adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics in this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method the error estimates are based on the solution of an appropriately defined adjoint problem we present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain the primal subdomains are discretized using either an interior penalty discontinuous galerkin method or a continuous galerkin method with weakly imposed dirichlet conditions the mixed subdomains are discretized using raviart thomas mixed finite elements the a posteriori error estimate also accounts for the errors due to adjoint inconsistent subdomain discretizations the coupling between the subdomain discretizations is achieved via a mortar space we show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

error analysis for optimal control problem governed by convection diffusion equations dg method in the current paper we study the convergence properties of the dgfe approximation of optimal control problem governed by convection diffusion equations we derive a posteriori error estimates and a priori error estimates for both the states ad joint and the control variable approximation for the optimal control problem these estimates are apparently not available in the literature

a posteriori error estimators for nonconforming finite element methods of the linear elasticity problem in this work we derive and analyze a posteriori error estimators for low order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes with hanging nodes allowed for local mesh refinement first it is shown that equilibrated neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution the first error estimator is then obtained by applying the equilibrated residual method with this set of neumann data from this implicit estimator we also derive two explicit error estimators one of which is similar to the one proposed by dorfler and ainsworth 2005 24 for the stokes problem it is established that all these error estimators are reliable and efficient in a robust way with respect to the lame constants the main advantage of our error estimators is that they yield guaranteed i e constant free upper bounds for the energy like error up to higher order terms due to data oscillation when a good estimate for the inf sup constant is available which is confirmed by some numerical results

nondeterministic functions and the existence of optimal proof systems we provide new characterizations of two previously studied questions on nondeterministic function classes q1 do nondeterministic functions admit efficient deterministic refinements q2 do nondeterministic function classes contain complete functions we show that q1 for the class npmv t is equivalent to the question whether the standard proof system for sat is p optimal and to the assumption that every optimal proof system is p optimal assuming only the existence of a p optimal proof system for sat we show that every set with an optimal proof system has a p optimal proof system under the latter assumption we also obtain a positive answer to q2 for the class npmv t an alternative view on nondeterministic functions is provided by disjoint sets and tuples we pursue this approach for disjoint np pairs and its generalizations to tuples of sets from np and conp with disjointness conditions of varying strength in this way we obtain new characterizations of q2 for the class npsv question q1 for npsv is equivalent to the question of whether every disjoint np pair is easy to separate in addition we characterize this problem by the question of whether every propositional proof system has the effective interpolation property again these interpolation properties are intimately connected to disjoint np pairs and we show how different interpolation properties can be modeled by np pairs associated with the underlying proof system

guaranteed a posteriori error estimator for mixed finite element methods of linear elasticity with weak stress symmetry in this paper we propose an a posteriori error estimator for the mixed finite element methods of the linear elasticity problem with the symmetry condition weakly imposed on the stress tensor the error estimator is constructed by making a proper decomposition of the stress error and using an argument similar to the hypercircle method it is shown that the resulting estimator yields a guaranteed upper bound on the stress error which relies on computable upper bounds of the constants in the first and second korn inequalities we also establish the local lower bound by using the discrete friedrichs inequality our approach is equivalent to the helmholtz decomposition of the stress error but requires assumptions neither on the regularity of the solution nor the geometry of the domain numerical results are provided to illustrate the efficiency of our error estimator

goal oriented error estimation based on equilibrated flux reconstruction for finite element approximations of elliptic problems abstract we propose an approach for goal oriented error estimation in finite element approximations of second order elliptic problems that combines the dual weighted residual method and equilibrated flux reconstruction methods for the primal and dual problems the objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution it is only assumed here that the discretization methods come with a priori error estimates and an equilibrated flux reconstruction algorithm a high order discontinuous galerkin dg method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes one contribution of the paper is to show how the order of the dg method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method a dg method or a mixed raviart thomas method is used for the solution of the primal problem numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

domain decomposition for linear elasticity with dg jumps and mortars in this article we present a domain decomposition method for solving a linear elasticity system data are transmitted by jumps as in a discontinuous galerkin method and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains a decoupling algorithm condenses the unknowns on the interface the matrix of the system is constructed by parallel computation numerical experiments confirming the theory are presented

numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques we consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary as a paradigm for such problems we simply refer to stokes flow after a brief review of the problem and of its well posedness we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints we exploit a penalty method to account for them we perform the analysis of the method in terms of consistency boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization since the additional terms introduced to account for the defective boundary conditions are non local we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy in contrast to alternative discretization methods based on lagrange multipliers accounting for the constraints on the boundary the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation indeed it does not require neither multipliers nor sub iterations or additional adjoint problems with respect to the reference problem at hand

an hp local discontinuous galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type in this paper an hp local discontinuous galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type on hp quasiuniform meshes using the brouwer fixed point theorem it is shown that the discrete problem has a solution and then using lipschitz continuity of the discrete solution map uniqueness is also proved a priori error estimates in broken h 1 norm and l 2 norm which are optimal in h suboptimal in p are derived these results are exactly the same as in the case of linear elliptic boundary value problems numerical experiments are provided to illustrate the theoretical results

robust equilibrated residual error estimator for diffusion problems conforming elements this paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem the estimator which is a modification of those in d braess and j schoberl math comput 77 2008 pp 651 672 r verfurth siam j numer anal 47 2009 pp 3180 3194 is based on the prager synge identity and on a local recovery of an equilibrated flux numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator when the distribution of diffusion coefficients is local quasi monotone it is shown theoretically that the estimator is robust with respect to the size of jumps

a residual based a posteriori error estimator for single phase darcy flow in fractured porous media in this paper we develop an a posteriori error estimator for a mixed finite element method for single phase darcy flow in a two dimensional fractured porous media the discrete fracture model is applied to model the fractures by one dimensional fractures in a two dimensional domain we consider raviart thomas mixed finite element method for the approximation of the coupled darcy flows in the fractures and the surrounding porous media we derive a robust residual based a posteriori error estimator for the problem with non intersecting fractures the reliability and efficiency of the a posteriori error estimator are established for the error measured in an energy norm numerical results verifying the robustness of the proposed a posteriori error estimator are given moreover our numerical results indicate that the a posteriori error estimator also works well for the problem with intersecting fractures

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

analysis of hp discontinuous galerkin methods for incompressible two phase flow in this paper we prove the convergence of a class of discontinuous galerkin methods for solving the fully coupled incompressible two phase flow problem in the non degenerate case estimates in both the mesh size and the polynomial degrees are obtained numerical convergence rates confirm the theoretical results

a posteriori error estimates of discontinuous galerkin methods for the signorini problem a reliable and efficient a posteriori error estimator is derived for a class of discontinuous galerkin dg methods for the signorini problem a common property shared by many dg methods leads to a unified error analysis with the help of a constraint preserving enriching map the error estimator of dg methods is comparable with the error estimator of the conforming methods numerical experiments illustrate the performance of the error estimator

solving the advection diffusion equation on unstructured meshes with discontinuous mixed finite elements and a local time stepping procedure explicit schemes are known to provide less numerical diffusion in solving the advection diffusion equation especially for advection dominated problems traditional explicit schemes use fixed time steps restricted by the global cfl condition in order to guarantee stability this is known to slow down the computation especially for heterogeneous domains and or unstructured meshes to avoid this problem local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local cfl condition have been developed in this paper a local time stepping approach is used with a numerical model based on discontinuous galerkin mixed finite element methods to solve the advection diffusion equation the developments are detailed for general unstructured triangular meshes numerical experiments are performed to show the efficiency of the numerical model for the simulation of i the transport of a solute on highly unstructured meshes and ii density driven flow where the velocity field changes at each time step the model gives stable results with significant reduction of the computational cost especially for the non linear problem moreover numerical diffusion is also reduced for highly advective problems copyright 2009 john wiley sons ltd

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

error indicators for mixed finite elements in 2 dimensional linear elasticity abstract we establish a posteriori error indicators for mixed finite elements in plane elasticity the error estimators refer to residuals of the strong equations and to jumps of the displacements on interelement boundaries for the bdm elements of lowest order the error indicators are computed with displacement fields which are obtained by a postprocessing procedure numerical examples show that adaptive mesh refinements based on these estimators lead to very efficient algorithms

