c1 continuous stress recovery in finite element analysis abstract a penalized discrete least squares pdls variational principle is employed to recover c 1 continuous smooth stress fields from finite element stresses the error functional involves discrete least squares with a penalty constraint to enforce c 1 continuity of the recovered stresses the recovery procedure uses a smoothing element analysis sea with element based interpolation functions to minimize the error functional sea pdls recovers a superconvergent stress field of higher accuracy and continuity than the underlying consistent finite element stress field general and specialized formulations of the functional are given and an appropriate discretization strategy for the sea is discussed numerical results for both one and two dimensional stress fields are presented and compared with the corresponding solutions for the superconvergent patch recovery spr method of zienkiewicz and zhu the results demonstrate that in addition to achieving a higher degree of continuity sea pdls produces generally more accurate stresses than spr although an additional global analysis is required results indicate that two coarse mesh analyses finite element and smoothing produce solutions comparable to a single finite element analysis with a much greater degree of refinement and more computational effort the procedure is unique in that the recovered stress field is essentially c 1 continuous the results are shown to be insensitive to the value of the penalty parameter above a threshold and hence uncertainty in the choice of an appropriate penalty value is eliminated

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

patch recovery based on superconvergent derivatives and equilibrium in this paper a postprocessing technique is developed for determining first order derivatives fluxes stresses at nodal points based on derivatives in superconvergent points it is an extension of the superconvergent patch recovery technique presented by zienkiewicz and zhu in contrast to that technique all flux or stress components are interpolated at the same time coupled by equilibrium equations at the superconvergent points the equilibrium equations and use of one order higher degree of interpolation polynomials of stress give a dramatic decrease in error of recovered derivatives even at boundaries

an improved variational method for finite element stress recovery and a posteriori error estimation abstract a new variational formulation is presented which serves as a foundation for an improved finite element stress recovery and a posteriori error estimation in the case of stress predictions interelement discontinuous stress fields from finite element solutions are transformed into a c 1 continuous stress field with c 0 continuous stress gradients these enhanced results are ideally suited for error estimation since the stress gradients can be used to assess equilibrium satisfaction the approach is employed as a post processing step in finite element analysis the variational statement used herein combines discrete least squares penalty constraint and curvature control functionals thus enabling automated recovery of smooth stresses and stress gradients the paper describes the mathematical foundation of the method and presents numerical examples including stress recovery in two dimensional structures and built up aircraft components and error estimation for adaptive mesh refinement procedures

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

dynamics of multi species competition predator system with impulsive perturbations and holling type iii functional responses abstract we investigate the dynamics of a class of multi species predator prey interaction models with holling type iii functional responses based on systems of nonautonomous differential equations with impulsive perturbations sufficient conditions for existence of a positive periodic solution are investigated by using a continuation theorem in coincidence degree theory which have been extensively applied in studying existence problems in differential equations and difference equations in addition sufficient criteria are established for the global stability and the globally exponential stability of the system by using the comparison principle and the lyapunov method

identification of space time distributed parameters in the gierer meinhardt reaction diffusion system we consider parameter identification for the classic gierer meinhardt reaction diffusion system the original gierer meinhardt model a gierer and h meinhardt kybernetik 12 1972 pp 30 39 was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology in our paper the parameters are extended in time and space and used as distributed control variables the methodology employs pde constrained optimization in the context of image driven spatiotemporal pattern formation we prove the existence of optimal solutions derive an optimality system and determine optimal solutions the results of numerical experiments in two dimensions are presented using the finite element method which illustrates the convergence of a variable step gradient algorithm for finding the optimal parameters of the system a practical target function is constructed for the optimal control algorithm corresponding to the actual image of a marine angel

combined effect of explicit time stepping and quadrature for curvature driven flows the flow of a closed surface of codimension 1 in bbb r n driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter varepsilon 0 conforming piecewise linear finite elements with mass lumping over a quasi uniform and weakly acute mesh of size h are further used for space discretization and combined with forward differences for time discretization with uniform time step tau the resulting explicit schemes are the basis for an efficient algorithm the so called dynamic mesh algorithm and exhibit finite speed of propagation and discrete ondegeneracy no iteration is required not even to handle the obstacle constraints the zero level set of the fully discrete solution is shown to converge past singularities to the true interface provided tau h 2 approx o varepsilon 4 and no fattening occurs if the more stringent relations tau h 2 approx o varepsilon 6 are enforced then an interface rate of convergence o varepsilon is derived in the vicinity of regular points along with a companion o varepsilon 1 2 for type i singularities for smooth flows an interface rate of convergence of o varepsilonq is proven provided tau h 2 approx o varepsilon 5 and exact integration is used for the potential term the analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature which bears some intrinsic interest lipschitz properties of viscosity solutions of the level set approach and discrete nondegeneracy these basic ingredients are also discussed

