a moving mesh method for the solution of the one dimensional phase field equations a moving mesh method is developed for the numerical solution of one dimensional phase change problems modelled by the phase field equations the computational mesh is obtained by equidistribution of a monitor function tailored for the functional variation of the phase field in the interfacial region existence and uniqueness of the discretised equations using a moving mesh are also established numerical results are given for classical and modified stefan test problems the numerical algorithm is relatively simple and is shown to be far more efficient than fixed grid methods

quantitative phase field modeling of dendritic growth in two and three dimensions we report the results of quantitative phase field simulations of the dendritic crystallization of a pure melt in two and three dimensions these simulations exploit a recently developed thin interface limit of the phase field model a karma and w j rappel phys rev e 53 r3017 1996 which is given here a detailed exposition this limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient simulations in one and two dimensions are first carried out to test the accuracy of phase field computations performed within this limit dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method both with and without interface kinetics simulations in three dimensions exploit in addition to the asymptotics a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies they are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three dimensional morphology of needle crystals without sidebranches for small crystalline anisotropy simulated values of ensuremath sigma are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy in contrast for large anisotropy simulated ensuremath sigma values are significantly larger than the predicted values this disagreement however does not signal a breakdown of solvability theory it is consistent with the finding that the amplitude of the mathrm cos 4 ensuremath varphi mode which measures the departure of the tip morphology from a shape of revolution increases with anisotropy this departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy finally the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law r ensuremath sim z with a slope dr dz that is nearly equal to the ratio of the two dimensional and three dimensional steady state tip velocities furthermore the evolution of the cross section of a three dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two dimensional growth shape in time in accord with the current theory of the three dimensional needle crystal shape

simulating finger phenomena in porous media with a moving finite element method the non equilibrium richards equation is solved using a moving finite element method in this paper the governing equation is discretized spatially with a standard finite element method and temporally with second order runge kutta schemes a strategy of the mesh movement is based on the work by li et al r li t tang p w zhang a moving mesh finite element algorithm for singular problems in two and three space dimensions journal of computational physics 177 2002 365 393 a beckett and mackenzie type monitor function is adopted to obtain high quality meshes around the wetting front a smoothing method which is based on the diffusive mechanism is used with the moving mesh technique high mesh quality and high numerical accuracy are obtained successfully the numerical convergence and the advantage of the algorithm are demonstrated by a series of numerical experiments

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

a residual based a posteriori error estimator for the stokes darcy coupled problem in this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow the flows are governed by the stokes and darcy equations respectively and the transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the finite element subspaces consider bernardi raugel and raviart thomas elements for the velocities piecewise constants for the pressures and continuous piecewise linear elements for a lagrange multiplier defined on the interface we derive a reliable and efficient residual based a posteriori error estimator for this coupled problem the proof of reliability makes use of suitable auxiliary problems diverse continuous inf sup conditions satisfied by the bilinear forms involved and local approximation properties of the clement interpolant and raviart thomas operator on the other hand helmholtz decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are the main tools for proving the efficiency of the estimator up to minor modifications our analysis can be extended to other finite element subspaces yielding a stable galerkin scheme

variational algorithms and pattern formation in dendritic solidification we present a completely new variational algorithm for computing dendritic solidification this algorithm reproduces the gibbs thomson relation as a balance between bulk and surface energy and is able to operate in the infinite mobility limit with no unphysical time step restriction it may be used with arbitrary non smooth surface energy functions and may include finite kinetic mobility we perform computations with isotropic and anisotropic surface energy from a small irregular initial seed we generate radial tip splitting structures for isotropic energy and parabolic dendrites with side branching for anisotropic energy for anisotropic energy the final structure is determined by the material and environmental properties the initial shape is forgotten for the parabolic dendrite tips we obtain agreement with the ivantsov solutions within a few percent and proper dimensional scaling of lengths and velocities with surface energy

coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

asymptotic analysis of the laminar viscous flow over a porous bed we consider the laminar viscous channel flow over a porous surface the size of the pores is much smaller than the size of the channel and it is important to determine the effective boundary conditions at the porous surface we study the corresponding boundary layers and by a rigorous asymptotic expansion we obtain saffman s modification of the interface condition observed by beavers and joseph the effective coefficient in the law is determined through an auxiliary boundary layer type problem whose computational and modeling aspects are discussed in detail furthermore the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size ep finally we give the interface condition linking the effective pressure fields in the porous medium and in the channel and we determine the jump of the effective pressures explicitly

error analysis for constrained first order system least squares finite element methods in this paper a general error analysis is provided for finite element discretizations of partial differential equations in a saddle point form with divergence constraint in particular this extends upon the work of j h adler and p s vassilevski springer proc math statist 45 springer new york 2013 pp 1 19 giving a general error estimate for finite element problems augmented with a divergence constraint and showing that these estimates are obtained for problems such as diffusion and stokes using the first order system least squares fosls finite element method the main result is that by enforcing the constraint on a mathbf h 1 equivalent fosls formulation one maintains optimal convergence of the fosls functional i e the energy norm of the error while guaranteeing the conservation of the divergence constraint i e mass conservation in some examples the error estimates and results depend on using finite elements for the constraint space that are inf sup stable when paired wi

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

an alternating crank nicolson method for decoupling the ginzburg landau equations the time dependent ginzburg landau model of superconductors consists of coupled nonlinear partial differential equations which presents difficulties in the numerical solution we present an alternating crank nicolson method for this model that leads to two decoupled algebraic subsystems one is linear and the other is semilinear both have nice numerical properties and can be solved by efficient matrix solvers we show the stability and convergence and derive error estimates for this scheme numerical results are also reported

moving mesh methods for singular problems on a sphere using perturbed harmonic mappings this work is concerned with developing moving mesh strategies for solving problems defined on a sphere to construct mappings between the physical domain and the logical domain it has been demonstrated that harmonic mapping approaches are useful for a general class of solution domains however it is known that the curvature of the sphere is positive which makes the harmonic mapping on a sphere not unique to fix the uniqueness issue we follow sacks and uhlenbeck ann of math 2 113 1981 pp 1 24 to use a perturbed harmonic mapping in mesh generation a detailed moving mesh strategy including mesh redistribution and solution updating on a sphere will be presented the moving mesh scheme based on the perturbed harmonic mapping is then applied to the moving steep front problem and the fokker planck equations with high potential intensities on a sphere the numerical experiments show that with a moderate number of grid points our proposed moving mesh algorithm can accurately resolve detailed features of singular problems on a sphere

a crank nicolson discontinuous finite volume element method for a coupled non stationary stokes darcy problem abstract in this paper we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a crank nicolson discretization in time and we obtain the solution at the first time step by applying a first order backward euler method to solve the coupling of time dependent stokes darcy problem the proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head whereas the pressure is approximated by the piecewise constant elements then the unique solvability of the approximate solution for the discrete problem is derived in addition the optimal error estimates of the full discretization in standard l 2 norm and broken h 1 norm are obtained finally a series of numerical experiments are provided to illustrate the features of the proposed method such as the optimal accuracy orders adaptive mesh refinement mass conservation capability to conveniently deal with complicated geometries with different types of boundary conditions and the applicability to the problems with realistic parameters in particular the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain it encompasses both fractures with high permeability which act as the prior flow conduit and fractures with low permeability as a flow barrier

a numerical investigation of blow up in reaction diffusion problems with traveling heat sources this paper studies the numerical solution of a reaction diffusion differential equation with traveling heat sources according to the fact that the locations of heat sources are known we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem several examples are provided to demonstrate the efficiency of the new moving mesh method especially in the case of two or three traveling heat sources moreover numerical results illustrate that the speed of the movement of the heat source is critical for blow up when there is only one traveling heat source for the case of two traveling heat sources blow up depends not only on the speed but also on the distance between the two traveling heat sources

