a unifying theory of a posteriori error control for discontinuous galerkin fem a unified a posteriori error analysis is derived in extension of carstensen numer math 100 617 637 2005 and carstensen and hu j numer math 107 3 473 502 2007 for a wide range of discontinuous galerkin dg finite element methods fem applied to the laplace stokes and lame equations two abstract assumptions a1 and a2 guarantee the reliability of explicit residual based computable error estimators the edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available the resulting reliable error estimate is applied to 16 representative dg fems from the literature the estimate recovers known results as well as provides new bounds to a number of schemes

finite element approximation of stochastic partial differential equations driven by poisson random measures of jump type the paper deals with stochastic partial differential equations driven by poisson random measures of jump type and their numerical approximation we investigate the accuracy of space and time approximation as space approximation we consider finite elements and as time approximation the implicit euler scheme

stochastic integrators with stationary independent increments lipschitzian stochastic differential equations driven by a process with stationary independent increments permit a priori growth and stability estimates up to any sure time and the solutions depend differentiably on parameters provided the levy measure of the driving term has suitable moments

discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

discontinuous galerkin finite element methods for incompressible non linear elasticity abstract a discontinuous galerkin finite element method dgfem for incompressible non linear elasticity is derived one of the limitations of the continuous galerkin finite element method cgfem when applied to incompressible elasticity is that the accuracy of the numerical solution may be adversely affected by the phenomenon known as locking this may prevent the use of a low order polynomial approximation to the displacement on each element we demonstrate using simulations that the dgfem presented here does not suffer from this drawback two further advantages in this setting of this dgfem over cgfems are that i highly anisotropic meshes meshes containing elements with a very high aspect ratio may be used without significantly degrading the accuracy of the solution and ii discontinuities in the elasticity parameters are handled more effectively

an a posteriori error estimate for hp adaptive dg methods for convection diffusion problems on anisotropically refined meshes we prove an a posteriori error estimate for hp adaptive discontinuous galerkin methods for the numerical solution of convection diffusion equations on anisotropically refined rectangular elements the estimate yields global upper and lower bounds of the errors measured in terms of a natural norm associated with diffusion and a semi norm associated with convection the anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure we present a series of numerical experiments to test the feasibility of this approach within a fully automated hp adaptive refinement algorithm

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

a discontinuous galerkin finite element discretization of the euler equations for compressible and incompressible fluids using the generalized variable formulation of the euler equations of fluid dynamics we develop a numerical method that is capable of simulating the flow of fluids with widely differing thermodynamic behavior ideal and real gases can be treated with the same method as an incompressible fluid the well defined incompressible limit relies on using pressure primitive or entropy variables in particular entropy variables can provide numerical methods with attractive properties e g fulfillment of the second law of thermodynamics we show how a discontinuous galerkin finite element discretization previously used for compressible flow with an ideal gas equation of state can be extended for general fluids we also examine which components of the numerical method have to be changed or adapted especially we investigate different possibilities of solving the nonlinear algebraic system with a pseudo time iteration numerical results highlight the applicability of the method for various fluids

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity analytical estimates this is the second of two papers in which we motivate introduce and analyze a new type of strategy for the stabilization of discontinuous galerkin dg methods in nonlinear elasticity problems the foremost goal behind it is to enhance the robustness of the method without deteriorating the accuracy of the resulting solutions its distinctive property is that for nonlinear elastic problems the stabilization term is solution dependent and hence it is termed an adaptive stabilization mechanism the key contribution of this paper is the construction of a stabilization strategy for which the method is perfectly stable since the stabilization parameters can be explicitly computed as part of the numerical solution this is accomplished through the main result of this paper which consists of a theorem that provides lower bounds for the size of the stabilization parameters numerical examples confirm the guaranteed stability of the resulting method however they also show that the computed lower bounds overestimate the minimum amount of stabilization needed negatively affecting the approximation properties of the method these results underscore the fact that a better understanding of the stabilization mechanisms is needed in order to construct a method that is both robust and efficient

