructed on quadrilateral meshes we prove that the condition numbers of the preconditioned systems are bounded by a constant independent of the mesh size and the number of levels a numerical example is given to indicate the asymptotically optimal performance of the algorithm

multigrid methods for two dimensional maxwell s equations on graded meshes in this paper we investigate the numerical solution for two dimensional maxwell s equations on graded meshes the approach is based on the hodge decomposition the solution u of maxwell s equations is approximated by solving standard second order elliptic problems quasi optimal error estimates for both u and i u in the l 2 norm are obtained on graded meshes we prove the uniform convergence of the w cycle and full multigrid algorithms for the resulting discrete problem the performance of these methods is illustrated by numerical results

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

universality in multidimensional maps bifurcations of cubic nonlinear symplectic mappings in two and four dimensions are discussed here series of bifurcations are studied by direct numerical calculation and by a renormalisation procedure it is shown that for period doubling bifurcations one finds the universal exponent of the quadratic area preserving map other exponents exist for higher multiplicities the renormalisation transformation has a fixed line in parameter space with an end point the latter implies that series of period doubling bifurcations may break off

a high resolution weighted ab semblance for dealing with amplitude variation with offset phenomenon abstractvelocity analysis is an essential step in seismic reflection data processing the conventional and fastest method to estimate how velocity changes with increasing depth is to calculate semblance coefficients traditional semblance has two problems low time and velocity resolution and an inability to handle amplitude variation with offset avo phenomenon although a method known as the ab semblance can arrive at peak velocities in the areas with an avo anomaly it has a lower velocity resolution than conventional semblance we have developed a weighted ab semblance method that can handle both problems simultaneously we have developed two new weighting functions to weight the ab semblance to enhance the resolution of velocity spectra in the time and velocity directions in this way we increase the time and velocity resolution while eliminating the avo problem the first weighting function is defined based on the ratio between the first and the second singular values of the time window to improve

preconditioners for nonconforming discretizations we prove an abstract norm equivalence for a two level method which allows us to reduce the construction of preconditioners for nonconforming finite element discretizations to known cases of conforming elements

a multigrid algorithm for the lowest order raviart thomas mixed triangular finite element method an optimal order multigrid method for the lowest order raviart thomas mixed triangular finite element is developed the algorithm and the convergence analysis are based on the equivalence between raviart thomas mixed methods and certain nonconforming methods both the dirichlet and singular neumann boundary value problems for second order elliptic equations are discussed

a non conforming piecewise quadratic finite element on triangles it is shown in this paper that non conforming finite elements on the triangle using second degree polynomials can be easily built and used indeed they appear as an enriched version of the standard piecewise quadratic six node element this work is divided into two parts in the first we present the basic properties of the element namely how it can be built and basic error estimates we also show that this element exhibits a very peculiar regularity property in the second part we apply our element to the approximation of viscous incompressible flows and more generally to the approximation of incompressible materials

multilevel finite element preconditioning for 3 refinement we develop a bpx type multilevel method for the numerical solution of second order elliptic equations in 2 using piecewise linear polynomials on a sequence of triangulations given by regular 3 refinement a multilevel splitting of the finest grid space is obtained from the nonnested sequence of spaces on the coarser triangulations using prolongation operators based on simple averaging procedures the main result is that the condition number of the corresponding bpx preconditioned linear system is uniformly bounded independent of the size of the problem the motivation to consider 3 refinement stems from the fact that it is a slower topological refinement than the usual dyadic refinement and that it alternates the orientation of the refined triangles therefore we expect a reduction of the amount of work when compared to the classical bpx preconditioner although both methods have the same asymptotical complexity numerical experiments confirm this statement

treatments of discontinuity and bubble functions in the multigrid method when multilevel finite element spaces are not nested different intergrid transfer operators would lead to different multigrid algorithms it is proposed in this paper that discontinuous functions be averaged to continuous functions and that the bubble functions be discarded in the multigrid transferring applications of the techniques to various problems are presented with convergence analysis numeric comparisons with other existing methods are provided

mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems we construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order pdes in the state variable e g pressure and its flux e g darcy velocity an important point is that no staggered grids or covolumes are used to stabilize the system only a single primary grid system is adopted and the degrees of freedom are imposed on the interfaces the approximate flux is sought in the lowest order raviart thomas space and the pressure field in the rotated q1 nonconforming space furthermore we demonstrate that the present finite volume method can be interpreted as a rotated q1 nonconforming finite element method for the pressure with a simple local recovery of flux numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method