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

discontinuous galerkin and mimetic finite difference methods for coupled stokes darcy flows on polygonal and polyhedral grids we study locally mass conservative approximations of coupled darcy and stokes flows on polygonal and polyhedral meshes the discontinuous galerkin dg finite element method is used in the stokes region and the mimetic finite difference method is used in the darcy region dg finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces optimal convergence estimates for the numerical scheme are derived results from computational experiments supporting the theory are presented

symmetric and nonsymmetric discontinuous galerkin methods for reactive transport in porous media for solving reactive transport problems in porous media we analyze three primal discontinuous galerkin dg methods with penalty namely symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg a cut off operator is introduced in dg to treat general kinetic chemistry error estimates in l2 h1 are established which are optimal in h and nearly optimal in p we develop a parabolic lift technique for sipg which leads to h optimal and nearly p optimal error estimates in the l2 l2 and negative norms numerical results validate these estimates we also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces

discontinuous galerkin finite element approximation of nonlinear non fickian diffusion in viscoelastic polymers we consider discrete schemes for a nonlinear model of non fickian diffusion in viscoelastic polymers the model is motivated by but not the same as that proposed by cohen et al in siam j appl math 55 pp 348 368 1995 the spatial discretisation is effected with both the symmetric and non symmetric interior penalty discontinuous galerkin finite element method and the time discretisation is of crank nicolson type we also discuss two means of handling the nonlinearity either implicitly which requires the solution of nonlinear equations at each time level or through a linearisation based on extrapolating from previous time levels the same optimal orders of convergence are proven in both cases and to verify this some numerical results are also given for the linearised scheme

superconvergence and a posteriori error estimates of the dg method for scalar hyperbolic problems on cartesian grids in this paper we analyze the discontinuous galerkin dg finite element method for the steady two dimensional transport reaction equation on cartesian grids we prove the l2 stability and optimal l2 error estimates for the dg scheme we identify a special numerical flux for which the l2 norm of the solution is of order p 1 when tensor product polynomials of degree at most p are used we further prove superconvergence towards a particular projection of the directional derivative the order of superconvergence is proved to be p 1 2 we also provide a very simple derivative recovery formula which is o h p 1 superconvergent approximation to the directional derivative moreover we establish an o h 2 p 1 global superconvergence for the solution flux at the outflow boundary of the domain these results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element finally we prove that the proposed a posteriori dg error estimates converge to the true errors in the l2 norm at o h p 1 rate and that the global effectivity index converges to unity at o h rate our results are valid without the flow condition restrictions we perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal

a general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids the overlapping covolume case we present a general framework for constructing and analyzing finite volume methods applied to the mixed formulation of second order elliptic problems on quadrilateral grids the control volumes or covolumes in the grids overlap an overlapping finite volume method of this type was first introduced by russell in t f russell tech report 3 reservoir simulation research corp tulsa ok 1995 and was tested for a variety of problems on rectangular and quadrilateral grids in z cai et al comput geosci 1 1997 pp 289 315 later in s h chou and d y kwak siam j numer anal 37 2000 pp 758 771 chou and kwak reformulated it as their mixed covolume method and proved optimal order error estimates using the covolume methodology from s h chou math comp 66 1997 pp 85 104 and s h chou and d y kwak siam j numer anal 35 1998 pp 494 507 however their treatment was restricted to the case of diagonal coefficient tensor and rectangular grids since a different approach was needed for the quadrilateral distorted rectangular case in this paper we give a new framework which can handle not only the rectangular anisotropic case but also the anisotropic and irregular grid cases in which the locally supported test functions are images of the natural unit coordinate vectors under the piola transformation our theory sheds light on how to create new test functions using quadratures and now covers russell s quadrilateral case

structure in approximation classes the study of the approximability properties of np hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking the last important breakthrough proves the apx completeness of several important optimization problems and thus reconciles two distinct views of approximation classes syntactic and computational s khanna et al in proc 35th ieee symp on foundations of computer science ieee computer society press los alamitos ca 1994 pp 819 830 in this paper we obtain new results on the structure of several computationally defined approximation classes in particular after defining a new approximation preserving reducibility to be used for as many approximation classes as possible we give the first examples of natural npo complete problems and the first examples of natural apx intermediate problems moreover we state new connections between the approximability properties and the query complexity of npo problems

