surface crouzeix raviart element for the laplace beltrami equation in this paper we are concerned with the nonconforming finite element discretization of geometric partial differential equations we construct a surface crouzeix raviart element on the linear approximated surface analogous to a flat surface the optimal convergence theory for the new nonconforming surface finite element method is developed even though the geometric error exists by taking an intrinsic viewpoint of manifolds we introduce a new superconvergent gradient recovery method for the surface crouzeix raviart element using only the information of discretized surface the potential of serving as an asymptotically exact a posteriori error estimator is also exploited a series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method

weak efficiency for multiobjective variational problems abstract sometimes to locate efficient solutions for multiobjective variational problems mvps is quite costly so in this paper we tackle the study of weakly efficient solutions for mvps a new concept of weak vector critical point which generalizes other ones already existent and a new class of pseudoinvex functions are introduced we will apply a new approach to prove that the new class of pseudoinvex functions is equivalent to the class of functions whose weak vector critical points are weakly efficient solution for mvps

geometrically consistent mesh modification a new paradigm of adaptivity is to execute refinement coarsening and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy we refer to this collectively as geometrically consistent gc mesh modification we discuss the concept of discrete gc show the failure of naive approaches and propose and analyze a simple algorithm that is gc and accuracy preserving

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

a posteriori error estimation based on numerical realization of the variational multiscale method this paper presents a numerical realization of the variational multiscale method with the objective of providing a reliable and easy to implement local error estimation technique the variational multiscale framework provides a systematic approach for solution scale decomposition into coarse scales captured by the mesh and fine or subgrid scales in the proposed work the coarse scale errors in the finite element solution are neglected in comparison to the fine scale errors the fine scale variational equation is then localized using a general localization function over an element or a patch of elements to develop a local error estimation technique based on the proposed framework a consistent formulation of a new subdomain error estimator for elliptic problems is derived without the necessity of introducing an error locality assumption the new subdomain error estimator is evaluated numerically within a mesh adaptivity algorithm and it is shown to produce very sharp error estimates that outperform the element residual method estimates

an adaptive strategy for elliptic problems including a posteriori controlled boundary approximation we derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary the numerical solution is assumed to be defined on a finite element mesh whose boundary vertices are located on the boundary of the continuous problem no assumption is made on a geometrically fitting shape a posteriori error estimates are given in the energy norm and the l 2 norm and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation furthermore a strategy is presented to compute the effect of the non discretized part of the domain on the error starting from a coarse mesh this especially implies that parts of the domain where the measured error is small stay non discretized the presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary the reliable computation of the a posteriori estimates and a mesh adaptation based on delaunay techniques numerical examples illustrate that errors outside the initial discretization will be detected

optimality and duality for nondifferentiable multiobjective variational problems abstract the concept of efficiency is used to formulate duality for nondifferentiable multiobjective variational problems wolfe and mond weir type vector dual problems are formulated by using the generalized schwarz inequality and a characterization of efficient solution we established the weak strong and converse duality theorems under generalized f convexity assumptions

variational principles and lagrangian functions for stochastic processes and their dissipative statistical descriptions abstract a variational lagrangian formulation for stochastic processes and for the evolution equations of the associated probability density functions is developed particular attention is dedicated to poisson kac processes possessing finite propagation velocity the variational formulation in terms of lagrangian and hamiltonian densities permits to address different forms of reversibility characterizing these processes with respect to processes driven by nowhere differentiable wiener fluctuations and associated with the concepts of dynamic and statistical reversibility the latter property i e the statistical reversibility implies that the extended markov operator associated with poisson kac and generalized poisson kac processes forms a group continuously parametrized with respect to time

on efficiency and duality for multiobjective variational control problems with f ρ convexity abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational control problems wolfe and mond weir type duals are formulated under the generalized f convexity on the functions involved weak and strong duality theorems are proved

surface diffusion of graphs variational formulation error analysis and simulation surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for graphs and derive a priori error estimates for a time continuous finite element discretization we also introduce a semi implicit time discretization and a schur complement approach to solve the resulting fully discrete linear systems after computational verification of the orders of convergence for polynomial degrees 1 and 2 we show several simulations in one dimension and two dimensions with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time such as infinite slopes and cracks

mixed type duality for multiobjective variational problems with generalized f ρ convexity a mixed type dual for multiobjective variational problems is formulated several duality theorems are established relating properly efficient solutions of the primal and dual variational problems under generalized f convexity static mixed type dual multiobjective problems are particular cases of these problems

