an optimal adaptive wavelet method for first order system least squares in this paper it is shown that any well posed 2nd order pde can be reformulated as a well posed first order least squares system this system will be solved by an adaptive wavelet solver in optimal computational complexity the applications that are considered are second order elliptic pdes with general inhomogeneous boundary conditions and the stationary navier stokes equations

finite element approximations of nonlinear eigenvalue problems in quantum physics in this paper we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics we derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations in particular we analyze the convergence and complexity of an adaptive finite element method in our analysis we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations we also present several numerical examples in quantum physics that support our theory

an application of the kantorovich theoremto nonlinear finite element analysis finite element solutions of strongly nonlinear elliptic boundary value problems are considered in this paper using the kantorovich theorem we show that if the frechet derivative of the nonlinear operator defined by the boundary value problem is an isomorphism at an exact solution then there exists a locally unique finite element solution near the exact solution moreover several a priori error estimates are obtained

an analysis of upwind schemes for the navier stokes equations we state a general result on approximation of branches of nonsingular solutions of nonlinear problems the abstract theory is then applied to the study of two mixed finite element schemes for the navier stokes equations using an upwind discretization of the convective terms

finite element discretization of the stokes and navier stokes equations with boundary conditions on the pressure we consider the stokes and navier stokes equations with boundary conditions of dirichlet type on the velocity on one part of the boundary and involving the pressure on the rest of the boundary we write the variational formulations of such problems next we propose a finite element discretization of them and perform the a priori and a posteriori analysis of the discrete problem some numerical experiments are presented in order to justify our strategy

a posteriori estimators for nonlinear elliptic partial differential equations many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of finite element discrete solutions to linear elliptic partial differential equations for each estimator there is a set of restrictions defined in such a way that the analysis of its performance is made possible usually the available estimators may be classified into two types i e the implicit estimators based on the solution of a local problem and the explicit estimators based on some suitable norm of the residual in a dual space regarding the performance an estimator is called asymptotically exact if it is a higher order perturbation of a norm of the exact error nowadays one may say that there is a larger understanding about the behavior of estimators for linear problems than for nonlinear problems the situation is even worse when the nonlinearities involve the highest derivatives occurring in the pde being considered strongly nonlinear pdes in this work we establish conditions under which those estimators originally developed for linear problems may be used for strongly nonlinear problems and how that could be done we also show that under some suitable hypothesis the estimators will be asymptotically exact whenever they are asymptotically exact for linear problems those results allow anyone to use the knowledge about estimators developed for linear problems in order to build new reliable and robust estimators for nonlinear problems

error indicators for the navier stokes equations in stream function and vorticity formulation this paper deals with a posteriori estimates for the finite element solution of the stokes problem in stream function and vorticity formulation for two different discretizations we propose error indicators and we prove estimates in order to compare them with the local error in a second step these results are extended to the navier stokes equations

numerical solutions for nonlinear elliptic problems based on first order system we present a theoretical analysis of numerical solutions for nonlinear elliptic problems our analysis is based on the abstract approximation theory for branches of nonsingular solutions developed by brezzi rappaz and raviart brr in most cases of finite element analysis the same spaces are used for the domain and range of the nonlinear operator concerning the application of brr theory this results in a loss of accuracy in the error estimates while numerical experiments show optimal convergence rates the main contribution of this paper is a theoretical analysis for the optimal convergence rates this is achieved by choosing different spaces for the domain and range of the nonlinear operator used in the application of brr theory previously this idea was used for an analysis of petrov galerkin formulation for the brr theory our analysis is developed for the first order least squares finite element methods and the mixed galerkin methods based on first order systems of equations

error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity this paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity the nonlinear equation which was derived and analyzed by huisken and ilmanen in j diff geom 59 353 437 represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature we first propose a finite element method for a regularized flow which involves a small parameter a rigorous analysis is presented to study well posedness and convergence of the scheme under certain mesh constraints and optimal rates of convergence are verified we then prove uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter both tend to zero computational results are provided to show the efficiency of the proposed finite element method and to numerically validate the jumping out phenomenon of the weak solution of the inverse mean curvature flow numerical studies are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter for optimal convergence of the proposed scheme finally a numerical convergence study for another approach recently proposed by r moser the inverse mean curvature flow and p harmonic functions preprint u bath 2005 for approximating the inverse mean curvature flow via p harmonic functions is also included