optimal strong convergence rates of numerical methods for semilinear parabolic spde driven by gaussian noise and poisson random measure abstract this paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation spde driven simultaneously by gaussian noise and poisson random measure more realistic in modeling real world phenomena the spde is discretized in space with the standard finite element method and in time with the linear implicit euler method or an exponential integrator more efficient and stable for stiff problems we prove the strong convergence of the fully discrete schemes toward the mild solution the results reveal how convergence orders depend on the regularity of the noise and the initial data in addition we exceed the classical orders 1 2 in time and 1 in space achieved in the literature when dealing with spde driven by poisson measure with less regularity assumptions on the nonlinear drift function in particular for trace class multiplicative gaussian noise we achieve convergence order o h 2 t 1 2 for additive trace class gaussian noise and an appropriate jump function we achieve convergence order o h 2 t numerical experiments to sustain the theoretical results are provided

improved error estimation of dynamic finite element methods for second order parabolic equations dynamic finite element schemes are analyzed for second order parabolic problems these schemes permit different finite element spaces at different time levels in order to efficiently capture time changing localized phenomena such as moving sharp fronts or layers the dynamical change of grids and interpolation polynomials is necessary and essential to many large scale transient problems standard characteristic and mixed finite element methods with dynamic function spaces are considered for linear and nonlinear problems in a unified framework with improved a priori error estimates and convergence results

exponential convergence of mixed hp dgfem for stokes flow in polygons we analyze mixed hp discontinuous galerkin finite element methods dgfem for stokes flow in polygonal domains in conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders we prove that the hp dgfem leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corners

hp adaptive composite discontinuous galerkin methods for elliptic eigenvalue problems on complicated domains in this paper we develop the a posteriori error estimation of hp adaptive discontinuous galerkin composite finite element methods dgfems for the discretization of second order elliptic eigenvalue problems dgfems allow for the approximation of problems posed on computational domains which may contain local geometric features the dimension of the composite finite element space is independent of the number of geometric features this is in contrast with standard finite element methods as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space computable upper bounds on the error for both eigenvalues and eigenfunctions are derived numerical experiments highlighting the practical application of the proposed estimators within an automatic hp adaptive refinement procedure will be presented

convergence in the incompressible limit of new discontinuous galerkin methods with general quadrilateral and hexahedral elements abstract standard low order finite elements which perform well for problems involving compressible elastic materials are known to under perform when nearly incompressible materials are involved commonly exhibiting the locking phenomenon interior penalty ip discontinuous galerkin methods have been shown to circumvent locking when simplicial elements are used the same ip methods however result in locking on meshes of quadrilaterals the authors have shown in earlier work that under integration of specified terms in the ip formulation eliminates the locking problem for rectangular elements here it is demonstrated through an extensive numerical investigation that the effect of using under integration carries over successfully to meshes of more general quadrilateral elements as would likely be used in practical applications and results in accurate displacement approximations uniform convergence with respect to the compressibility parameter is shown numerically additionally a stress approximation obtained here by postprocessing shows good convergence in the incompressible limit

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

hp interpolation of nonsmooth functions and an application to hp a posteriori error estimation the quasi interpolation operators of clement and scott zhang type are generalized to the setting of the hp fem finite element method new polynomial lifting and inverse estimates are presented the classical residual based a posteriori error estimator is generalized to the hp fem

energy norm a posteriori error estimates for discontinuous galerkin approximations of the linear elasticity problem we present a residual based a posteriori error estimate in an energy norm of the error in a family of discontinuous galerkin approximations of linear elasticity problems the theory is developed in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

mixed discontinuous galerkin approximation of the maxwell operator we introduce and analyze a discontinuous galerkin discretization of the maxwell operator in mixed form here all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order for piecewise constant coefficients the method is shown to be stable and optimally convergent with respect to the mesh size numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented

