a posteriori error analysis for stabilised finite element approximations of transport problems we develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments two alternative dual problems are considered one is based on the formal adjoint of the hyperbolic differential operator the other on the transposition of the bilinear form for the stabilised finite element method we show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes

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

robust a posteriori estimator for advection diffusion reaction problems we propose an almost robust residual based a posteriori estimator for the advection diffusion reaction model problem the theory is developed in the one dimensional setting the numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators in particular the mentioned norm possesses features that allow us to obtain a meaningful a posteriori estimator robust up to a log pe factor where pe is the global peclet number of the problem various numerical tests are performed in one dimension to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature we also consider a possible two dimensional extension of our result and only present a few basic numerical tests indicating that the estimator seems to preserve the good features of the one dimensional setting

robustness in a posteriori error analysis for fem flow models we derive a residual based a posteriori error estimator for a stabilized finite element discretization of certain incompressible oseen like equations we focus our attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem and mesh parameters we also consider a scalar reaction convection diffusion problem and a divergence free projection problem in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction convection diffusion phenomena and separately of the incompressibility constraint

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

discrete interpolation norms with applications we describe norm representations for interpolation spaces generated by finite dimensional subspaces of hilbert spaces these norms are products of integer and noninteger powers of the grammian matrices associated with the generating pair of spaces for the interpolation space we include a brief description of some of the algorithms which allow the efficient computation of matrix powers we consider in some detail the case of fractional sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques numerical experiments are included

variational multiscale a posteriori error estimation for multi dimensional transport problems this paper presents an explicit a posteriori error estimator for the multi dimensional transport equation based on an approximation to the variational multiscale theory the technique is adequate for methods with a local error distribution such as stabilized methods yielding efficiencies close to one for both the hyperbolic and diffusive limit regarding the hyperbolic limit previous work is extended to multi dimensions adequate norms are proposed for the computation of the error and proper error intrinsic time scales are calculated for the bilinear quadrilateral and the linear triangle furthermore the model considers the element interface error along the element edges enabling the error prediction in the diffusive limit the success of the method can be explained by the fact that in stabilized methods the element local problem captures most of the error and the proposed error intrinsic time scales are an approximation to the solution of the dual problem the proposed technology which can be implemented straightforwardly in existing codes is extremely economical because it is a simple explicit postprocessing

proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory we study the large n limit of a sparse random block matrix ensemble it depends on two parameters the average connectivity z and the size of the blocks d which is the dimension of an euclidean space in the limit of large d with frac z d fixed we prove the conjecture that the spectral distribution of the sparse random block matrix converges in the case of the adjacency block matrix to the one of the effective medium approximation in the case of the laplacian block matrix to the marchenko pastur distribution we extend previous analytical computations of the moments of the spectral density of the adjacency block matrix and the lagrangian block matrix valid for all values of z and d

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

a robust a posteriori estimator for the residual free bubbles method applied to advection diffusion problems we develop the a posteriori error analysis for the rfb method applied to the linear advection diffusion problem the numerical error measured in suitable norms is estimated in terms of the numerical residual the robustness is investiged in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error

the multiscale approach to error estimation and adaptivity abstract a new explicit a posteriori error estimator is investigated which emanates from the variational multiscale theory the error estimator uses an approximation of the green s function which propagates the error according to the dual problem the technique predicts the error in the l2 norm from element residuals both interior and inter element jumps therefore leading to a very economical procedure finally the methodology is applied to fluid flow transport showing that for convection dominated flows the efficiency index is independent of the diffusion coefficient

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

an adaptive algorithm for the crank nicolson scheme applied to a time dependent convection diffusion problem an a posteriori upper bound is derived for the nonstationary convection diffusion problem using the crank nicolson scheme and continuous piecewise linear stabilized finite elements with large aspect ratio following lozinski et al 2009 13 a quadratic time reconstruction is used a space and time adaptive algorithm is developed to ensure the control of the relative error in the l 2 h 1 norm numerical experiments illustrating the efficiency of this approach are reported it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step even in the convection dominated regime and in the presence of boundary layers

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

a posteriori error estimates for finite volume approximations of elliptic equations on general surfaces abstract in this paper we present a residual based a posteriori error estimate for the finite volume discretization of steady convection diffusion reaction equations defined on surfaces in r 3 which are often implicitly represented as level sets of smooth functions reliability and efficiency of the proposed a posteriori error estimator are rigorously proved numerical experiments are also conducted to verify the theoretical results and demonstrate the robustness of the error estimator

robust a posteriori error estimates for stationary convection diffusion equations we analyze a posteriori error estimators for finite element discretizations of convection dominated stationary convection diffusion equations using locally refined isotropic meshes the estimators are based on either the evaluation of local residuals or the solution of discrete local problems with dirichlet or neumann boundary conditions all estimators yield global upper and lower bounds for the error measured in a norm that incorporates the standard energy norm and a dual norm of the convective derivative they are fully robust in the sense that the ratio of the upper and lower bounds is uniformly bounded with respect to the size of the convection the estimates are also uniform with respect to the size of the zero order reaction term and also hold for the limit case of vanishing reaction

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100