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

duality for nondifferentiable static multiobjective variational problems involving generalized f ϱ convex functions using the generalized schwarz inequality we establish some weak duality theorems for nondifferentiable static multiobjective variational problems involving generalized f convex functions later in the sequel we introduce three dual models for the nondifferentiable static multiobjective fractional variational problems and derive weak duality results for such programs

isogeometric analysis of high order partial differential equations on surfaces we consider the numerical approximation of high order partial differential equations pdes defined on surfaces in the three dimensional space with particular emphasis on closed surfaces we consider computational domains that can be represented by b splines or nurbs as for example the sphere and we spatially discretize the pdes by means of nurbs based isogeometric analysis in the framework of the standard galerkin method we numerically solve benchmark laplace beltrami problems of the fourth and sixth order as well as the corresponding eigenvalue problems with the goal of analyzing the role of the continuity of the nurbs basis functions on closed surfaces in this respect we show that the use of globally high order continuous basis functions as allowed by the construction of periodic nurbs leads to the efficient solution of the high order pdes finally we consider the numerical solution of high order phase field problems on closed surfaces namely the cahn hilliard and crystal equations

a priori anisotropic mesh adaptation on implicitly defined surfaces mesh adaptation on surfaces demands particular care due to the important role played by the surface fitting we propose an adaptive procedure based on a new error analysis which combines a rigorous anisotropic estimator for the l 1 norm of the interpolation error with an anisotropic heuristic control of the geometric error we resort to a metric based adaptive algorithm which employs local operations to modify the initial mesh according to the information provided by the error analysis an extensive numerical validation corroborates the robustness of the error analysis as well as of the adaptive procedure

an adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems an efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator an adaptive algorithm with variable time step sizes and space meshes is proposed and studied which at each time step delays the mesh coarsening until the final iteration of the adaptive procedure allowing only mesh and time step size refinements before it is proved that at each time step the adaptive algorithm is able to reduce the error indicators and thus the error below any given tolerance within a finite number of iteration steps the key ingredient in the analysis is a new coarsening strategy numerical results are presented to show the competitive behavior of the proposed adaptive algorithm

symmetric duality for multiobjective fractional variational problems with generalized invexity the concept of symmetric duality for multiobjective fractional problems has been extended to the class of multiobjective variational problems weak strong and converse duality theorems are proved under generalized invexity assumptions a close relationship between these problems and multiobjective fractional symmetric dual problems is also presented

duality for multiobjective variational problems abstract wolfe and mond weir type duals for multiobjective variational problems are formulated under convexity assumptions on the functions involved weak and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems a close relationship between these variational problems and nonlinear multiobjective programming problems is also indicated

parametric fem for geometric biomembranes we consider geometric biomembranes governed by an l 2 gradient flow for bending energy subject to area and volume constraints helfrich model we give a concise derivation of a novel vector formulation based on shape differential calculus and corresponding discretization via parametric fem using quadratic isoparametric elements and a semi implicit euler method we document the performance of the new parametric fem with a number of simulations leading to dumbbell red blood cell and toroidal equilibrium shapes while exhibiting large deformations

an improved a priori error analysis of nitsche s method for robin boundary conditions in a previous paper juntunen and stenberg in math comput 78 1353 1374 2009 we have extended nitsche s method in abhandlungen aus dem mathematischen seminar der universitat hamburg 1970 71 for the poisson equation with general robin boundary conditions the analysis required that the solution is in h s with s 3 2 here we give an improved error analysis using a technique proposed by gudi math comput 79 2169 2189 2010