uniformly convergent interior penalty methods using multilinear approximations for problems in elasticity interior penalty methods for linear elasticity when used with bilinear elements are effective in the context of compressible materials however they may produce poor approximations with these elements in the nearly incompressible regime we propose a new general interior penalty formulation and prove that it is uniformly convergent with respect to the compressibility parameter for multilinear elements the new formulation is a modification of well known methods nonsymmetric incomplete and symmetric interior penalty galerkin through underintegration of selected edge terms

dynamic domain decomposition and grid modification for parabolic problems abstract a dynamic grid modification and domain decomposition method is given and analyzed for parabolic problems this method allows one to apply different domain decompositions and different grids and interpolation polynomials on the subdomains at different time levels when necessary the procedure relies on an implicit galerkin method in the subdomains and explicit flux calculation on the inter domain boundaries in addition a dynamic finite element scheme is proposed and analyzed which is applicable to general parabolic problems these methods are well suited to large scale time dependent problems involving localized phenomena such as sharp fronts or layers which also change with time convergence and stability analyses in the l 2 norm are given numerical experiments are provided to check the performance of the methods and make comparison with other methods

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

an a posteriori error analysis of an augmented discontinuous galerkin formulation for darcy flow in this paper we develop an a posteriori error analysis for an augmented discontinuous garlerkin formulation applied to the darcy flow more precisely we derive a reliable and efficient a posteriori error estimator which consists of residual terms finally we present several numerical experiments showing the robustness of the method and the theoretical properties of the estimator thus confirming the capability of the corresponding adaptive algorithms to localize the inner layers the singularities and or the large stress regions of the exact solution

a linear nonconforming finite element method for nearly incompressible elasticity and stokes flow abstract we introduce a new triangular element for nearly incompressible elasticity and incompressible fluid flow the method consists of conforming linear elements for one of the displacement or velocity for flows component and linear non conforming elements for the other component the element is proved to give an optimal approximation and this is also confirmed by several numerical examples

analysis of some moving space time finite element methods two space time finite element methods for solving time dependent partial differential equations are defined and analyzed the methods are based on the use of isoparametric finite elements to implicitly define the time discretization on a moving mesh in the space dimensions one method allows for adding and deleting knots in a continuous fashion while the other allows for discontinuous changes in the mesh static rezone a detailed convergence analysis for a model parabolic equation with a possibly large convection term is presented here the authors obtain symmetric best approximation error estimates similar to those obtained by dupont math comp 39 1982 pp 85 107 for the semidiscrete case

a new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location this work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body beyond the obvious merits of its applications this problem also presents significant mathematical challenges the near incompressibility inherent in the model of linear elasticity in the body gives rise to the locking effect and necessitates a unique treatment of both the direct and inverse problems a general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least squares mols objective functional the mols functional is shown to be convex thus overcoming the nonconvexity of the classical output least squares ols functional and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters generalized derivative formulas for the coefficient to solution map are also given along with a complete convergence analysis

a geometric structure preserving discretization scheme for incompressible linearized elasticity abstract in this paper we present a geometric discretization scheme for incompressible linearized elasticity we use ideas from discrete exterior calculus dec to write the action for a discretized elastic body modeled by a simplicial complex after characterizing the configuration manifold of volume preserving discrete deformations we use hamilton s principle on this configuration manifold the discrete euler lagrange equations are obtained without using lagrange multipliers the main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field we explicitly derive the governing equations for the two dimensional case where the discrete spaces for the displacement field are constructed by p 1 polynomials over primal meshes for incompressibility constraint p 0 polynomials over dual meshes for the kinetic energy and p 1 polynomials over support volumes for the elastic energy and the discrete space of the pressure field is constructed by p 0 polynomials over primal meshes we test the efficiency and robustness of this geometric scheme using some numerical examples in particular we do not see any volume locking and or checkerboarding of pressure in our numerical examples this suggests that our choice of discrete solution spaces is compatible

galerkin finite element methods for stochastic parabolic partial differential equations we study the finite element method for stochastic parabolic partial differential equations driven by nuclear or space time white noise in the multidimensional case the discretization with respect to space is done by piecewise linear finite elements and in time we apply the backward euler method the noise is approximated by using the generalized l2 projection operator optimal strong convergence error estimates in the l2 and dot h 1 norms with respect to the spatial variable are obtained the proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem the computational analysis and numerical example are given