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

a domain decomposition method for the steady state navier stokes darcy model with beavers joseph interface condition this paper proposes and analyzes a robin type multiphysics domain decomposition method ddm for the steady state navier stokes darcy model with three interface conditions in addition to the two regular interface conditions for the mass conservation and the force balance the beavers joseph condition is used as the interface condition in the tangential direction the major mathematical difficulty in adopting the beavers joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system y cao et al comm math sci 8 2010 pp 1 25 y cao et al siam j numer anal 47 2010 pp 4239 4256 in this paper the well posedness of the navier stokes darcy model with beavers joseph condition is analyzed by using a branch of nonsingular solutions by following the idea in y cao et al numer math 117 2011 pp 601 629 the three physical interface conditions are utilized together to construct the robin type boundary conditions on th

accuracy enhancement for non isoparametric finite element simulations in curved domains application to fluid flow abstract among a few known techniques the isoparametric version of the finite element method for meshes consisting of curved triangles or tetrahedra is the one most widely employed to solve pdes with essential conditions prescribed on curved boundaries it allows to recover optimal approximation properties that hold for elements of order greater than one in the energy norm for polytopic domains however besides a geometric complexity this technique requires the manipulation of rational functions and the use of numerical integration we consider a simple alternative to deal with dirichlet boundary conditions that bypasses these drawbacks without eroding qualitative approximation properties in the present work we first recall the main principle this technique is based upon by taking as a model the solution of the poisson equation with quadratic lagrange finite elements then we show that it extends very naturally to viscous incompressible flow problems although the technique applies to any higher order velocity pressure pairing as an illustration a thorough study thereof is conducted in the framework of the stokes system solved by the classical taylor hood method

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

sparsity and its applications this volume is based on a series of lectures prepared by a group of internationally renowned experts in the field for a meeting on the general theme of sparsity and its applications at the university of technology loughborough in april 1983 an introductory section briefly surveys the field of sparsity research and summarises the main ideas discussed in each contribution the contributions themselves examine the design analysis implementation and application of computer algorithms with a view to producing mathematical software which can be used as an efficent tool in different areas of science and engineering technology

cluster robust bootstrap inference in quantile regression models abstractin this article i develop a wild bootstrap procedure for cluster robust inference in linear quantile regression models i show that the bootstrap leads to asymptotically valid inference on the entire quantile regression process in a setting with a large number of small heterogeneous clusters and provides consistent estimates of the asymptotic covariance function of that process the proposed bootstrap procedure is easy to implement and performs well even when the number of clusters is much smaller than the sample size an application to project star data is provided supplementary materials for this article are available online

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

on balanced moving mesh methods moving mesh methods are a widely used approach in the numerical solution of pdes where the original pdes are transformed from a physical domain to a computational domain the objective is to utilize a uniform mesh in the computational domain to get a non uniform physical mesh that better captures the behavior of the solution the movement of the physical mesh points can be governed by a moving mesh pde associated with a corresponding monitor function and both the original pdes and the moving mesh pdes must be solved simultaneously the motivation for this paper is to study a balanced moving mesh method where the aim is to strike a balance between the properties of the solution of the original pde and that of the moving mesh pde we focus on particular choices of the monitor function that give both a well behaved mesh transformation and a well behaved solution in the computational domain both theoretical analysis and numerical experiments are presented as illustrations

first order system least squares for the signorini contact problem in linear elasticity a first order system least squares formulation for the signorini problem modeling frictionless contact in linear elasticity is studied in addition to the displacement field the stress tensor is used as an independent process variable a contact boundary term is added to the usual least squares functional in order to achieve coercivity and continuity in appropriate norms the discrete functional is shown to constitute an a posteriori error estimator on which an adaptive refinement strategy may be based as finite element spaces standard conforming piecewise polynomials for the displacement approximation are combined with raviart thomas elements for the rows in the stress tensor computational results for a test problem of hertzian contact illustrate the effectiveness of our least squares approach

