diffusion on self avoiding walk networks two properties of self avoiding walk networks constructed by connecting the nearest neighbour points of self avoiding walks on the square lattice are studied numerically the average end to end linear part of such networks is investigated and is found to be proportional to the length of the walk secondly random walks on this structure are studied and the associated spectral dimension of such networks is estimated to be 1 03 or 0 03 which implies that the effect of such connections affects the spectral dimension only weakly if at all

domain decomposition algorithms for mixed methods for second order elliptic problems domain decomposition algorithms for mixed finite element methods for linear second order elliptic problems in r 2 and r 3 are developed a convergence theory for two level and multilevel schwarz methods applied to the algorithms under consideration is given it is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems numerical experiments are presented to illustrate the present techniques

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

multigrid method and multilevel additive preconditioner for mixed element method for non self adjoint and indefinite problems in this paper a v cycle multigrid method is proposed for mixed element method for non self adjoint and indefinite second order elliptic problems and the uniform convergence of the v cycle multigrid method is proven under minimal regularity assumption meanwhile a multilevel additive preconditioner is given for these problems and an optimal convergence rate for preconditioned gmres method is obtained under minimal regularity assumption

to what class of fractals does the alexander orbach conjecture apply discussion de la decouverte numerique recente d alexander et orbach selon laquelle pour un agregat infini naissant en percolation le rapport d f d w dimension fractale de l agregat dimension fractale d une marche aleatoire dans l agregat est approximativement independant de d pour d 1

monte carlo study of random walks on a 2d gasket fractal in an external field using monte carlo simulation random walks on a two dimensional sierpinski gasket in the presence of an external field were studied it was observed that the random walk motion of a particle displayed a crossover from anomalous diffusion to drift for a non zero bias field and the crossover time tcr was a decreasing function of the external bias field the associated dynamic exponents obtained in a computer simulation agree with the predictions of stinchcombe s scaling treatment 1985

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

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

a new finite size scaling approach to random walks we present a new finite size scaling method for the random walks rw superseding a previously widely used renormalization group approach which is shown here to be inconsistent the method is valid in any dimension and is based on the exact solution for the two point correlation function and on finite size scaling as an example the phase diagram is derived for the random walk in two dimensions with a surface bulk interaction where the system has either a surface or a defect line we also discuss an initial calculation of the corresponding phase diagram for the case of a critically diluted lattice

on the constant positive linear dependence condition and its application to sqp methods in this paper we introduce a constant positive linear dependence condition cpld which is weaker than the mangasarian fromovitz constraint qualification mfcq and the constant rank constraint qualification crcq we show that a limit point of a sequence of approximating karush kuhn tucker kkt points is a kkt point if the cpld holds there we show that a kkt point satisfying the cpld and the strong second order sufficiency conditions ssosc is an isolated kkt point we then establish convergence of a general sequential quadratical programming sqp method under the cpld and the ssosc finally we apply these results to analyze the feasible sqp method proposed by panier and tits in 1993 for inequality constrained optimization problems we establish its global convergence under the ssosc and a condition slightly weaker than the mangasarian fromovitz constraint qualification and we prove superlinear convergence of a modified version of this algorithm under the ssosc and a condition slightly weaker than the linear independence constraint qualification

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

resistance and related scaling properties of self avoiding random walks three scaling exponents for random and self avoiding chains are studied by direct renormalisation these describe the end to end resistance of a conducting chain the arc length of the shortest path traceable between the chain ends and the arc length of chain in no loop the scaling behaviours of these quantities are identical to first order in epsilon 4 d d is the space dimension for a long self avoiding chain the authors predict that each one one is proportional to the chain length

random walk on self avoiding walk in external bias diffusion drift and trapping the effect of external bias on random walks rws on self avoiding walks on square lattices in the presence of bridges is studied by monte carlo simulation the qualitative as well as quantitative features of the latter problem are compared and contrasted with those of rws on percolation clusters in the presence of external bias

natural preconditioning and iterative methods for saddle point systems the solution of quadratic or locally quadratic extremum problems subject to linear ized constraints gives rise to linear systems in saddle point form this is true whether in the continuous or the discrete setting so saddle point systems arising from the discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as do for example interior point methods and the sequential quadratic programming approach to nonlinear optimization this survey concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods these preconditioners are related to the original extremum problem and their effectiveness in terms of rapidity of convergence is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matri