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

hp optimal discontinuous galerkin methods for linear elliptic problems the aim of this paper is to overcome the well known lack of p optimality in hp version discontinuous galerkin dg discretizations for the numerical approximation of linear elliptic problems for this purpose we shall present and analyze a class of hp dg methods that is closely related to other dg schemes however combines both p optimal jump penalty as well as lifting stabilization we will prove that the resulting error estimates are optimal with respect to both the local element sizes and polynomial degrees

an analysis of the moving finite element procedure the moving finite element procedure of miller siam j numer anal 18 1981 pp 1019 1057 is analysed and broken down into a number of steps the central step is a mapping from element velocities to nodal velocities which is exact for a class of one dimensional scalar nonlinear partial differential equation pdes for more general pdes the mapping is preceded by an element by element projection step in higher dimensions the mapping is followed by node by node projection for first order pdes the procedure gives rise to ordinary differential equation systems approximating the method of characteristics for second order operators the behaviour of the nodes can only be described qualitatively the central mapping is a legendre transformation from the data to the nodal velocities while the full procedure is an approximate legendre transformation the same transformation can be used to construct a dual pde which may be numerically easier to solve the structure is used to analyse nodal movement and to giv

an equation error approach for the elasticity imaging inverse problem for predicting tumor location the primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body this nonlinear inverse problem not only represents an important and interesting application it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility due to the locking effect classical finite element methods are not effective for incompressible elasticity equations therefore special treatment is necessary for both the direct and inverse problems to study the inverse problem in an optimization framework we propose an extension of the equation error approach we focus on two cases namely when the material parameter is sufficiently smooth and when it is may be discontinuous for the latter case we extend the total variation regularization method to the elasticity imaging inverse problem we give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem we give sufficient details on the discrete formulas as well as on the implementation issues numerical examples for smooth and discontinuous coefficients are given

a posteriori error control for dpg methods a combination of ideas in least squares finite element methods with those of hybridized methods recently led to discontinuous petrov galerkin dpg finite element methods they minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard locally computable dual norm this paper establishes a general a posteriori error analysis for the natural norms of the dpg schemes under conditions equivalent to a priori stability estimates it is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors the presented abstract framework for a posteriori error analysis applies to known dpg discretizations of laplace and lame equations and to a novel dpg method for the stress velocity formulation of stokes flow with symmetric stress approximations since the error control does not rely on the discrete equations it applies to inexactly computed or otherwise perturbed solutions within the discrete space

the euler scheme for levy driven stochastic differential equations in relation with monte carlo methods to solve some integro differential equations we study the approximation problem of ee g x t by ee g bar x t n where x t 0 leq t leq t is the solution of a stochastic differential equation governed by a levy process z t bar x n t is defined by the euler discretization scheme with step fractn with appropriate assumptions we show that the error ee g x t ee g bar x t n can be expanded in powers of frac1n if the levy measure of z has finite moments of order high enough otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the levy measure the simulation of the increments of z t is also discussed

numerical analysis of semilinear stochastic evolution equations in banach spaces the solution of stochastic evolution equations generally relies on numerical computation here usually the main idea is to discretize the spde spatially obtaining a system of sdes that can be solved by e g the euler scheme in this paper we investigate the discretization error of semilinear stochastic evolution equations in lp spaces resp banach spaces the space discretization may be done by galerkin approximation for the time discretization we consider the implicit euler the explicit euler scheme and the crank nicholson scheme in the last section we give some examples i e we consider an spdes driven by nuclear wiener noise approximated by wavelets and delay equation approximated by finite differences