on the influence of reflective boundary conditions on the statistics of poisson kac diffusion processes abstract we analyze the influence of reflective boundary conditions on the statistics of poisson kac diffusion processes and specifically how they modify the poissonian switching time statistics after addressing simple cases such as diffusion in a channel and the switching statistics in the presence of a polarization potential we thoroughly study poisson kac diffusion in fractal domains diffusion in fractal spaces highlights neatly how the modification in the switching time statistics associated with reflections against a complex and fractal boundary induces new emergent features of poisson kac diffusion leading to a transition from a regular behavior at shorter timescales to emerging anomalous diffusion properties controlled by walk dimensionality of the fractal set

symmetric duality for multiobjective variational problems abstract a mond weir type symmetric dual for a multiobjective variational problem is formulated weak and strong duality theorems under generalized convexity assumptions are proved for properly efficient solutions under an additional condition on the kernel function that occurs in the formulation of the problems a self duality theorem is proved a close relationship between these variational problems and symmetric dual nonlinear multiobjective programming problems is also incorporated

fully reliable localized error control in the fem if the first task in numerical analysis is the calculation of an approximate solution the second is to provide a guaranteed error bound and is often of equal importance the standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some saturation property for coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds the aim of this paper is to provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well the main argument is a localization via a partition of unity which leads to problems on small domains two fully reliable estimates are established the sharper one solves an analytical interface problem with residuals following babuska and rheinboldt siam j numer anal 15 1978 pp 736 754 the second estimate is a modification of the standard residual based a posteriori estimate with explicit constants from local analytical eigenvalue problems for some class of triangulations we show that the efficiency constant is smaller than 2 5 according to our numerical experience the overestimation of our computable estimates proved to be reasonably small with an overestimation by a factor between 2 5 and 4 only

an adaptive finite element method for the laplace beltrami operator on implicitly defined surfaces we present an adaptive finite element method for approximating solutions to the laplace beltrami equation on surfaces in mathbb r 3 which may be implicitly represented as level sets of smooth functions residual type a posteriori error bounds which show that the error may be split into a residual part and a geometric part are established in addition implementation issues are discussed and several computational examples are given

some applications of persistent random walks and the telegrapher s equation abstract a persistent random walk can be regarded as a multidimensional markov process the bias free telegraphers equation is 2 p t 2 1 t p t v 2 2 p it can be regarded as interpolating between the wave equation t and the diffusion equation t 0 previously it has found application in thermodynamics cf the review in rev mod phys 61 1989 41 62 1990 375 more recent applications are reviewed in the present article

partition of unity for localization in implicit a posteriori finite element error control for linear elasticity the partition of unity for localization in adaptive finite element method fem for elliptic partial differential equations has been proposed in carstensen and funken siam j sci comput 2000 21 1465 1484 and is applied therein to the laplace problem a direct adaptation to linear elasticity in this paper yields a first estimator l based on patch oriented local weighted interface problems the global korn inequality with a constant c korn yields reliability u u h korn l for any finite element approximation u h to the exact displacement u in order to localize this inequality further and so to involve the global constant c korn directly in the local computations we deduce a new error estimator l the latter estimator is based on local weighted interface problems with rigid body motions rbm as a kernel and so leads to effective estimates only if rbm are included in the local fe test functions therefore the excluded first order fem has to be enlarged by rbm which leads to a partition of unit method pum with rbm called p 1 rbm or to second order fems called p 2 fem for p 1 rbm and p 2 fem or even higher order schemes one obtains the sharper reliability estimate u u h l efficiency holds in the strict sense of l u u h the local weighted interface problems behind the implicit error estimators l and l are usually not exactly solvable and are rather approximated by some fem on a refined mesh and or with a higher order fem the computable approximations l l and l are shown to be reliable in the sense of u u h min l l osc the oscillations osc are known functions of the given data and higher order terms if the data are smooth for first order fem the mathematical proofs are based on weighted korn inequalities and inverse estimates combined with standard arguments the numerical experiments for uniform and adapted fem on benchmarks such as an l shape problem cook s membrane or a slit problem validate the theoretical estimates and also concern numerical bounds for c korn and the locking phenomena

a class of nondifferentiable continuous programming problems abstract optimality conditions duality and converse duality results are obtained for a class of continuous programming problems with a nondifferentiable term in the integrand of the objective function the proofs are based on a fritz john theorem for constrained optimization in abstract spaces the results generalize various well known results in variational problems with differentiable functions and also give a dynamic analogue of certain nondifferentiable programming problems