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

an application of the abstract multilevel theory to nonconforming finite element methods a multilevel product algorithm and its additive version or a multilevel preconditioner are proposed for the crouzeix raviart nonconforming finite element methods for second order elliptic equations the authors show that the preconditioned matrix has a condition number bounded uniformly from above if the domain can be partitioned into overlapping convex subdomains a uniform convergence for the multilevel product algorithm is also established the results are regularity free for the continuous problem and can be applied to problems with rough solutions and local grid refinement numerical results that illustrate the theory are presented

analysis of preconditioners for saddle point problems mixed finite element formulations give rise to large sparse block linear systems of equations the solution of which is often sought via a preconditioned iterative technique in this work we present a general analysis of block preconditioners based on the stability conditions inherited from the formulation of the finite element method the babuska brezzi or inf sup conditions the analysis is motivated by the notions of norm equivalence and field of values equivalence of matrices in particular we give sufficient conditions for diagonal and triangular block preconditioners to be norm and field of values equivalent to the system matrix

resistance of the edwards walk an analytical and numerical study of the scaling of edwards random walks defined as the zero component limit n to 0 of an n component g0 mod phi 2 mod 2 theory is done in three dimensions the hausdorf dimension 1 nu and the resistance exponent x is calculated using renormalisation group methods in 4 in dimensions the numerical study is done by defining a monte carlo algorithm on the cubic lattice to generate ensembles of edwards walks

anomalous diffusion on regular and random models for diffusion limited aggregation families of regular fractal models for witten sander dla clusters in d dimensions are proposed resembling the computer generated aggregates the authors evaluate random walk dimensionalities exactly using resistance scaling ideas the applicability of the conflicting conjectures of alexander orbach 1982 ao dw 3df 2 and aharony stauffer 1984 as dw df 1 to dla is then examined one finds that the dendritic nature of the fractals presented ensures that as is satisfied exactly in all dimensions an rsrg fugacity transformation method is applied to the simplest of the regular fractals in d 2 the result for dw is compared with the exact resistance scaling result to obtain an estimate of the error introduced by the rsrg the rg method is then generalised to treat the random dla problem the resulting estimate for dw is in fortuitously good agreement with as the corresponding monte carlo data and the results above for the regular fractal models

convergence and domain decomposition algorithm for nonconforming and mixed methods for nonselfadjoint and indefinite problems in this paper the existence uniqueness and convergence of the solutions of nonconforming and mixed methods with non quasi uniform partition are proven for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption moreover uniform convergence of the solution of nonconforming method is obtained meanwhile the domain decomposition algorithm which suits non quasi uniform partition is proposed for nonconforming and mixed methods

on linear monotone iteration and schwarz methods for nonlinear elliptic pdes the schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains the solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain in this paper proofs of convergence of some schwarz alternating methods for nonlinear elliptic problems which are known to have solutions by the monotone method also known as the method of subsolutions and supersolutions are given in particular an additive schwarz method for scalar as well some coupled nonlinear pdes are shown to converge to some solution on finitely many subdomains even when multiple solutions are possible in the coupled system case each subdomain pde is linear decoupled and can be solved concurrently with other subdomain pdes these results are applicable to several models in population biology

spectral dimension and the shortest path of saw s with multi neighbour interactions the spectral dimension and the shortest path of self avoiding walks saws with bridge length interaction range b 1 2 3 2 are studied numerically the spectral dimension is calculated by performing exact multi neighbour random walks on the monte carlo generated saw configurations it is found that the spectral dimension is not affected by the finite range of interactions finite bridge length and approach to ds 1 both in dimensions d 2 and 3 the shortest path length sn of the n step saw with local bridges is also investigated it is shown that sn n a n delta b c n the authors numerical simulation results indicate that the exponent delta is independent of the dimensionality and is about 3 16 in dimensions d 2 5