finite element approximation of a spatially structured metapopulation pde model abstract we present a new fully spatially structured pde metapopulation model for predator prey dynamics in d 3 space dimensions a nonlinear reaction diffusion system of rosenzweig macarthur form models predator prey dynamics in two high quality patches embedded in a low quality subdomain where species can diffuse convect and die our model substantially generalizes and improves earlier fully structured metapopulation models after a nondimensionalization procedure in order to approximate the metapopulation model we present a fully discrete galerkin finite element method in two space dimensions which is a generalization of the finite element method analyzed in a previous single patch predator prey model the numerical solutions are illustrated for some test cases using matlab numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch in an appendix we also give an outline for the proof of the well posedness of the model

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

a three node mindlin plate element with improved transverse shear abstract a displacement methodology for mindlin elements recently employed in the development of an efficient four node quadrilateral min4 is the basis for a three node explicitly integrated triangular element min3 the approach herein referred to as anisoparametric is founded upon two kinematic requirements that are auxiliary to the standard convergence criteria the initial complete quadratic deflection field is constrained by continuous shear edge constraints to achieve an isoparametric like three node configuration much success of the element is due to an innovative shear correction factor concept several numerical experiments are performed in the linear elasto static regime to ascertain the element performance

on a prey predator reaction diffusion system with holling type iii functional response in this paper we study a prey predator model defined by an initial boundary value problem whose dynamics is described by a holling type iii functional response we establish global existence and uniqueness of the strong solution we prove that if the initial data are positive and satisfy a certain regularity condition the solution of the problem is positive and bounded on the domain q 0 t x w and then we deduce the continuous dependence on the initial data a numerical approximation of the system is carried out with a spectral method coupled with the fourth order runge kutta time solver the biological relevance of the comparative numerical results is also presented

counting function for a sphere of anisotropic quartz we calculate the leading weyl term of the counting function for a monocrystalline quartz sphere in contrast to other studies of counting functions the anisotropy of quartz is a crucial element in our investigation hence we do not obtain a simple analytical form but we carry out a numerical evaluation to this end we employ the radon transform representation of the green s function we compare our result to a previously measured unique data set of several tens of thousands of resonances

a three level finite element approximation of a pattern formation model in developmental biology this paper concerns a second order three level piecewise linear finite element scheme 2 sbdf ruuth in j math biol 34 148 176 1995 for approximating the stationary turing patterns of a well known experimental substrate inhibition reaction diffusion thomas system thomas in analysis and control of immobilized enzyme systems pp 115 150 1975 a numerical analysis of the semi discrete in time approximations leads to semi discrete a priori bounds and an optimal error estimate the analysis highlights the technical challenges in undertaking the numerical analysis of multi level ge 3 3 schemes we illustrate the effectiveness of the numerical method by repeating an important classical experiment in mathematical biology namely to approximate the turing patterns of the thomas system over a schematic mammal skin domain with fixed geometry at various scales we also make some comments on the correct procedure for simulating turing patterns in general reaction diffusion systems

a variational method for finite element stress recovery and error estimation abstract a variational method for obtaining smoothed stresses from a finite element derived non smooth stress field is presented the method is based on minimizing a functional involving discrete least squares error plus a penalty constraint that ensures smoothness of the stress field an equivalent accuracy criterion is developed for the smoothing analysis which results in a c1 continuous smoothed stress field possessing the same order of accuracy as that found at the superconvergent optimal stress points of the original finite element analysis application of the smoothing analysis to residual error estimation is also demonstrated

