on the uniqueness of static perfect fluid solutions in general relativity following earlier work of masood ul alam we consider a uniqueness problem for non rotating stellar models given a static asymptotically flat perfectfluid spacetime with barotropic equation of state p and given another such spacetime which is spherically symmetric and has the same p and the same surface potential we prove that both are identical provided p satisfies a certain differential inequality this inequality is more natural and less restrictive than the conditions required by masood ul alam

least squares mixed finite element methods for non selfadjoint elliptic problems i error estimates a least squares mixed finite element method for general second order non selfadjoint elliptic problems in two and three dimensional domains is formulated and analyzed the finite element spaces for the primary solution approximation u h and the flux approximation sigma h consist of piecewise polynomials of degree k and r respectively the method is mildly nonconforming on the boundary the cases k r and k 1 r are studied it is proved that the method is not subject to the lbb condition optimal l 2 and h 1 error estimates are derived for regular finite element partitions numerical experiments confirming the theoretical rates of convergence are presented

asymptotically simple solutions of the vacuum einstein equations in even dimensions we show that a set of conformally invariant equations derived from the fefferman graham tensor can be used to construct global solutions of vacuum einstein equations in all even dimensions this gives in particular a new simple proof of friedrich s result on the future hyperboloidal stability of minkowski space time and extends its validity to even dimensions

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

least squares for second order elliptic problems abstract in this paper we introduce and analyze two least squares methods for second order elliptic differential equations with mixed boundary conditions these methods extend to problems which involve oblique derivative boundary conditions as well as nonsymmetric and indefinite problems as long as the original problem has a unique solution with the methods to be developed neumann and oblique boundary conditions are imposed weakly and thus avoid compatibility conditions on the finite element subspaces the resulting least squares approximations are unconditionally stable no conditions on the step size h and will be shown to converge at an optimal rate the first least squares method involves a discrete computable h 1 norm of the residual and stabilization terms consisting of the jumps at the interelement boundaries and a weighted elementwise l 2 norm of the residual over the finite elements this method is developed without the introduction of additional problem variables the second method involves the use of the flux as an additional unknown although this method is similar to the least squares method for first order systems introduced in 7 it differs in that discontinuous finite elements are allowed it is also more general in that it extends to the oblique boundary problem

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

the l 2 norm error estimates for the div least squares method this paper studies l 2 norm error estimates for the div least squares method for which the associated homogeneous least squares functional is equivalent to the h mbox div times h 1 norm for the respective dual and primal variables least squares of this type for the second order elliptic equations elasticity and the stokes equations are an active area of research and error estimates in the h mbox div times h 1 norm were previously established in this paper we establish optimal l 2 norm error estimates for the primal variable under the minimum regularity requirement through a refined duality argument

goal oriented local a posteriori error estimators for h div least squares finite element methods we propose a goal oriented local a posteriori error estimator for h div least squares ls finite element methods our main interest is to develop an a posteriori error estimator for the flux approximation in a preassigned region of interest d subset omega the estimator is obtained from the ls functional by scaling residuals with proper weight coefficients the weight coefficients are given in terms of local mesh size h t and a function omega d depending on the distance to d this new error estimator measures the pollution effect from the outside region of d and provides a basis for local refinement in order to efficiently approximate the solution in d numerical experiments show superior performances of our goal oriented a posteriori estimators over the standard ls functional and global error estimators

pointwise error estimates for first order div least squares finite element methods and applications to superconvergence and a posteriori error estimators first order div least squares finite element methods for second order elliptic partial differential equations are considered while there has been significant progress in error estimates for the methods to the best of our knowledge these estimates are based on the global norm of the error in this paper we provide highly localized pointwise error estimates for the primary variable u by establishing that the least squares solutions are locally higher order perturbations of the standard galerkin solutions as elementary consequences we identify the superconvergent points the set of superconvergent points for the primary function of the least squares solution is the same as that of the standard galerkin solution also we present a class of average type a posteriori error estimators for the method and conditions are given under which they are asymptotically exact or equivalent estimators in the maximum norm on each single element of the underlying mesh

computer simulation of a model for irreversible gelation a random mixture of bifunctional and tetrafunctional units is placed on a simple cubic lattice permanent bonds between these units are then formed by the random motion of active centres radicals resulting in a model for the gelation of polyacrylamide and similar processes the largest of the clusters formed by this kinetic percolation process is identified with the gel fraction the mean cluster size with the weight average degree of polymerisation the critical exponent gamma of the mean cluster size is roughly the same as for random percolation that is different from that of the classical theory of flory 1941 and stockmayer 1943 but the corresponding amplitude ratio that is the ratio of average molecular weight on both sides of the gel point differs strongly from its value in random percolation thus this kinetic gelation model seems to belong to a universality class of its own different from both that of random percolation and that of classical gelation theories

weak coupling of solutions of first order least squares method a theoretical analysis of a first order least squares finite element method for second order self adjoint elliptic problems is presented we investigate the coupling effect of the approximate solutions u h for the primary function u and h for the flux a u we prove that the accuracy of the approximate solution u h for the primary function u is weakly affected by the flux a u that is the bound for u u h 1 is dependent on but only through the best approximation for multiplied by a factor of meshsize h similarly we provide that the bound for h h div is dependent on u but only through the best approximation for u multiplied by a factor of the meshsize h this weak coupling is not true for the non selfadjoint case we provide the numerical experiment supporting the theorems in this paper