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

convergence and quasi optimality of adaptive fem with inhomogeneous dirichlet data we consider the solution of a second order elliptic pde with inhomogeneous dirichlet data by means of adaptive lowest order fem as is usually done in practice the given dirichlet data are discretized by nodal interpolation as model example serves the poisson equation with mixed dirichlet neumann boundary conditions for error estimation we use an edge based residual error estimator which replaces the volume residual contributions by edge oscillations for 2d we prove convergence of the adaptive algorithm even with optimal convergence rate for 2d and 3d we show convergence if the nodal interpolation operator is replaced by the l 2 projection or the scott zhang quasi interpolation operator as a byproduct of the proof we show that the scott zhang operator converges pointwise to a limiting operator as the mesh is locally refined this property might be of independent interest besides the current application finally numerical experiments conclude the work

isogeometric analysis for surface pdes with extended loop subdivision abstract we investigate the isogeometric analysis for surface pdes based on the extended loop subdivision approach the basis functions consisting of quartic box splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly and construct the solution space for dependent variables as well which is consistent with the concept of isogeometric analysis the subdivision process is equivalent to the h refinement of nurbs based isogeometric analysis the performance of the proposed method is evaluated by solving various surface pdes such as surface laplace beltrami harmonic biharmonic triharmonic equations which are defined on the limit surfaces of extended loop subdivision for different initial control meshes numerical experiments show that the proposed method has desirable performance in terms of the accuracy convergence and computational cost for solving the above surface pdes defined on both open and closed surfaces the proposed approach is proved to be second order accuracy in the sense of l 2 norm with theoretical and or numerical results which is also outperformed over the standard linear finite element by several numerical comparisons

on the techniques for constructing admissible stress fields in model verification performances on engineering examples robust global goal oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation in the context of computational mechanics the construction of admissible stress fields ie stress tensors which verify the equilibrium equations is required to set up strict and guaranteed error bounds using residual based error estimators and plays an important role in the quality of the error estimates this work focuses on the different procedures used in the calculation of admissible stress fields which is a crucial and technically complicated point the three main techniques that currently exist called the element equilibration technique eet the star patch equilibration technique spet and the element equilibration star patch technique eespt are investigated and compared with respect to three different criteria namely the quality of associated error estimators computational cost and easiness of practical implementation into commercial finite element codes the numerical results which are presented focus on industrial problems they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators which have the same convergence rate as the exact global error is consistent two and three dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches the analysis of the results reveals that the spet is more accurate than eet and eespt methods but the corresponding computational cost is higher overall the numerical tests prove the interest of the hybrid method eespt and show that it is a correct compromise between quality of the error estimate practical implementation and computational cost furthermore the influence of the cost function involved in the eet and the eespt is studied in order to optimize the estimators

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

optimality conditions and duality for multiobjective variational problems with generalized b invexity we extend the concepts of b type i and generalized b type i functions to the continuous case and we use these concepts to establish sufficient optimality conditions and duality results for multiobjective variational programming problems

a necessary and sufficient condition for duality in multiobjective variational problems abstract in this paper we move forward in the study of duality and efficiency in multiobjective variational problems we introduce new classes of pseudoinvex functions and prove that not only it is a sufficient condition to establish duality results but it is also necessary moreover these functions are characterized in order that all kuhn tucker or fritz john points are efficient solutions recent papers are improved we provide an example to show this improvement and illustrate these classes of functions and results

adaptive discontinuous galerkin methods on surfaces we present a dual weighted residual based a posteriori error estimate for a discontinuous galerkin approximation of a surface partial differential equation we restrict our analysis to a linear second order elliptic problem posed on hypersurfaces in mathbb r 3 r3 which are implicitly represented as level sets of smooth functions we show that the error in the energy norm may be split into a residual part and a higher order geometric part upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator we also present a novel geometric driven refinement strategy for pdes on surfaces which considerably improves the performance of the method on complex surfaces