adaptive moving grid methods for two phase flow in porous media a b s t r a c t in this paper we present an application of the moving mesh method for approximating numerical solutions of the two phase flow model in porous media the numerical schemes combine a mixed finite element method and a finite volume method which can handle the nonlinearities of the governing equations in an efficient way the adaptive moving grid method is then used to distribute more grid points near the sharp interfaces which enables us to obtain accurate numerical solutions with fewer computational resources the numerical experiments indicate that the proposed moving mesh strategy could be an effective way to approximate two phase flows in porous media

a moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

identifying social interactions through conditional variance restrictions this paper proposes a new method for identifying social interactions using conditional variance restrictions the method provides a consistent estimate of the social multiplier when social interactions take the linear in means form manski 1993 when social interactions are not of the linear in means form the estimator under certain conditions continues to form the basis of a consistent test of the no social interactions null with correct large sample size the methods are illustrated using data from the tennessee class size reduction experiment project star the application suggests that differences in peer group quality were an important source of individual level variation in the academic achievement of project star kindergarten students

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

a fully implicit fully adaptive time and space discretisation method for phase field simulation of binary alloy solidification a fully implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2d is presented in addition we combine a second order fully implicit time discretisation scheme with variable step size control to obtain an adaptive time and space discretisation method the superiority of this method compared to widely used fully explicit methods with respect to cpu time and accuracy is shown due to the high nonlinearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step we use a nonlinear multigrid solver which shows almost h independent convergence behaviour

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

first order system least squares for coupled stokes darcy flow the coupled problem with stokes flow in one subdomain and a darcy flow model in a second subdomain is studied in this paper both flow problems are treated as first order systems involving pseudostress and velocity in the stokes case and using a flux pressure formulation in the darcy subdomain as process variables respectively the beavers joseph saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems a combination of h mathrm div conforming raviart thomas and standard h 1 conforming elements is used for the stokes as well as for the darcy subsystem the homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates it also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

inferring optimal peer assignment from experimental data this article studies the problem of optimally dividing individuals into peer groups to maximize social gains from heterogeneous peer effects the specific setting analyzed here concerns efficient ways of allocating roommates in college dormitories using confidential data on a sample of dartmouth college freshmen who were randomly assigned to be roommates the article derives efficient roommate pairing rules based on demographic and academic background which maximize different aggregate outcomes segregation by precollege academic standing and by race are seen to minimize mean enrolment into sororities and maximize mean enrolment into fraternities segregation has no effect on mean and median freshman year grade point average gpa but increases the higher and decreases the lower percentiles of the gpa distribution for both men and women efficiency loss due to legal constraints on allocations e g race blindness is shown to be significant and also larger for women for whom peer effects are more nonl

parametric raviart thomas elements for mixed methods on domains with curved surfaces the finite element approximation on curved boundaries using parametric raviart thomas spaces is studied in the context of the mixed formulation of poisson s equation as a saddle point system it is shown that optimal order convergence is retained on domains with piecewise c k 2 boundary for the parametric raviart thomas space of degree k geq 0 under the usual regularity assumptions this extends the analysis in f bertrand s munzenmaier and g starke siam j numer anal 52 2014 pp 3165 3180 from the first order system least squares formulation to mixed approaches of saddle point type in addition a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two dimensional case moreover the appropriate treatment of inhomogeneous flux boundary conditions is discussed the results are confirmed by computational results which also demonstrate that optimal order convergence is not achieved in general if standard raviart th