on inter element forces in the fem displacement formulation and implications for stress recovery this paper describes a new technique for the determination of the inter element forces and tractions as well as stress state at nodes as a post processing step after the solution of standard fe displacement calculation the work is motivated in the context of a broader development of a procedure to simulate fracture processes using a discrete approach without the need of double noded interface elements the technique easily implementable is based on the double minimization of an objective function representing the error between the inter element stress tractions and the projection of the best fit stress tensor t along the planes of the interfaces converging at an element corner node the formulation is illustrated with some basic examples in which the resulting stress tensors and inter element forces are compared to theoretical solutions and to the results obtained by using a traditional stress average smoothing method

a multigrid preconditioned numerical scheme for a reaction diffusion system reaction diffusion operators have been used to model many engineering and biological systems in this study we consider a reaction diffusion system modeling various engineering and life science problems there are many algorithms to approximate such mathematical models most of the algorithms are conditionally stable and convergent for a big time step size a krylov subspace type solver for such models converges slowly or oscillates because of the presence of the diffusion term here we study a multigrid preconditioned generalized minimal residual method gmres for such a model we start with a five point scheme for the spatial integration and a method of lines for the temporal integration of the system of pdes then we implement a multigrid iterative algorithm for the full discrete model and show some numerical results to demonstrate the dominance of the solver we analyze the convergence rate of such a multigrid iterative preconditioning algorithm reaction diffusion systems arise in many mathematical models and thus this study has many applicabilities

numerical analysis of a first order in time implicit symplectic scheme for predator prey systems abstract the numerical solution of reaction diffusion systems modelling predator prey dynamics using implicit symplectic imsp schemes is relatively new when applied to problems with chaotic dynamics they perform well both in terms of computational effort and accuracy however until the current paper a rigorous numerical analysis was lacking we analyse the semi discrete in time approximations of a first order imsp scheme applied to spatially extended predator prey systems we rigorously establish semi discrete a priori bounds that guarantee positive and stable solutions and prove an optimal a priori error estimate this analysis is an improvement on previous theoretical results using standard implicit explicit imex schemes the theoretical results are illustrated via numerical experiments in one and two space dimensions using fully discrete finite element approximations

lumped finite elements for reaction cross diffusion systems on stationary surfaces all the authors am is cv mf thank the isaac newton institute for mathematical sciences for its hospitality during the programme coupling geometric pdes with physics for cell morphology motility and pattern formation epsrc ep k032208 1 this work am has received funding from the european union s horizon 2020 research and innovation programme under the marie sklodowska curie grant agreement no 642866 am and cv acknowledge support from the engineering and physical sciences research council ep j016780 1 on modelling analysis and simulation of spatial patterning on evolving biological surfaces and the leverhulme trust research project grant rpg 2014 149 on unravelling new mathematics for 3d cell migration am was partially supported by a fellowship from the simons foundation am is a royal society wolfson research merit award holder generously funded by the wolfson foundation

optimal control of a nutrient phytoplankton zooplankton fish system we consider the mathematical formulation analysis and numerical solution of an optimal control problem for a nonlinear nutrient phytoplankton zooplankton fish reaction diffusion system we study the existence of optimal solutions derive an optimality system and determine optimal solutions in the original spatially homogeneous formulation m scheffer oikos 62 1991 pp 271 282 the dynamics of plankton were investigated as a function of parameters for nutrient levels and fish predation rate on zooplankton in our paper the model is spatially extended and the parameter for fish predation treated as a multiplicative control variable the model has implications for the biomanipulation of food webs in eutrophic lakes to help improve water quality in order to illustrate the control of irregular spatiotemporal dynamics of plankton in the model we implement a semi implicit in time finite element method with mass lumping and present the results of numerical experiments in two space dimensions

superconvergent derivatives a taylor series analysis an analysis based on the use of taylor series expansions is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite element interpolant or approximation the analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates superconvergence moreover it leads directly to a class of post processing strategies for the derivatives and offers a more direct constructive approach to the subject

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

finite element approximation of spatially extended predator prey interactions with the holling type ii functional response we study the numerical approximation of the solutions of a class of nonlinear reaction diffusion systems modelling predator prey interactions where the local growth of prey is logistic and the predator displays the holling type ii functional response the fully discrete scheme results from a finite element discretisation in space with lumped mass and a semi implicit discretisation in time we establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions the class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics expressed in nondimensional form the theoretical techniques were adapted from a previous study of an idealised reaction diffusion system garvie and blowey in eur j appl math 16 5 621 646 2005