higher order quasi monte carlo integration for bayesian pde inversion abstract we analyze combined quasi monte carlo quadrature and finite element approximations in bayesian estimation of solutions to countably parametric operator equations with holomorphic dependence on the parameters as considered in schillings and schwab 2014 such problems arise in numerical uncertainty quantification and in bayesian inversion of operator equations with distributed uncertain inputs such as uncertain coefficients uncertain domains or uncertain source terms and boundary data we show that the parametric bayesian posterior densities belong to a class of weighted bochner spaces of functions of countably many variables with a particular structure of the qmc quadrature weights up to a problem dependent and possibly large finite dimension s product weights can be used and beyond this dimension weighted spaces with so called spod weights recently introduced in dick et al 2014 are used to describe the solution regularity we establish error bounds for higher order quasi monte carlo quadrature for the bayesian estimation based on dick et al 2016 it implies in particular regularity of the parametric solution and of the countably parametric bayesian posterior density in spod smoothness driven product and order dependent weighted spaces of integrand functions this in turn implies that the quasi monte carlo quadrature methods in dick et al 2014 are applicable to these problem classes with dimension independent convergence rates o n 1 p of n point hoqmc approximated bayesian estimates where 0 p 1 depends only on the sparsity class of the uncertain input in the bayesian estimation fast component by component cbc for short construction gantner and schwab 2016 allow efficient deterministic bayesian estimation with up to 1 0 4 parameters

stabilized finite element method for navier stokes equations with physical boundary conditions this paper deals with the numerical approximation of the 2d and 3d navier stokes equations satisfying nonstandard boundary conditions this lays on the finite element discretisation of the corresponding stokes problem which is achieved through a three fields stabilized mixed formulation a priori and a posteriori error bounds are established for the nonlinear problem ascertaining the convergence of the method finally numerical tests are presented including mesh refinement via error indicators

a posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification a posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy space discretization with continuous piecewise linear finite elements is considered the l 2 in time h 1 in space error is bounded above and below by an error estimator based on the equation residual numerical results show that the effectivity index is close to one an adaptive finite element algorithm is proposed and a solutal dendrite is computed c 2002 elsevier science b v all rights reserved

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

nonlinear discontinuous petrov galerkin methods the discontinuous petrov galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper its lowest order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation a reduced formulation and a weighted nonlinear least squares method quasi optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dpg framework for the approximation of a regular solution the variational model example allows for a built in guaranteed error control despite inexact solve the subtle uniqueness of discrete minimizers is monitored in numerical examples

closed form solutions of euler bernoulli beams with singularities abstract the problem of the integration of the static governing equations of the uniform euler bernoulli beam with discontinuities is studied in particular two types of discontinuities have been considered flexural stiffness and slope discontinuities both the above mentioned discontinuities have been modeled as singularities of the flexural stiffness by means of superimposition of suitable distributions generalized functions to a uniform one dimensional field closed form solutions of governing differential equation requiring the knowledge of the boundary conditions only are proposed and no continuity conditions are enforced at intermediate cross sections where discontinuities are shown the continuity conditions are in fact embedded in the flexural stiffness model and are automatically accounted for by the proposed integration procedure finally the proposed closed form solution for the cases of slope discontinuity is compared with the solution of a beam having an internal hinge with rotational spring reproducing the slope discontinuity

a priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems the effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied under similar assumption on the quadrature formula as for linear problems optimal error estimates in the l 2 and the h 1 norms are proved the numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh the analysis is valid for both simplicial and rectangular finite elements of arbitrary order numerical experiments corroborate the theoretical convergence rates

local boundary value problems for the error in fe approximation of non linear diffusion systems in this work we investigate the a posteriori error estimation for a class of non linear multicomponent diffusion operators which includes the stefan maxwell equations the local error indicators for the global error are based on local boundary value problems which are chosen to approximate either the global residual of the finite element approximation or the global linearized error equation using representative numerical examples it is shown that the error indicators based on the latter approach are more accurate for estimating the global error for this problem class as the problem becomes more non linear and can even produce better adaptive mesh refinement amr in addition we propose a new local error indicator for the error in output functionals that is accurate inexpensive to compute and is suitable for amr as demonstrated by numerical examples

the volume of the moduli space of flat connections on a nonorientable 2 manifold we compute the riemannian volume of the moduli space of flat connections on a nonorientable 2 manifold for a natural class of metrics we also show that witten s volume formula for these moduli spaces may be derived using haar measure and we give a new proof of witten s volume formula for the moduli space of flat connections on a 2 manifold using haar measure

a posteriori analysis of the finite element discretization of some parabolic equations nous considerons la discretisation d equations paraboliques soit lineaires soit semi lineaires par un schema d euler implicite en temps et par elements finis en espace l idee de cet article est de construire des indicateurs d erreur lies a l approximation en temps et en espace et de prouver leur equivalence avec l erreur dans le but de travailler avec des pas de temps adaptatifs et des maillages d elements finis adaptes a la solution