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

a posteriori error estimators for the first order least squares finite element method in this paper we propose a posteriori error estimators for certain quantities of interest for a first order least squares finite element method in particular we propose an a posteriori error estimator for when one is interested in email protected s email protected 0 where email protected u our a posteriori error estimators are obtained by assigning proper weight in terms of local mesh size h t to the terms of the least squares functional an a posteriori error analysis yields reliable and efficient estimates based on residuals numerical examples are presented to show the effectivity of our error estimators

maximal hypersurfaces in stationary asymptotically flat spacetimes existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike near infinity the first class consists of strongly causal asymptotically flat spacetimes which contain no black hole or white hole but may contain ergoregions where the killing orbits fail to be timelike the second class of spacetimes possess a black hole and a white hole with the black and white hole horizons intersecting in a compact 2 surfaces

a posteriori error estimates for the primary and dual variables for the div first order least squares finite element method abstract we propose a posteriori error estimators for first order div least squares ls finite element method for linear elasticity stokes equations and general second order scalar elliptic problems our main interest is obtaining a posteriori error estimators for the dual variables fluxes strains stress etc which are main quantity of interest in many applications we also provide a posteriori error estimators for the primary variable these estimators are obtained from the local least squares functional by assigning weight coefficients scaling the respective residuals the weight coefficients are given in terms of local meshsize h k we establish the global upper bounds and local lower bounds for the estimators the estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution the indicators and estimators are identical numerical experiments show a superior performance of our a posteriori estimators for user specific norm over the standard ls functional

finite element approximation for grad div type systems in the plane this paper studies a 3 times 3 linear system of differential equations of grad div type with appropriate boundary conditions in constructing numerical approximate solutions to this boundary val

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

on finite element approximation of the gradient for solution of poisson equation a nonconforming mixed finite element method is presented for approximation of w with w f w r 0 convergence of the order left nabla w u h right 0 omega mathcal o h 2 is proved when linear finite elements are used only the standard regularity assumption on triangulations is needed

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

monte carlo and series study of corrections to scaling in two dimensional percolation corrections to scaling for percolation cluster numbers in two dimensions are studied by monte carlo simulations of very large systems up to 17 109 lattice sites and by series analysis both series and monte carlo work suggests that the value of the correction to scaling exponent is slightly lower at the percolation threshold than away from it moreover the corrections to scaling observed at pc omega equivalent to 0 64 might be due to the mixing of scaling fields rather than to the irrelevant scaling fields the monte carlo results are compatible with finite size scaling and finite size scaling corrections are estimated technical problems associated with monte carlo simulation of very large systems are discussed in an appendix

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

the multipole structure of stationary space times a definition of multipole moments for stationary asymptotically flat solutions of einstein s equations is proposed it is shown that these moments characterize a given space time uniquely conversely they can be arbitrarily prescribed i e they generate power series for the field variables which satisfy the field equations to all orders despite their apparently rather different origin they are shown to be identical with the geroch hansen ones

monte carlo experiments on percolation the influence of boundary conditions the slow convergence of the cluster size distribution observed in recent monte carlo simulations of two dimensional site percolation is shown to be due to the use of free boundary conditions when periodic boundaries are employed the convergence is improved considerably to the extent that the integrated size distribution for large typically 1010 site lattices is essentially flat over a range of cluster sizes spanning three orders of magnitude the existence of this plateau originally predicted by scaling theory leads to an improved estimate of the critical probability for the square lattice the observed correction to scaling is not readily characterised by a single additive term a fact which helps one to understand the wide range of published exponent estimates

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

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

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

critical behaviour of a kinetic gelation model with a constant initiator creation rate the authors present results from a computer simulation study of a radical initiated kinetic growth model in which the radicals initiators are created with a constant rate rc simulations of the growth process were carried out on simple cubic lattices as large as 100 100 100 for two values of the initiator creation rate finite size scaling analyses of the bulk properties yield critical exponents which are the same to within error bars as those found in similar kinetic gelation models and in percolation studies the critical amplitude ratio c c is consistent with the values obtained in the kinetic gelation studies with fixed initiator concentrations but is clearly different from percolation values the cluster size distribution shows the same monotonic behaviour as seen in percolation and is markedly different from the oscillatory behaviour seen in an investigation of a similar gelation model the scaling properties of the cluster size distribution however cannot be described using simple droplet theory instead they use a generalised scaling form which produces a good fit to the cluster data

on existence and behaviour of asymptotically flat solutions to the stationary einstein equations we show existence and uniqueness of asymptotically flat solutions to the stationary einstein equations ins 3 b r whereb r is a ball of radiousr 0 when a small enough continuous complex function u on s is given regularity and decay estimates imply that these solutions are analytic in the interior ofs and also at infinity when suitably conformally rescaled