grid modification for second order hyperbolic problems a family of galerkin finite element methods is presented to accurately and efficiently solve the wave equation that includes sharp propagating wave fronts the new methodology involves different finite element discretizations at different time levels thus at any time level relatively coarse grids can be applied in regions where the solution changes smoothly while finer grids can be employed near wave fronts the change of grid from time step to time step need not be continuous and the number of grid points at different time levels can be arbitrarily different the formulation is applicable to general second order hyperbolic equations stability results are proved and a priori error estimates are established for several boundary conditions our error estimates consist of three parts the time finite difference discretization error the spatial finite element discretization error and the error due to the projections of the approximated solution from old grids onto new grids

grid modification for the wave equation with attenuation the wave equation with attenuation due to a linear friction is approximated by a new mixed finite element method which allows one to use different grids and basis functions at different times when necessary this method enables one to track sharp moving wave fronts more efficiently and accurately error estimates with optimal convergent rates are established unconditional stability is also proved for this method

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

error estimation and adaptivity for incompressible hyperelasticity summary a galerkin fem is developed for nonlinear incompressible hyper elasticity that takes account of nonlinearities in both the strain tensor and the relationship between the strain tensor and the stress tensor by using suitably defined linearised dual problems with appropriate boundary conditions a posteriori error estimates are then derived for both linear functionals of the solution and linear functionals of the stress on a boundary where dirichlet boundary conditions are applied a second higher order method for calculating a linear functional of the stress on a dirichlet boundary is also presented together with an a posteriori error estimator for this approach an implementation for a 2d model problem with known solution where the entries of the strain tensor exhibit large rapid variations demonstrates the accuracy and sharpness of the error estimators finally using a selection of model problems the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity copyright 2014 john wiley sons ltd

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

locking free adaptive discontinuous galerkin fem for linear elasticity problems an adaptive discontinuous galerkin finite element method for linear elasticity problems is presented we develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials absence of volume locking furthermore we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes

adaptive finite element simulation of incompressible flows by hybrid continuous discontinuous galerkin formulations in this work we design hybrid continuous discontinuous finite element spaces that permit discontinuities on nonmatching element interfaces of nonconforming meshes then we develop an equal order stabilized finite element formulation for incompressible flows over these hybrid spaces which combines the element interior stabilization of supg type continuous galerkin formulations and the jump stabilization of discontinuous galerkin formulations optimal stability and convergence results are obtained for the adaptive setting we use a standard error estimator and marking strategy numerical experiments show the optimal accuracy of the hybrid algorithm for both uniformly and adaptively refined nonconforming meshes the outcome of this work is a finite element formulation that can naturally be used on nonconforming meshes as discontinuous galerkin formulations while keeping the much lower cpu cost of continuous galerkin formulations

different domain decompositions at different times for capturing moving local phenomena abstract a conservative galerkin domain decomposition method for time dependent problems is given and analyzed this method allows one to apply different domain decompositions at different time levels when necessary in order to capture time changing local phenomena such as propagating fronts or moving layers error estimates in the energy and l 2 norms are established numerical results are presented

semi discretization of stochastic partial differential equations on r 1 by a finite difference method the paper concerns finite difference scheme for the approximation of partial differential equations in r 1 with additional stochastic noise by replacing the space derivatives in the original stochastic partial differential equation spde for short with difference quotients we obtain a system of stochastic ordinary differential equations we study the difference between the solution of the original spde and the solution to the corresponding equation obtained by discretizing the space variable the need to approximate the solution in r 1 with functions of compact support requires us to introduce a scale of weighted sobolev spaces employing the weighted lp theory of spde a sup norm error estimate is derived and the rate of convergence is given

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form