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

on combining feasibility descent and superlinear convergence in inequality constrained optimization extension of quasi newton techniques from unconstrained to constrained optimization via sequential quadratic programming sqp presents several difficulties among these are the possible inconsistency away from the solution of first order approximations to the constraints resulting in infeasibility of the quadratic programs and the task of selecting a suitable merit function to induce global convergence in ths case of inequality constrained optimization both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints and that yield monotonically decreasing objective function values feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not well defined outside the feasible set it has been recently shown that this can be achieved while preserving local two step superlinear convergence in this note the essential ingredients for an sqp based method exhibiting the desired properties are highlighted correspondingly a class of such algorithms is described and analyzed tests performed with an efficient implementation are discussed

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

shortest path of saws with bridges series results recent monte carlo simulation results of yang and chakrabarti 1990 suggest that the shortest path sn of an n stepped self avoiding walk saw with finite range interactions of bridges has a finite size scaling behaviour sn n approximately a n delta b c n where the exponent delta is superuniversal delta approximately 0 19 for all dimensions d studied 2 or d or 5 the authors report the small n series enumeration results for sn for saws with nearest neighbour bridges on square up to n 18 triangular up to n 11 and simple cubic up to n 12 lattices the estimated values of delta for different lattices approximately 0 22 or 0 01 for d 2 and approximately 0 26 or 0 01 for d 3 have been compared with the above monte carlo estimate and the indication of a superuniversal behaviour for delta has been discussed

two level additive schwarz preconditioners for nonconforming finite element methods two level additive schwarz preconditioners are developed for the nonconforming p1 finite element approximation of scalar second order symmetric positive definite elliptic boundary value problems the morley finite element approximation of the biharmonic equation and the divergence free nonconforming p1 finite element approximation of the stationary stokes equations the condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap

cluster renormalisation study of site lattice animals in two and three dimensions the author studies site lattice animals using the cluster renormalisation cr approach previously applied to bond animals the exponent v characterising the asymptotic behaviour of the mean square radius of gyration of site animals and the fractal dimension df 1 v are determined in two and three dimensions he finds v 0 649 or 0 009 and v approximately 0 51 in two and three dimensions respectively in addition he studies restricted valence site animals on a triangular lattice using a single parameter cr approach

diffusion on lattice animals and percolation clusters a renormalisation group approach a position space renormalisation group method is developed to study the scaling properties of random walks on fractals calculations in two dimensions for site lattice animals and site percolation clusters yield numerical results for the critical exponent defined by the asymptotic behaviour of the root mean square displacement of the walker the renormalisation group results are consistent with recent monte carlo calculations

mixed finite element methods for quasilinear second order elliptic problems a mixed finite element method is developed to approximate the solution of a quasilinear second order elliptic partial differential equation the existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived

a feasible sequential linear equation method for inequality constrained optimization in this paper by means of the concept of the working set which is an estimate of the active set we propose a feasible sequential linear equation algorithm for solving inequality constrained optimization problems at each iteration of the proposed algorithm we first solve one system of linear equations with a coefficient matrix of size m m where m is the number of constraints to compute the working set we then solve a subproblem which consists of four reduced systems of linear equations with a common coefficient matrix unlike existing qp free algorithms the subproblem is concerned with only the constraints corresponding to the working set the constraints not in the working set are neglected consequently the dimension of each subproblem is not of full dimension without assuming the isolatedness of the stationary points we prove that every accumulation point of the sequence generated by the proposed algorithm is a kkt point of the problem moreover after finitely many iterations the working set becomes independent of the iterates and is essentially the same as the active set of the kkt point in other words after finitely many steps only those constraints which are active at the solution will be involved in the subproblem under some additional conditions we show that the convergence rate is two step superlinear or even q superlinear we also report some preliminary numerical experiments to show that the proposed algorithm is practicable and effective for the test problems

