co volume methods for degenerate parabolic problems a complementary volume co volume technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems such as the stefan problem it is shown that these algorithms give rise to a discrete semigroup theory that parallels the continuous problem in particular the discrete stefan problem gives rise to nonlinear semigroups in both the discretel 1 andh 1 spaces

adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of biot s consolidation model abstract this paper is concerned with the analysis of coupled mixed finite element methods applied to the biot s consolidation model we consider two mixed formulations that use the stress tensor and darcy velocity as primary variables as well as the displacement and pressure the first formulation is with a symmetric stress tensor while the other enforces the symmetry of the stress weakly through the introduction of a lagrange multiplier the well posedness of the two formulations is shown through galerkin s method and suitable a priori estimates the two formulations are then discretized with the backward euler scheme in time and with two mixed finite elements in space we present next a general and unified a posteriori error analysis which is applicable for any flux and stress conforming discretization our estimates are based on h 1 conforming reconstruction of the pressure and a suitable h 1 d conforming reconstruction of the displacement both are continuous and piecewise affine in time these reconstructions are used to infer a guaranteed and fully computable upper bound on the energy type error measuring the differences between the exact and the approximate pressure and displacement the error components resulting from the spatial and the temporal discretization are distinguished they are then used to design an adaptive space time algorithm numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm

a posteriori error estimation based on potential and flux reconstruction for the heat equation we derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous galerkin various finite volume and mixed finite element methods in space and the backward euler scheme in time extensions to conforming and nonconforming finite element spatial discretizations are also outlined our estimates are based on a h 1 conforming reconstruction of the potential continuous and piecewise affine in time and a locally conservative mathbf h mathrm div conforming reconstruction of the flux piecewise constant in time they yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative local in time lower bounds are also derived for nonconforming methods on time varying meshes the lower bounds require a mild parabolic type constraint on the meshsize

guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems abstract we derive a posteriori error estimates for a class of second order monotone quasi linear diffusion type problems approximated by piecewise affine continuous finite elements our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual as well as a global error lower bound up to a generic constant independent of the nonlinear operator they are thus fully robust with respect to the nonlinearity thanks to the choice of the error measure they are also locally efficient albeit in a different norm and hence suitable for adaptive mesh refinement moreover they allow to distinguish estimate separately and compare the discretization and linearization errors hence the iterative newton raphson fixed point linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error this can lead to important computational savings as performing an excessive number of unnecessary linearization iterations can be avoided a strategy combining the linearization stopping criterion and adaptive mesh refinement is proposed and numerically tested for the p laplacian

a posteriori error estimates for combined finite volume finite element discretizations of reactive transport equations on nonmatching grids abstract we derive in this paper guaranteed and fully computable a posteriori error estimates for vertex centered finite volume type discretizations of transient linear convection diffusion reaction equations our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew symmetric part of the differential operator lower bounds global in space but local in time are also derived these lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting on the basis of the derived estimates we propose an adaptive algorithm which enables to automatically achieve a user given relative precision this algorithm also leads to efficient calculations as it balances the time and space error contributions as an example we apply our estimates to the combined finite volume finite element scheme including such features as use of mass lumping for the time evolution or reaction terms of upwind weighting for the convection term and discretization on nonmatching meshes possibly containing nonconvex and non star shaped elements a collection of numerical experiments illustrates the efficiency of our estimates and the use of the space time adaptive algorithm

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

adaptive asynchronous time stepping stopping criteria and a posteriori error estimates for fixed stress iterative schemes for coupled poromechanics problems abstract in this paper we develop adaptive iterative coupling schemes for the biot system modeling coupled poromechanics problems we particularly consider the space time formulation of the fixed stress iterative scheme in which we first solve the problem of flow over the whole space time interval then exploiting the space time information for solving the mechanics two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in space and the backward euler scheme in time therefrom adaptive fixed stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions these ingredients are used to derive a posteriori error estimates for the fixed stress algorithms distinguishing the different error components namely the spatial discretization the temporal discretization and the fixed stress iteration components precisely at the iteration k 1 of the adaptive algorithm we prove that our estimate gives a guaranteed and fully computable upper bound on the energy type error measuring the difference between the exact and approximate pressure and displacement these error components are efficiently used to design adaptive asynchronous time stepping and adaptive stopping criteria for the fixed stress algorithms numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms

stress and flux reconstruction in biots poro elasticity problem with application to a posteriori error analysis we derive equilibrated reconstructions of the darcy velocity and of the total stress tensor for biots poro elasticity problem both reconstructions are obtained from mixed finite element solutions of local neumann problems posed over patches of elements around mesh vertices the darcy velocity is reconstructed using raviartthomas finite elements and the stress tensor using arnoldwinther finite elements so that the reconstructed stress tensor is symmetric both reconstructions have continuous normal component across mesh interfaces using these reconstructions we derive a posteriori error estimators for biots poro elasticity problem and we devise an adaptive spacetime algorithm driven by these estimators the algorithm is illustrated on test cases with analytical solution on the quarter five spot problem and on an industrial test case simulating the excavation of two galleries