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

convergence analysis of variational and non variational multigrid algorithms for the laplace beltrami operator we design and analyze variational and non variational multigrid algorithms for the laplace beltrami operator on a smooth and closed surface in both cases a uniform convergence for the v cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid the main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface in addition the vanishing mean value constraint is imposed on each level thereby avoiding singular quadratic forms without adding additional computational cost numerical results supporting our analysis are reported in particular the algorithms perform well even when applied to surfaces with a large aspect ratio

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

numerical methods for nonlinear second order hyperbolic partial differential equations i time linearized finite difference methods for 1 d problems abstract a three parameter family of finite difference methods for one dimensional hyperbolic equations with damping and nonlinearities based on the introduction of a new dependent variable and three implicitness parameters for the time discretization and the diffusion and reaction terms is presented and shown to result in two time level semi discrete equations second order accurate spatial discretizations and time linearization of the nonlinear source terms are shown to result in three time level linear finite difference equations which are second order accurate in time whenever the reaction and diffusion terms are allocated in the proportion 1 2 1 to the past current and future time levels second order accurate in time time linearized three point compact operator methods for the spatial derivatives which only involve three time levels and three grid points and are fourth order accurate in space are also presented and their linear stability analyzed an exhaustive comparison between analytical and numerical results indicates that the accuracy of time linearized methods increases as the time step and grid spacing are decreased and is nearly independent of the order of the spatial discretization when there is a sufficient number of grid points to resolve the wave structure and the best accuracy is achieved when the three implicitness parameters are equal to 1 2 it is also shown that the accuracy of time linearized methods is a strong function of the wave speed as the wave speed is increased and the wave equation tends to a parabolic one the methods are still three time level techniques which may not preserve the positivity of the solution whereas a crank nicolson discretization of the corresponding parabolic equation does preserve this property provided that there are no mathematical incompatibilities between the initial and the boundary conditions it is also shown that for one dimensional wave equations with monotonic kinetics and time dependent forcing there are very few differences between second and first order time accurate discretizations of the diffusion terms provided that the time step and grid size are sufficiently small and the discretization of the reaction terms plays a less important rule than that of the diffusion terms

duality for a class of multiobjective control problems with generalized invexity abstract b mond and i smart j math anal appl 136 1988 325 333 defined a kind of invexity and discussed the duality and sufficiency in scalar control problems with such invexity d bhatia and p kumar j math anal appl 189 1995 676 692 defined another kind of invexity corresponding generalized invexity and discussed the duality for multiobjective control problems with such generalized invexity in this paper the duality results for multiobjective control problems with mond and smart s generalized invexity are discussed

statistics of a filtered telegraph signal equations are derived which describe the time evolution of the probability density and corresponding characteristic function of a telegraph signal which has passed through a detuned lorentzian filter a closed form expression for the characteristic function is obtained for the tuned case and the predicted joint statistics and correlation properties are reviewed in the context of earlier results low order correlation properties for the more general detuned case are calculated it is shown that the stationary single interval statistics can be generated by a random phasor moving in a space of fractional dimensions and that a simple transform of variable leads to distributions which are stable in this space

guaranteed computable bounds for the exact error in the finite element solution part ii bounds for the energy norm of the error in two dimensions this paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution these bounds are constructed in terms of the solutions of local residual problems with equilibrated residual loads and are rather sharp even for coarse meshes he sharpness of the bounds can be further improved by employing few iterations of a relatively inexpensive iterative scheme he main result is that the bounds are guaranteed for the nergy norm of the exact error unlike the bounds which ave been proposed in 13 14 which are guaranteed only for the nergy norm of the error with respect to an enriched truth esh finite element solution copyright 2000 john wiley sons ltd

generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems withv invexity abstract a mond weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form is considered and various duality results viz weak strong and converse duality theorems are developed for conditionally properly efficient solutions these results are obtained under v invexity assumptions and its generalizations on objective and constraint functions this work extends many results on variational problems established earlier