an analysis of nonconforming multi grid methods leading to an improved method for the morley element we recall and slightly refine the convergence theory for nonconforming multi grid methods for symmetric positive definite problems developed by bramble pasciak and xu we derive new results to verify the regularity and approximation assumption and the assumption on the smoother from the analysis it will appear that most efficient multi grid methods can be expected for fully regular problems and for prolongations for which the energy norm of the iterated prolongations is uniformly bounded guided by these observations we develop a new multi grid method for the biharmonic equation discretized with morley finite elements or equivalently for the stokes equations discretized with the p0 nonconforming p1 pair numerical results show that the new method is superior to standard ones

a two level algorithm for the weak galerkin discretization of diffusion problems this paper analyzes a two level algorithm for the weak galerkin wg finite element methods based on local raviart thomas rt and brezzi douglas marini bdm mixed elements for two and three dimensional diffusion problems with dirichlet condition we first show the condition numbers of the stiffness matrices arising from the wg methods are of o h 2 we use an extended version of the xu zikatanov xz identity to derive the convergence of the algorithm without any regularity assumption finally we provide some numerical results

flux recovery from primal hybrid finite element methods a flux recovery technique is introduced and analyzed for the computed solution of the primal hybrid finite element method for second order elliptic problems the recovery is carried out over a single element at a time while ensuring the continuity of the flux across the interelement edges and the validity of the discrete conservation law at the element level our construction is general enough to cover all degrees of polynomials and grids of triangular or quadrilateral type we illustrate the principle using the raviart thomas spaces but other well known related function spaces such as the brezzi douglas marini bdm or brezzi douglas fortin marini bdfm space can be used as well an extension of the technique to the nonlinear case is given numerical results are presented to confirm the theoretical results

an optimal order nonconforming multigrid method for the biharmonic equation an optimal order multigrid method for solving the biharmonic equation using morley nonconforming finite elements is developed

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

multigrid analysis of finite element methods with numerical integration the author analyzes multigrid convergence rates when elliptic boundary value problems are discretized using finite element methods with numerical integration the resulting discrete problem does not fall into the standard variational framework for analyzing multigrid methods since the bilinear forms on different grid levels are not suitably related to each other he first discusses extensions of the standard variational multigrid theory and then apply these results to the case of numerical quadrature in particular it is shown that the v cycle algorithm has a convergence rate independent of grid size under suitable conditions

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

random noise attenuation using local signal and noise orthogonalization abstractwe have developed a novel approach to attenuate random noise based on local signal and noise orthogonalization in this approach we first removed from a seismic section using one of the conventional denoising operators and then applied a weighting operator to the initially denoised section to predict the signal leakage energy as well as retrieve it from the initial noise section the weighting operator was obtained by solving a least squares minimization problem via shaping regularization with a smoothness constraint next the initially denoised section and the retrieved signal were combined to form the final denoised section the proposed denoising approach corresponded to orthogonalizing the initially denoised signal and noise in a local manner we evaluated the denoising performance using local similarity to test the orthogonalization property of the estimated signal and noise we calculated the local similarity map between the denoised signal section and removed noise section low values o

a shorter proof of the existence of the feigenbaum fixed point we use the leray schauder fixed point theorem to prove the existence of an analytic fixed point for the period doubling accumulation renormalization operator our argument does not however show that the linearization of the renormalization operator at this fixed point is hyperbolic

nonconforming multigrid method for nonsymmetric and indefinite problems abstract the convergence of w cycle multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems using p 1 nonconforming finite elements is proved