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

adaptive regularization linearization and discretization and a posteriori error control for the two phase stefan problem we consider in this paper the time dependent two phase stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward euler temporal discretizations regularization of the enthalpy temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual as well as on the l2 l2 error of the temperature and the l2 h1 error of the enthalpy moreover they allow to distinguish the space time regularization and linearization error components an adaptive algorithm is proposed which ensures computational savings through the online choice of a sufficient regularization parameter a stopping criterion for the linearization iterations local space mesh refinement time step adjustment and equilibration of the spatial and temporal errors we also prove the efficiency of our estimate our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization as an example we apply it to the vertex centered finite volume finite element with mass lumping and quadrature and newton methods numerical results illustrate the effiectiveness of our estimates and the performance of the adaptive algorithm

an a posteriori error estimate for vertex centered finite volume discretizations of immiscible incompressible two phase flow in this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium we take into account the capillary pressure which leads to a coupled system of two equations parabolic and elliptic the parabolic equation may become degenerate i e the nonlinear diffusion coefficient may vanish over regions that are not known a priori we first show that under appropriate assumptions the energy type norm differences between the exact and the approximate nonwetting phase saturations the global pressures and the kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals we then bound the dual norm of the residuals by fully computable a posteriori estimators our analysis covers a large class of conforming vertex centered finite volume type discretizations with fully implicit time stepping as an example we focus here on two approaches a mathematical scheme derived from the weak formulation and a phase by phase upstream weighting engineering scheme finally we show how the different error components namely the space discretization error the time discretization error the linearization error the algebraic solver error and the quadrature error can be distinguished and used for making the calculations efficient

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

error estimates for the finite volume discretization for the porous medium equation we analyze the convergence of a numerical scheme for a class of degenerate parabolic problems modelling reactions in porous media and involving a nonlinear possibly vanishing diffusion the scheme involves the kirchhoff transformation of the regularized nonlinearity as well as an euler implicit time stepping and triangle based finite volumes we prove the convergence of the approach by giving error estimates in terms of the discretization and regularization parameter

a framework for robust a posteriori error control in unsteady nonlinear advection diffusion problems we derive a framework for a posteriori error estimates in unsteady nonlinear possibly degenerate advection diffusion problems our estimators are based on a space time equilibrated flux reconstruction and are locally computable they are derived for the error measured in a space time mesh dependent dual norm stemming from the problem and meshes at hand augmented by a jump seminorm measuring possible nonconformities in space owing to this choice a guaranteed and globally efficient upper bound is achieved as well as robustness with respect to nonlinearities advection dominance domain size final time and absolute and relative size of space and time steps local in time and in space efficiency is also shown for a localized upper bound of the error measure in order to apply the framework to a given numerical method two simple conditions local space time mass conservation and an approximation property of the reconstructed fluxes need to be verified we show how to do this for the interior penalty

error estimates for the multidimensional two phase stefan problem in this paper we derive rates of convergence for regularizations of the multidimen sional two phase stefan problem and use the regularized problems to define backward dif ference in time and c piecewise linear in space galerkin approximations we find an l2 rate of convergence of order in the e regularization and an l rate of convergence of order h2 e ai 6f in the galerkin estimates which leads to the natural choices a4 3 a a4 3 and a resulting 0 i2 3 l2 rate of convergence of the numerical scheme to the solution of the differential equation an essentially 0 h rate is demonstrated when e 0 and a h2 in our galerkin scheme under a boundedness hypothesis on the galerkin approxima tions the latter result is consistent with computational experience 1 introduction given a smoothly bounded domain s c rn we consider the equation in distribution form l h au fiu 0

the numerical study of a microscale model for lithium ion batteries abstract we consider a thermodynamic consistent microscale model from for the transport processes of lithium ion concentration and charge in lithium ion batteries a fully convergent finite element discretisation in space of arbitrary order with uniform time stepping is discussed for this strongly nonlinear coupled system of nonlinear pdes we analyse the error for higher order finite element methods on a representative geometry which shows already the difficulties when dealing with a realistic microstructure moreover we derive explicit analytical formulas for the exact solutions for the elliptic subproblem and the complete time dependent problem for a non trivial geometry these results show the need for appropriate boundary approximations as well as adaptive refinement strategies in addition we incorporate a two phase intercalation model for the electrode this moving boundary model also called stefan problem has also been implemented simulations of our battery model including the two phase model will be shown