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

afem for the laplace beltrami operator on graphs design and conditional contraction property we present an adaptive finite element method afem of any polynomial degree for the laplace beltrami operator on c 1 graphs in r d d 2 we first derive residual type a posteriori error estimates that account for the interaction of both the energy error in h 1 and the surface error in w 1 1 due to approximation of we devise a marking strategy to reduce the total error estimator namely a suitably scaled sum of the energy geometric and inconsistency error estimators we prove a conditional contraction property for the sum of the energy error and the total estimator the conditional statement encodes resolution of in w 1 1 we conclude with one numerical experiment that illustrates the theory

random walker and the telegrapher s equation a paradigm of a generalized hydrodynamics the telegrapher s equation te is the continuum limit of a persisting random walker we find that the te reproduces the original spectrum almost exactly for all wavelengths far beyond the validity of the expansion this surprising property is used as a paradigm towards the derivation of a generalized hydrodynamics applications to other problems are explored

sufficient optimality criteria and duality for multiobjective variational control problems with v invexity abstract the sufficient optimality criteria for a class of multiobjective variational control problems under the v invexity assumption are presented duality results are proved under a variety of v invexity assumptions multiobjective fractional variational control problems are also studied

isogeometric analysis for second order partial differential equations on surfaces we consider the numerical solution of second order partial differential equations pdes on lower dimensional manifolds specifically on surfaces in three dimensional spaces for the spatial approximation we consider isogeometric analysis which facilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by b splines or nurbs our analysis addresses linear nonlinear time dependent and eigenvalues problems involving the laplace beltrami operator on surfaces moreover we propose a priori error estimates under h refinement in the general case of second order pdes on the lower dimensional manifolds we highlight the accuracy and efficiency of isogeometric analysis with respect to the exactness of the geometrical representations of the surfaces

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

generalized type i invexity and duality in multiobjective variational problems abstract in this paper we introduce new classes of generalized v type i invex functions for variational problems and consider a multiobjective variational problem vp a number of sufficiency results are established using lagrange multiplier conditions under various types of generalized v type i invexity requirements duality results are obtained for mond weir type duals under the above generalized v type i invexity assumptions and their generalizations

nonstationary markovian replication process causing diverse diffusions we introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions a nonstationary markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker s displacement and the generalized telegrapher equation with time varying coefficients and we find that diffusivity can be determined by temporal changes of replication of an immediate step by controlling the replications we realize diverse diffusions such as alternating diffusion superdiffusion subdiffusion and marginal diffusion which originate from oscillating increasing decreasing and slowly increasing or decreasing replications with time respectively

superconvergence and gradient recovery of linear finite elements for the laplace beltrami operator on general surfaces superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the laplace beltrami equation on general surfaces with mildly structured triangular meshes for a large class of practically useful grids the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the laplace beltrami equation and as a result various postprocessing gradient recovery including simple and weighted averaging local and global l 2 projections and zienkiewicz and zhu z z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution numerical experiments are presented to confirm the theoretical results

ten remarks on nonconvex minimisation for phase transition simulations nonconvex minimisation problems are encountered in many applications such as phase transitions in solids or liquids but also in optimal design tasks or micromagnetism in contrast to rubber type elastic materials and many other variational problems in continuum mechanics the minimal energy may be not attained in the sense of sobolev functions the nonrank 1 convex minimisation problem m is ill posed the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures their limit is a measure called gradient young measure gym and describes the effective energy density w qc the quasiconvex envelope of the original energy density w this gives rise to a relaxed minimisation problem r which is well posed in the sense that the minimum is attained the paper compares computational aspects of the two problems m and r neither adaptive finite element methods nor effective solvers may work for discrete versions of m but theoretical and numerical evidence supports that they work for r the drawback of r is that the relaxed energy density w qc is not always given by a known closed form formula instead a numerical relaxation has to be involved which is of the form m and hence yields a computational challenge other semiconvexity notions are requested and give rise to a new category of algorithms the demand for quasiconvexification algorithms is highly motivated by recent models in finite plasticity as the time discretisation in the latter models typically leads to nonrank 1 convex minimisation problems 2004 published by elsevier b v

duality with bf type i functions for a class of nondifferentiable multiobjective variational problems a class of bf type i functions and its extensions are introduced in the continuous case an example is presented in support utilizing these new concepts sufficient optimality conditions and duality results are presented for multiobjective variational problems involving arbitrary norms

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985