a multigrid version of a simple finite element method for the stokes problem we consider a finite element method for the stokes problem on a rectangular domain based on piecewise bilinear velocities and piecewise constant pressures on a uniform rectangular grid it is shown that by a simple stabilization strategy the method can be implemented in a convergent multigrid procedure introduction one of the most natural ways of discretizing the stokes equations on a two dimensional rectangular domain is to apply the finite element techniques with continuous piecewise bilinear velocities and piecewise constant pressures on a rectangular grid this method is easily extended to more general quadrilateral meshes and it has proved to be quite effective in practice the convergence analysis of the method was first carried out in 10 on a rectangular domain in 12 the estimates of 10 are improved and extended to more general quadrilateral meshes so far the above method has been used mainly in connection with direct band solvers usually combined with penalty perturbation techniques to eliminate the pressure 9 in this paper we consider an iterative variant of the method based on multigrid techniques such an algorithm seems attractive especially since the un derlying finite difference equations are relatively simple multigrid methods for the stokes problem and for more general elliptic systems were considered previously by hackbusch 8 and verfurth 14 in 14 it is shown that a convergent multigrid algorithm for the stokes problem can be constructed by appropriately scaling the variables provided that the underlying finite element scheme satisfies the babu ka brezzi stability condition 2 5 in our case however the finite element method is not stable in this sense and we have not been able to show that the straightforward application of the algorithm proposed in 14 yields a convergent process therefore we suggest first modifying the finite element method in such a way that it becomes stable in the ordinary sense the usual way of stabilizing an unstable mixed method is to add more velocity or primary variables a classical example of this is the quadratic linear velocity pres sure element of crouzeix and raviart 6 where added bubble functions act as stabilizers in the velocity space increasing the dimension of the velocity space would be an alternative in our case however we propose another method which seems to yield a simpler algorithm it is based on adding an extra stabilizing term into the

constrained full waveform inversion by model reparameterization the waveform inversion problem is inherently ill posed traditionally regularization schemes are used to address this issue for waveform inversion where the model is expected to have many details reflecting the physical properties of the earth regularization and data fitting can work in opposite directions the former smoothing and the latter adding details to the model we propose constraining estimated velocity fields by reparameterizing the model this technique also called model space preconditioning is based on directional laplacian filters it preserves most of the details of the velocity model while smoothing the solution along known geological dips preconditioning also yields faster convergence at early iterations the laplacian filters have the property to smooth or kill local planar events according to a local dip field by construction these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models we illustrate with 2d syn

an impes scheme for a two phase flow in heterogeneous porous media using a structured grid abstract we develop a numerical scheme for a two phase immiscible flow in heterogeneous porous media using a structured grid finite element method which has been successfully used for the computation of various physical applications involving elliptic interface equations li et al 2003 2004 chou et al 2010 kwak et al 2010 chang and kwak 2011 the proposed method is based on the implicit pressure explicit saturation procedure to solve the pressure equation we use an ifem based on the rannacher turek rannacher and turek 1992 nonconforming space which is a modification of the work in kwak et al 2010 where broken p 1 nonconforming element of crouzeix raviart 1973 1 was developed for the darcy velocity we apply the mixed finite volume method studied in chou et al 2003 and kwak et al 2010 on the basis of immersed finite element method ifem in this way the darcy velocity of the flow can be computed cheaply locally after we solve the pressure equation the computed darcy velocity is used to solve the saturation equation explicitly the whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media in fact we apply a new version of multigrid algorithm to solve the pressure equation for which the cpu time grows like o n numerical tests for analytic problems show that our method is almost optimal in all variables problem with sources sinks is also computed and it seems to work well

a multilevel algorithm for mixed problems we describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise e g from mixed finite element approximations of the stokes equation the main difficulty besides the indefiniteness is the lack of regularity of the solution of the corresponding continuous problem this is overcome by introducing a scale of mesh dependent norms the convergence rate of the described algorithm is bounded independently of the meshsize for convenience we only discuss jacobi relaxation as smoothing operator in detail in the last section we comment on lanczos type smoothing procedures

a note on convergence of v cycle nonconforming multigrid methods1this research was supported by the doctoral program foundation of china universities and by nsf and china postdoctoral science foundation 1 in this paper the nonconforming multigrid methods are studied i e the fine mesh spaces and coarse mesh spaces also use the same nonconforming finite element contrary to the observation by chen convergence of multigrid method for nonconforming method pre print 1996 even if quadratic forms are not changed on coarse meshes v cycle nonconforming multigrid methods still can converge the key of our methods is the choice of a good intergrid operator moreover optimal convergent order of our v cycle nonconforming multigrid methods is obtained