mixed finite element domain decomposition for nonlinear parabolic problems abstract fully discrete mixed finite element method is considered to approximate the solution of a nonlinear second order parabolic problem a massively parallel iterative procedure based on domain decomposition technique is presented to solve resulting nonlinear algebraic equations robin type boundary conditions are used to transmit information between subdomains the convergence of the iteration for each time step is demonstrated optimal order error estimates are also derived numerical examples are given

a superlinearly convergent feasible method for the solution of inequality constrained optimization problems when iteratively solving optimization problems arising from engineering design applications it is sometimes crucial that all iterates satisfy a given set of hard inequality constraints and generally desirable that the objective function value improve at each iteration in this paper we propose an algorithm of the successive quadratic programming sqp type which unlike other algorithms of this type does enjoy such properties under mild assumptions the new algorithm is shown to converge from any initial point locally superlinearly numerically tested it has proven to be competitive with the most successful currently available nonlinear programming algorithms while the latter do not exhibit the desired properties

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

an analysis of monotonicity conditions in the mixed hybrid finite element method on unstructured triangulations in this paper we consider the mixed hybrid finite element method on unstructured triangular grids and evaluate its monotonicity properties by using a non standard set of basis functions for the velocity approximation space the mixed hybrid discretization of the steady state diffusion equation produces a system matrix that depends only on the inner product of the outward normals to the edges of the triangulation and not on the choice of the velocity space basis this property is used to study the characteristics of the system matrix it is well known that this matrix is of type m if the angles of the triangulation are not bigger than 2 an m matrix has a nonnegative inverse i e all the elements are nonnegative this implies the existence of a discrete maximum principle and thus monotonicity of the discretization we show that when the triangulation is of delaunay type and satisfies the property that no circumcenters of boundary elements with dirichlet conditions lie outside the domain the inverse of the final matrix is always positive even in the presence of obtuse angles

cell renormalisation of growth processes true self avoiding walks and growing animals the authors present a generating function formalism for growth processes in general and introduce a new cell renormalisation scheme as examples they treat the true self avoiding walk and the eden process or growing animals in two and three dimensions using small cells the results in both cases indicate a substantial increase in the fractal dimension d compared with their equilibrium counterparts the case of the eden process may however suggest a peculiar convergence behaviour as the cell size b tends to infinity

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

uniform convergence and preconditioning methods for projection nonconforming and mixed methods for nonselfadjoint and indefinite problems abstract the purpose of this paper is to prove the existence uniqueness and uniform convergence of the solutions of so called projection nonconforming and mixed element methods and the equivalence between projection nonconforming element method and mixed element method with nonquasi uniform partition for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption meanwhile the optimal error estimate of the solution of mixed element method is obtained under h 2 smoothness hypothesis for nonselfadjoint and indefinite elliptic problems without h 2 regularity finally the discrete compactness result for nonconforming element space with nonquasi uniform partition is proven and some preconditioning methods for projection nonconforming and mixed element methods with nonquasi uniform partition are given it is proven that the h 1 condition number of preconditioned operator is uniformly bounded and its singular values cluster in a relatively small finite interval

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part i algorithms and numerical results we describe sequential and parallel algorithms based on the schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the raviart thomas finite element spaces these lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block gauss seidel or block jacobi methods with overlapping blocks the indefiniteness requires special treatment the sub blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of dryja and widlund for standard elliptic problems if there is sufficient overlap between the subdomains the algorithm converges with a rate independent of the mesh size the number of subdomains and discontinuities of the coefficients extensions of the above algorithms to the case of local grid refinement is also described convergence theory for these algorithms will be presented in a subsequent paper