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

first order system least squares on curved boundaries higher order raviart thomas elements with this paper our investigation of the finite element approximation on curved boundaries using raviart thomas spaces in the context of first order system least squares methods is continued and extended to the higher order case it is shown that the optimal order of convergence is retained from the lowest order case if a parametric version of raviart thomas elements is used this is illustrated numerically for an elliptic boundary value problem involving a circular boundary curve

least squares mixed finite element solution of variably saturated subsurface flow problems a least squares mixed finite element formulation is applied to the nonlinear elliptic problems arising in each time step of an implicit euler discretization for variably saturated flow problems this approach simultaneously constructs approximations to the flux in raviart thomas spaces and to the hydraulic potential by standard h1 conforming linear finite elements two important properties of the least squares approach are investigated in detail the local least squares functional provides an a posteriori error estimator and gauss newton methods are robust iterative solvers for the resulting nonlinear least squares problems computational experiments conducted for a realistic water table recharge problem illustrate the effectiveness of this approach

efficient r adaptive isogeometric analysis with winslow s mapping and monitor function approach abstract motivated by the moving mesh method in conventional finite element analysis in this paper we propose an r adaptive isogeometric analysis framework by winslow s mapping and monitor function approach we construct a new monitor function based on the absolute curvature metric of the isogeometric solution surface in the proposed r adaptive framework the redistribution of the inner control points is performed to increase the density of control points in the regions where curvature based or error based monitor functions are maximum the new locations of the inner control points are obtained by solving governing pdes using isogeometric collocation method which is more efficient in solving large scale problems by employing winslow s mapping we ensure that the generated r adaptive parameterization is valid for isogeometric analysis and can be used as optimal initial grid for subsequent h refinements multiple two dimensional heat conduction examples are solved which confirm the effectiveness of the proposed method

first order system least squares method for the interface problem of the stokes equations the first order system least squares method for the stokes equation with discontinuous viscosity and singular force along the interface is proposed and analyzed first interface conditions are derived by introducing a physical meaningful variable such as the velocity gradient the stokes equation transformed into a first order system of equations then the continuous and discrete norm least squares functionals using legendre and chebyshev weights for the first order system are defined we showed that continuous and discrete homogeneous least squares functionals are equivalent to appropriate product norms the spectral convergence of the proposed method is given a numerical example is provided to support the method and its analysis

a simple moving mesh method for one and two dimensional phase field equations a simple moving mesh method is proposed for solving phase field equations the numerical strategy is based on the approach proposed in li et al j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution the phase field equations are discretized by a finite volume method and the mesh moving part is realized by solving the conventional euler lagrange equations with the standard gradient based monitors numerical results demonstrate the accuracy and effectiveness of the proposed algorithm

nurbs based least squares finite element methods for fluid and solid mechanics summary this contribution investigates the performance of a least squares finite element method based on non uniform rational b splines nurbs basis functions the least squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions thus the introduction of auxiliary variables is avoided but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives the methodology is applied to the incompressible navier stokes equations and to linear as well as nonlinear elastic solid mechanics the numerical examples presented feature convective effects and incompressible or nearly incompressible material the numerical results which are obtained with equal order interpolation and without any stabilisation techniques are smooth and accurate it is shown that for p and h refinement the theoretical rates of convergence are achieved copyright 2014 john wiley sons ltd

adaptive least squares finite element methods for linear transport equations based on an h div flux reformulation abstract in this paper we study the least squares finite element methods lsfem for the linear hyperbolic transport equations the linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions while the normal component of the flux across the mesh faces needs to be continuous traditional lsfems using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions in order to separate the continuity requirements a new flux variable is introduced with this reformulation the continuities of the flux and the solution can be handled separately in natural h div l 2 conforming finite element spaces several variants of the methods are developed to handle the inflow boundary condition strongly or weakly with the reformulation the new lsfems can handle discontinuous solutions and boundary conditions much better than the traditional lsfems with continuous polynomial approximations with least squares functionals as a posteriori error estimators the adaptive methods can naturally identify error sources including singularity and non matching discontinuity for discontinuity aligned mesh no extra error is introduced if an r t 0 p 0 pair is used to approximate the flux and the solution the new adaptive lsfems can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods extensive numerical tests are performed to show the effectiveness of the methods developed in the paper