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

a nonconforming multigrid method for the stationary stokes equations an optimal order w cycle multigrid method for solving the stationary stokes equations is developed using p1 nonconforming divergence free finite elements

optimal order nonnested multigrid methods for solving finite element equations i on quasi uniform meshes we prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested when a coarse mesh is not a submesh of the finer one the coarse level correction usually does not have the a projection property and does amplify the iterative error in some components nevertheless the low frequency components of the error can still be caught by the coarse level correction since the amplified highfrequency errors will be damped out by the fine level smoothing efficiently the optimal work order of the standard multigrid method can still be maintained however unlike the case of nested meshes a nonnested multigrid method with one smoothing does not converge in general no matter whether it is a v cycle or a w cycle method it is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones since nonnested multigrid methods accept quite arbitrarily related meshes we may then combine the efficiencies of adaptive refinements and of multigrid algorithms

robust multigrid solvers for the biharmonic problem in isogeometric analysis abstract in this paper we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis iga in this framework one typically sets up b splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function with this approach it is feasible to set up h 2 conforming discretizations we propose two multigrid methods for such a discretization one based on gauss seidel smoothing and one based on mass smoothing we prove that both are robust in the grid size the latter is also robust in the spline degree numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches particularly of a hybrid approach combining both smoothers

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

a conjugate gradient method and a multigrid algorithm for morley s finite element approximation of the biharmonic equation the numerical solution of the linear equations arising from morley s nonconforming displacement method is studied a suitable preconditioning for the conjugate gradient method is described furthermore the nonconformity of the discretization necessitates some non standard ingredients of multigrid algorithms

conservative p 1 conforming and nonconforming galerkin fems effective flux evaluation via a nonmixed method approach given a p1 conforming or nonconforming galerkin finite element method gfem solution ph which approximates the exact solution p of the diffusion reaction equation nabla cdot mathcal k nabla p alpha p f with full tensor variable coefficient mathcal k we evaluate the approximate flux uh to the exact flux bf u mathcal k nabla p by a simple but physically intuitive formula over each finite element the flux is sought in the continuous in normal component or the discontinuous raviart thomas space a systematic way of deriving such a formula is introduced this direct method retains local conservation property at the element level typical of mixed methods finite element or finite volume type but avoids solving an indefinite linear system in short the present method retains the best of the gfem and the mixed method but without their shortcomings thus we view our method as a conservative gfem and demonstrate its equivalence to a certain mixed finite volume box method the equivalence theorems explain how the pressure can decouple basically cost free from the mixed formulation the accuracy in the flux is of first order in the h dive omega norm numerical results are provided to support the theory

multigrid algorithms for nonconforming and mixed methods for nonsymmetric and indefinite problems in this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for nonsymmetric and or indefinite elliptic problems we show that a simple cal v cycle multigrid iteration using conforming coarse grid corrections converges at a uniform rate provided that the coarsest level in the multilevel iteration is sufficiently fine but independent of the number of multigrid levels various types of smoothers for the nonsymmetric and indefinite problems are discussed extensive numerical results are presented

preconditioning nonconforming finite element methods this paper is concerned with the h sup 1 condition number and singular value distribution of preconditioned nonself adjoint discrete elliptic operators in particular a wide class of nonconforming finite element methods are investigated the main results give a precise interval on which the singular values cluster results of this kind have an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations

an optimal order multigrid method for p1 nonconforming finite elements etude d une methode multigrille d ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l aide d elements finis non conformes

a multilevel algorithm for the biharmonic problem a finite element discretization of the mixed variable formulation of the biharmonic problem is considered a multilevel algorithm for the numerical solution of the discrete equations is described convergence is proved under the assumption ofh 3 regularity

preconditioningp1 nonconforming finite elements condition numbers and singular value distributions this work deals with the condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are discretizations of second order elliptic operatorsa andb usingp 1 nonconforming finite elements of crouzeix and raviart b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval these reults are being extended to the aforementioned nonconforming finite elements it will be shown that for quasiuniform grids theb h singular values are bounded above and below by positive constants which are independent of the grid sizeh moreover they also cluster in a smaller but usually estimable interval issues of implementation are also discussed

analytic regularity of stokes flow on polygonal domains in countably weighted sobolev spaces we investigate the analytic regularity of the stokes problem in a polygonal domain r2 with straight sides and piecewise analytic data we establish a shift theorem in weighted sobolev spaces of arbitrary order with explicit control of the order dependence of the constants the shift theorem in the framework of countably weighted sobolev spaces implies in particular interior analyticity and gevrey type analytic regularity near the corners