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

anisotropy in field assisted diffusion on a gasket fractal the random walk on the two dimensional gasket fractal has been studied with various forms of constant bias fields applied using the monte carlo method crossover from anomalous diffusion to drift was observed in the horizontal and upward bias fields while crossover from anomalous to normal diffusion was found in the downward bias field the scaling relation r bs t approximately tkf bstk was confirmed and the scaling exponent of f x approximately xbeta at large x was obtained as beta 1 175 or 0 004 for the horizontal bias field and beta 0 319 or 0 002 for the downward bias field in good agreement with the theoretical work by parrondo et al 1990

preconditioning projection nonconforming element method for the lowest order raviart thomas mixed triangular element method this paper is concerned with the h 1 condition number and the b h singular value distribution of preconditioned operators b h 1a h 0 h 1 where a h is the projection nonconforming element discretization of the nonself adjoint and possibly indefinite second order elliptic operator a b h is the nonconforming element discretization of the self adjoint and positive definite second order elliptic operator b the projection nonconforming element method is equivalent to the lowest order raviart thomas mixed triangular element method for second order elliptic problems it is proved that the h 1 condition number of operators b h 1a h 0 h 1 are uniformly bounded and the b h singular values cluster in a positive finite interval finally a simple v cycle multigrid implementation of b h 1 is given

random walk on self avoiding walk a model for conductivity of linear polymers random walks on self avoiding walks saws are studied using monte carlo techniques on a square lattice with nearest neighbour hopping along the chain and between saw points which are nearest neighbours on the embedding lattice the average of the square of the end to end distance for random walks of t steps on saws of length n is fitted to the scaling forms rt2 varies as ndelta tk for t or approximately ntheta where theta approximately 2 nu s k nu s being the average end to end distance exponent for saws the observed value of the exponent delta is supported by the authors real space renormalisation group result for the conductivity of saw chains the exponent k has been related to the effective fractal dimension of the saw chain

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part ii convergence theory in this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric indefinite linear systems arising from mixed finite element discretizations of elliptic problems the algorithms include schwarz methods and iterative refinement methods on locally refined grids the implementation of schwarz and iterative refinement algorithms have been discussed in part i a discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived

corrigendum on the constant positive linear dependence condition and its application to sqp methods we correct an assertion in the quoted paper qi and wei siam j optim 10 2000 pp 963 981

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

bdm mixed methods for a linear elliptic problem abstract a new mixed formulation is introduced to approximate the solution of a nonlinear elliptic problem based on the brezzi douglas marini mixed finite elements the existence and uniqueness of solutions of the mixed formulation and its discretization are demonstrated optimal error estimates in l 2 l and h s are derived numerical results are presented to compare the present approach with the standard one and to justify the theoretical results obtained here

scaling treatment of anisotropic diffusion exact scaling treatments are given of anisotropic diffusion in pure and diluted chains and in a fractal a dynamic decimation method is used to obtain detailed dynamic scaling descriptions the results for chains show crossover to drift or localised behaviour induced by anisotropy or dilution the fractal results include bias induced crossover with exponent unity from anomalous diffusion behaviour exponent log2 5 to drift behaviour and scaling corrections exponent log2 5 3 arising from rotational anisotropy

resistance of random walks unrestricted lattice random walks in which a unit conductor is placed along each bond traversed are considered the mean end to end resistance is studied as a function of the number of steps in the walk and the spatial dimension a critical scaling law is found whose exponent is consistently given by four different calculational schemes

diffusion and fracton dimensionality on fractals and on percolation clusters the authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions the results for the fractal support the rammal and toulouse proposition 1983 that ds n dn varies as b n s n the results for percolation are in excellent agreement with the alexander and orbach conjecture 1982 that the fracton dimensionality d 4 3

a parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods a parallelizable interative procedure based on domain decomposition techniques is defined and analyzed for mixed finite element methods for elliptic equations with the analysis being presented for the decomposition of the domain into the individual elements associated with the mixed method or into larger subdomains applications to time dependent problems are indicated

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

the notion of a strange fractal phenomenon disorder induced walk it is argued that a random system coupled with a walk may cause an unusual phenomenon which shares some of the opposite properties of fractals

homogeneity and spectral dimension of aggregation fractals the fracton or spectral dimension ds of two sutherland type ghost aggregation fractals is obtained for high space dimension by a scaling argument if as seems probable these fractals are homogeneous the results ds 2ln2 ln3 and ds 4 square root 17 1 support recent conjectures that ds is close to 4 3 for all homogeneous fractals however they are not consistent with the stronger conjecture that ds is exactly 4 3 for all such fractals