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

spectral implementation of an adaptive moving mesh method for phase field equations phase field simulations have been extensively applied to modeling microstructure evolution during various materials processes however large scale simulations of three dimensional 3d microstructures are still computationally expensive among recent efforts to develop advanced numerical algorithms the semi implicit fourier spectral method is found to be particularly efficient for systems involving long range interactions as it is able to utilize the fast fourier transforms fft on uniform grids in this paper we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy while maintaining a uniform grid in the computational domain for the spectral implementation this approach not only provides more accurate treatment at the interfaces requiring higher resolution but also retains the numerical efficiency of the semi implicit fourier spectral method numerical examples using the new adaptive moving mesh semi implicit fourier spectral method are presented for both two and three space dimensional microstructure simulations and they are compared with those obtained by other methods by maintaining a similar accuracy the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size

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

adaptive mesh refinement computation of solidification microstructures using dynamic data structures we study the evolution of solidification microstructures using a phase field model computed on an adaptive finite element grid we discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase field model at low undercooling in particular we show that the computational complexity of solving any phase boundary problem scales with the interface arclength when using an adapting mesh moreover the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions which would otherwise require lattices greater than 217 217elements we examine the convergence properties of our algorithm we also present two dimensional time dependent calculations of dendritic evolution with and without surface tension anisotropy we benchmark our results for dendritic growth with microscopic solvability theory finding them to be in good agreement with theory for high undercoolings at low undercooling however we obtain higher values of velocity than solvability theory at low undercooling where transients dominate in accord with a heuristic criterion which we derive

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

multilevel boundary functionals for least squares mixed finite element methods for least squares mixed finite element methods for the first order system formulation of second order elliptic problems a technique for the weak enforcement of boundary conditions is presented this approach is based on least squares boundary functionals which are equivalent to the h 1 2 and h1 2 norms on the trace spaces of lowest order raviart thomas elements for the flux and standard continuous piecewise linear elements for the pressure respectively continuity and coercivity of the resulting bilinear form is proved implying optimal order convergence of the resulting galerkin approximation the boundary least squares functional is implemented using multilevel principles and the technique is tested numerically for a model problem

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

least squares spectral method for the two dimensional stokes interface problems abstract this paper deals with the pseudo spectral least squares method to approximate the solution of the stokes interface problem that is the stokes equations with discontinuous viscosity and singular source term the stokes interface equations are adjusted in each sub domain along with the interface conditions then the equations are reformulated as a first order system by introducing stress tensor as a new independent variable the continuous and discrete norm least squares functionals using legendre and chebyshev weights for the first order system are defined and shown to be equivalent to appropriate product norms the spectral convergence of the proposed method is presented numerical examples are also given to demonstrate the method and analysis

two preconditioners based on the multi level splitting of finite element spaces the hierarchical basis preconditioner and the recent preconditioner of bramble pasciak and xu are derived and analyzed within a joint framework this discussion elucidates the close relationship between both methods special care is devoted to highly nonuniform meshes exclusively local properties like the shape regularity of the finite elements are utilized

efficient computation of dendritic growth with r adaptive finite element methods this paper deals with the application of a moving grid method to the solution of a phase field model for dendritic growth in two and three dimensions a mesh is found as the solution of an optimization problem that automatically includes the boundary conditions and is solved using a multi grid approach the governing equations are discretized in space by linear finite elements and a split time level scheme is used to numerically integrate in time one novel aspect of the method is the choice of a regularized monitor function the moving grid method enables us to obtain accurate numerical solutions with much less degree of freedoms it is demonstrated numerically that the tip velocity obtained by our method is in good agreement with the previously published results

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