an adaptive finite element method for two phase stefan problems in two space dimensions i stability and error estimates a simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two phase stefan problems in 2d a typical triangula tion is coarse away from the discrete interface where discretization parameters satisfy a parabolic relation whereas it is locally refined in the vicinity of the dis crete interface so that the relation becomes hyperbolic several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations mesh selection is based upon equidistributing pointwise interpolation errors be tween consecutive meshes and imposing that discrete interfaces belong to the so called refined region consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated a general theory for interpolation between noncompatible meshes is set up in lp based norms the resulting scheme is stable in various sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods 3 2 2 on quasi uniform meshes namely 0 r as opposed to 0 x to achieve the same global asymptotic accuracy here r 0 is the uniform time step 112 a rate of convergence of essentially 0 x is derived in the natural energy spaces provided the total number of mesh changes is restricted to 0 x which in turn is compatible with the mesh selection procedure an auxiliary quasi optimal pointwise error estimate for the laplace operator is proved as well numerical results illustrate the scheme s efficiency in approximating both solutions and interfaces

solution of porous medium type systems by linear approximation schemes the convergence of semi discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type the enthalpy formulation and variational technique are used the semi discretization used reduces the original parabolic p d e to linear elliptic p d e the algebraic correction arising from nonlinearities is treated by newton like iterations in finite steps some numerical experiments are discussed and compared with the analytical solutions

a moving mesh finite element method for the two dimensional stefan problems an r adaptive moving mesh method is developed for the numerical solution of an enthalpy formulation of two dimensional heat conduction problems with a phase change the grid is obtained from a global mapping of the physical to the computational domain which is designed to cluster mesh points around the interface between the two phases of the material the enthalpy equation is discretised using a semiimplicit galerkin finite element method using linear basis functions the moving finite element method is applied to problems where the phase front is cusp shaped and where the interface changes topology

explicit error bounds in a conforming finite element method the goal of this paper is to define a procedure for bounding the error in a conforming finite element method the new point is that this upper bound is fully explicit and can be computed locally numerical tests prove the efficiency of the method it is presented here for the case of the poisson equation and a first order finite element approximation

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

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

a posteriori error estimates including algebraic error and stopping criteria for iterative solvers for the finite volume discretization of a second order elliptic model problem we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system we show that the algebraic error can be bounded by constructing an equilibrated raviart thomas nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector next claiming that the discretization error and the algebraic one should be in balance we construct stopping criteria for iterative algebraic solvers an attention is paid in particular to the conjugate gradient method which minimizes the energy norm of the algebraic error using this convenient balance we also prove the efficiency of our a posteriori estimates i e we show that they also represent a lower bound up to a generic constant for the overall energy error a local version of this result is also stated this makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

elliptic reconstruction and a posteriori error estimates for parabolic problems it is known that the energy technique for a posteriori error analysis of finite element discretizations of parabolic problems yields suboptimal rates in the norm l infty 0 t l 2 omega in this paper we combine energy techniques with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order and regularity for piecewise polynomials of degree higher than one this technique may be regarded as the dual a posteriori counterpart of wheeler s elliptic projection method in the a priori error analysis

the development of the green s function for the boltzmann equation we review the particle like and wave like property of the boltzmann equation this property leads to a sequence of developments on the mathematical theory of the green s function for the boltzmann equation

diffusion limited reactions of hard core particles in one dimension we investigate three different methods to tackle the problem of diffusion limited reactions annihilation of hard core classical particles in one dimension we first extend an approach devised by lushnikov sov phys jetp 64 811 1986 and calculate for a single species the asymptotic long time and or large distance behavior of the two point correlation function based on a work by grynberg and stinchcombe phys rev e 50 957 1994 phys rev lett 74 1242 1995 76 851 1996 which was developed to treat stochastic adsorption desorption models we provide in a second step the exact two point one and two time correlation functions of lushnikov s model we then propose a formulation of the problem in terms of path integrals for pseudo fermions this formalism can be used to advantage in the multispecies case especially when applying perturbative renormalization group techniques

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

on a numerical approach to stefan like problems this paper is devoted to the numerical analysis of a bidimensional two phase stefan problem we approximate the enthalpy formulation byc 0 piecewise linear finite elements in space combined with a semi implicit scheme in time under some restrictions related to the finite element mesh and to the timestep we prove positivity stability and convergence results various numerial tests are presented and discussed in order to show the accuracy of our scheme

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

a posteriori error estimation and adaptivity for degenerate parabolic problems two explicit error representation formulas are derived for degenerate parabolic pdes which are based on evaluating a parabolic residual in negative norms the resulting upper bounds are valid for any numerical method and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient they are applied to a practical space time discretization consisting of c 0 piecewise linear finite elements over highly graded unstructured meshes and backward finite differences with varying time steps two rigorous a posteriori error estimates are derived for this scheme and used in designing an efficient adaptive algorithm which equidistributes space and time discretization errors via refinement coarsening a simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach and hints at the potentials and reliability of the proposed method