an adaptive finite element algorithm for a two dimensional stationary stefan like problem an efficient adaptive algorithm is presented for a stationary regularized stefan problem in 2d the adaptive criteria relies upon a posteriori estimates based on the residual equation since the problem we are studying is a non linear diffusion convection problem these error estimates are relatively crude in the sense that the constant relating the true error to the error indicator is unknown thus instead of trying to build a mesh such that the true error is below a given tolerance the local error indicator is equidistributed in a way such that the final number of triangles is close to a desired value at each iteration of the adaptive algorithm the new triangulation is obtained from the previous one by adding or deleting vertices according to the error indicator and a global mesh regeneration is performed using a delaunay mesh generator numerical experiments for two practical situations show the efficiency and the robustness of our approach

sparse grid reduced basis bayesian inversion we extend the reduced basis rb accelerated bayesian inversion methods for affine parametric linear operator equations which are considered in 16 17 to non affine nonlinear parametric operator equations we generalize the analysis of sparsity of parametric forward solution maps in 20 and of bayesian inversion in 48 49 to the fully discrete setting including petrov galerkin high fidelity hifi discretization of the forward maps we develop adaptive stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold the nonaffinity and nonlinearity with respect to w r t the distributed uncertain parameters and the unknown solution is collocated specifically by the so called empirical interpolation method eim for the corresponding bayesian inversion problems computational efficiency is enhanced in two ways first expectations w r t the posterior are computed by adaptive quadratures with dimension independent convergence rates proposed in 49 the present work generalizes 49 to account for the impact of the pg discretization in the forward maps on the convergence rates of the quantities of interest qoi for short second we propose to perform the bayesian estimation only w r t a parsimonious rb approximation of the posterior density based on the approximation results in 49 the infinite dimensional parametric deterministic forward map and operator admit n term rb and eim approximations which converge at rates which depend only on the sparsity of the parametric forward map in several numerical experiments the proposed algorithms exhibit dimension independent convergence rates which equal at least the currently known rate estimates for n term approximation we propose to accelerate bayesian estimation by first offline construction of reduced basis surrogates of the bayesian posterior density the parsimonious surrogates can then be employed for online data assimilation and for bayesian estimation they also open a perspective for optimal experimental design

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

the linearization method for the numerical analysis of finite element solutions to quasi linear elliptic partial differential equations linearization methods have been used in the numerical analysis of finite element solutions to nonlinear partial differential equations pdes for quite a long time frequently essential properties such as differentiability of the nonlinear operator and boundedness and invertibility properties for linearized operators are assumed without deeper considerations in this work we describe the main difficulties and build a framework which is able to circumvent them allowing for the consideration of larger classes of problems a priori error estimates are developed using that framework and a model problem is presented which illustrates the main results of this paper

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

some aspects of adaptive grid technology related to boundary and interior layers we consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions as the presence of these layer structures suggests reliable and accurate solution of this class of problems using finite difference finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms when the nature and structure of the layer is known mesh grading can be achieved during the grid generation by specifying an appropriate grading function however in many applications the location and nature of the layer behavior is not known in advance consequently adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach in this paper we provide a brief overview of the main adaptive grid strategies in the context of problems with layers associated error indicators that guide the refinement feedback control grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement amr the question of pollution effects in computation of local error indicators the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered the application of amr for layer problems is illustrated by means of case studies from semiconductor device transport drift diffusion nonlinear reaction diffusion layers due to surface capillary effects and shockwaves in compressible gas dynamics

a posteriori error estimation for elasto plastic problems based on duality theory abstract in this paper we introduce a new approach to a posteriori error estimation for elasto plastic problems based on the duality theory of the calculus of variations we show that in spite of the prevailing view duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts the error in the constitutive law and the error in the equilibrium equations this representation is both physically meaningful and computationally important the duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the galerkin orthogonality condition in particular they encompass the familiar smoothening or gradient averaging techniques commonly used in practice we prove that the duality error estimators are equivalent to the error when the approximate solution is the exact solution of the discretised problem moreover in the case of linear elasticity the known residual type error estimators can be obtained from the duality error estimators numerical results for a model elasto plastic problem show the accuracy of the duality error estimators

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

adaptive streamline diffusion methods for compressible flow using conservation variables we consider the streamline diffusion finite element method applied to compressible flow using conservation variables we propose some adaptive algorithms and present related numerical results

analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems a fully discrete analysis of the finite element heterogeneous multiscale method for a class of nonlinear elliptic homogenization problems of nonmonotone type is proposed in contrast to previous results obtained for such problems in dimension d leq2 for the h 1 norm and for a semi discrete formulation w e p ming and p zhang j amer math soc 18 2005 no 1 121 156 we obtain optimal convergence results for dimension d leq3 and for a fully discrete method which takes into account the microscale discretization in addition our results are also valid for quadrilateral finite elements optimal a priori error estimates are obtained for the h 1 and l 2 norms improved estimates are obtained for the resonance error and the newton method used to compute the solution is shown to converge numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method for various nonlinear problems