velocity analysis using similarity weighted semblance weighted semblance can be used for improving the performance of traditional semblance for specific data sets we have developed a novel approach for prestack velocity analysis using weighted semblance the novelty came from a different weighting criteria in which the local similarity between each trace and a reference trace is used on the one hand low similarity corresponded to a noise point or a point indicating incorrect moveout which should be given a small weight on the other hand high similarity corresponded to a point indicating correct moveout that should be given a high weight our approach could also be effectively used for analyzing avo anomalies with increased resolution compared with ab semblance synthetic and field common midpoint gathers demonstrated higher resolution using the approach we developed applications of the proposed method on a prestack data set further confirmed that the stacked data using the similarity weighted semblance could obtain better energy focused events which indicated a more precise velocity picking

higher dimensional nonnested multigrid methods nonnested multigrid methods are shown to be optimal order solvers for systems of finite element equations arising from elliptic boundary problems in any space dimension results are derived for lagrange type elements of arbitrary degree

a generalized bpx multigrid framework covering nonnested v cycle methods more than a decade ago bramble pasciak and xu developed a framework in analyzing the multigrid methods with nonnested spaces or non inherited quadratic forms it was subsequently known as the bpx multigrid framework which was widely used in the analysis of multigrid and domain decomposition methods however the framework has an apparent limit in the analysis of nonnested v cycle methods and it produces a variable v cycle or nonuniform convergence rate v cycle methods or other nonoptimal results in analysis thus far this paper completes a long time effort in extending the bpx multigrid framework so that it truly covers the nonnested v cycle we will apply the extended bpx framework to the analysis of many v cycle nonnested multigrid methods some of them were proven previously only for two level and w cycle iterations some numerical results are presented to support the theoretical analysis of this paper

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

multilevel algorithms for mixed problems ii treatment of the mini element we present a multilevel algorithm for the mixed finite element approximation of the stokes problem by the mini element a local l 2 projection of the bubble functions and a strengthened cauchy inequality are essential tools for the convergence analysis of the multigrid algorithm

a renormalisation group method for iterated maps on the real axis a renormalisation group method for one dimensional iterated maps displaying period doublings is studied the renormalisation transformation involves variable rescaling similar to the doubling transformation of feigenbaum 1978 1979 but is defined on the space of parameters of a given mapping numerical calculations are carried out for maps with different analytic dependence near the extremum

multigrid computation of vector potentials in this paper we analyze the use of edge finite elements and the multigrid method to approximate the problem of computing a static magnetic field in a cavity we show how the mixed finite element discrete problem may be reformulated as a symmetric positive definite system this involves the proof of a discrete analogue of the second kind friedrichs inequality then we show how the multigrid method may be applied to compute the magnetic vector potential the difficulty here is that the bilinear forms on different meshes are not related in the standard way two different intergrid transfer operators are defined in the paper for the multigrid method the convergence theory and the numerical tests include both algorithms

an optimal order multigrid method for biharmonic c finite element equations in this paper we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the hsieh clough tocher c1 macro finite elements or several otherc1 finite elements since the multiplec1 finite element spaces considered are not nested the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method this method converges with the optimal computational order

renormalization group analysis of bifurcations in area preserving maps abstract two dimensional area preserving maps can be represented by a generating function the action high orders of successive period doubling bifurcations are studied by writing a renormalization group scheme for this action fixed points and eigenvalues of this scheme are found and interpreted

multigrid and multilevel methods for nonconforming q 1 elements in this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second order elliptic problems using nonconforming rotated q 1 finite elements in two space dimensions in particular for the case of square partitions and the laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the w cycle with any number of smoothing steps and close to optimal condition number estimates for v cycle preconditioners this is in contrast to most of the other nonconforming finite element discretizations where only results for w cycles with a sufficiently large number of smoothing steps and variable v cycle multigrid preconditioners are available some numerical tests including also a comparison with a preconditioner obtained by switching from the nonconforming rotated q 1 discretization to a discretization by conforming bilinear elements on the same partition illustrate the theory

on a hierarchical basis multilevel method with nonconforming p1 elements we develop a hierarchical basis multilevel method for symmetric second order elliptic boundary value problems in two dimensional polygonal domains based on nonconforming p1 triangular elements the main result is that the condition number of the hierarchical discretization is bounded byo k wherek is the number of refinement levels for comparison yserentant s result yields the sharp boundo k 2 for the corresponding hierarchical discretization with conforming linear elements