anisotropic and dynamic mesh adaptation for discontinuous galerkin methods applied to reactive transport mesh adaptation strategies are proposed for discontinuous galerkin methods applied to reactive transport problems with emphasis on dynamic and anisotropic adaptation they include an anisotropic mesh adaptation scheme and two isotropic methods using an l 2 l 2 norm error estimator and a hierarchic error indicator these dynamic mesh adaptation approaches are investigated using benchmark cases numerical results demonstrate that the three approaches resolve time dependent transport adequately without slope limiting for both long term and short term simulations it is shown that these results apply to problems where either diffusion or advection dominates in different subdomains moreover for these schemes mass conservation is retained locally during dynamic mesh modification comparison studies indicate that the anisotropic mesh adaptation provides the most efficient meshes and has the least numerical diffusion among the three adaptation approaches 2005 elsevier b v all rights reserved

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

comparing reductions to np complete sets under the assumption that np does not have p measure 0 we investigate reductions to np complete sets and prove the following 1 adaptive reductions are more powerful than nonadaptive reductions there is a problem that is turing complete for np but not truth table complete 2 strong nondeterministic reductions are more powerful than deterministic reductions there is a problem that is snp complete for np but not turing complete 3 every problem that is many one complete for np is complete under length increasing reductions that are computed by polynomial size circuits the first item solves one of lutz and mayordomo s twelve problems in resource bounded measure 1999 we also show that every many one complete problem for ne is complete under one to one length increasing reductions that are computed by polynomial size circuits

on the solution of the coupled navier stokes and darcy equations abstract this paper introduces and analyzes two mathematical models for coupling the incompressible navier stokes equations with the porous media flow equations a numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium is proposed existence and uniqueness results under small data condition of the numerical solution are proved optimal a priori error estimates are derived numerical examples comparing the two models under varying physical parameters are provided

a posteriori error analysis for locally conservative mixed methods in this work we present a theoretical analysis for a residual type error estimator for locally conservative mixed methods this estimator was first introduced by braess and verfurth for the raviart thomas mixed finite element method working in mesh dependent norms we improve and extend their results to cover any locally conservative mixed method under minimal assumptions in particular avoiding the saturation assumption made by braess and verfurth our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries the main results are applied to the p1 nonconforming finite element method and the interior penalty discontinuous galerkin method as well as the mixed finite element method

hp adaptive ipdg tdg fem for parabolic obstacle problems for a parabolic obstacle problem two equivalent hp fem discretization methods based on interior penalty discontinuous galerkin in space and discontinuous galerkin in time are presented the first approach is based on a variational inequality vi formulation and the second approach on a mixed method in which the non penetration condition is resolved by a lagrange multiplier the discrete lagrange multiplier is a linear combination of biorthogonal basis functions allowing to write the discrete vi constraints as a set of complementarity problems employing a penalized fischer burmeister non linear complementarity function the discrete mixed problem can be solved by a locally q quadratic converging semi smooth newton ssn method the hierarchical a posteriori error estimator for the vi formulation which under the saturation assumption is both efficient and reliable allows hp adaptivity the numerical experiments show improved convergence compared to uniform and h adaptive meshes furthermore an a priori error estimate is given for the vi formulation

guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems abstract in this work we analyze a posteriori error estimator for a general class of mixed finite element methods of elliptic problems which guarantees an upper bound on the vector error thus extending the recent result obtained for the lowest order triangular raviart thomas mixed finite element method in m ainsworth a posteriori error estimation for lowest order raviart thomas mixed finite elements siam j sci comput 30 2007 08 189 204 the error estimator is constructed through a variant of stenberg s postprocessing procedure and the guaranteed upper bound is readily established by making use of the argument similar to the hypercircle method however the proof of the lower bound given in the above reference does not seem to apply to other kinds of mixed elements so we employ a different technique using the discrete friedrichs inequality to establish the lower bound of the error estimator

one way functions and circuit complexity abstract a finite function f is a mapping of 0 1 n into 0 1 m where is a symbol to be thought of as undefined a family of finite functions is said to be one way in a circuit complexity sense if it can be computed with polynomial size circuits but every family of inverses of these functions cannot in this paper we show that provided functions that are not one to one are allowed one way functions exist if and only if the satisfiability problem sat does not have polynomial size circuits a family of functions fi x can be checked if some family of polynomial size circuits with inputs x and y can determine if fi x y a family of functions fi x can be evaluated if some family of polynomial size circuits with input x can compute fi x can all families of total functions that can be checked also be evaluated we show that this is true if and only if the nonuniform versions of the complexity classes p and up co up are equal a family of functions fi is one way for constant depth circuits if fi can be computed with unbounded famin circuits of polynomial size and constant depth but every family of inverses fi 1 cannot we give two provably one way functions in fact permutaions for constant depth circuits the second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth

on γ reducibility versus polynomial time many one reducibility abstract we prove that each element of a class of functions denoted by npc t p whose graphs can be accepted in nondeterministic polynomial time can be evaluated in deterministic polynomial time if and only if reducibility is equivalent to polynomial time many one reducibility we also modify the proof technique used to obtain part of this result to obtain the stronger result that if every reduction can be replaced by a polynomial time turing reduction then every function in npc t p can be evaluated in deterministic polynomial time

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

on the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients the successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients an a posteriori error indicator a refinement coarsening strategy and the choice of various parameters the objective of the paper is to examine the influence of these factors on the performance of adaptive finite element methods for a model problem the linear elliptic equation with strongly discontinuous coefficients we derive a new a posteriori error estimator which depends locally on the oscillations of the coefficients around singular points extensive numerical experiments are reported to support our theoretical results and to show the competitive behaviors of the proposed adaptive algorithm

a comparison of a posteriori error estimators for mixed finite element discretizations by raviart thomas elements we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations in particular we present and analyze four different kinds of error estimators a residual based estimator a hierarchical one error estimators relying on the solution of local subproblems and on a superconvergence result respectively finally we examine the relationship between the presented error estimators and compare their local components

quasi optimal a priori interface error bounds and a posteriori estimates for the interior penalty method in this work we show quasi optimal interface error estimates for solutions obtained by the symmetric interior penalty discontinuous galerkin method it is proved that the numerical solution restricted to an interface converges with order lvert ln h rvert h k 1 under suitable regularity requirements where the logarithmic factor is only present in the lowest order case i e k 1 for this case we also derive and analyze two a posteriori error estimators which demonstrate that the jump terms of the discrete fluxes are not essential to obtain local efficiency and reliability we support our analysis by numerical results and demonstrate that the interface approximation can be locally postprocessed to obtain discrete solutions of order h k 1 2 in the energy norm

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity motivation formulation and numerical examples the goal of this paper is to motivate introduce and demonstrate a novel approach to stabilizing discontinuous galerkin dg methods in nonlinear elasticity problems the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities with the goal of better approximating the solution consequently it is called an adaptive stabilization strategy the need for such a strategy is motivated through two and three dimensional examples in nonlinear elasticity the proposed scheme is simple to implement and compute and its performance is demonstrated with two and three dimensional numerical examples the accuracy of the proposed method is compared against a conforming method of the same order and a dg method with a traditional form of stabilization results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization however conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here a two dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one

on sparse hard sets for counting classes abstract in this paper we study one word decreasing self reducible sets which are introduced by lozano and toran 1991 these are the usual self reducible sets with the peculiarity that the self reducibility machine makes at most one query and this is lexicographically smaller than the input we show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one word decreasing self reducible using this fact we can prove that for any class k chosen from pp np c p mod2p mod3 p it holds that 1 if there is a sparse pbtt hard set for k then k p and 2 if there is a sparse snbtt hard set for k then k np co np this generalizes the result of ogiwara and watanabe 1991 to the mentioned complexity classes

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

a posteriori error estimation and dynamic adaptivity for symmetric discontinuous galerkin approximations of reactive transport problems a symmetric discontinuous galerkin scheme is applied for solving reactive transport problems an explicit a posteriori error estimator in l 2 l 2 is derived for the semi discrete scheme applied to transport with general kinetic reactions explicit a posteriori error estimators in terms of linear functionals and negative norms are also obtained for transport problems with simple reactions an adaptive approach based on the a posteriori error estimators is proposed to modify the mesh dynamically numerical examples demonstrate the efficiency and effectivity of the estimators moreover it is shown in this study that the proposed adaptivity eliminates the need of slope limiters 2005 elsevier b v all rights reserved

a posteriori error estimators for stabilized p1 nonconforming approximation of the stokes problem in this paper we derive and analyze some a posteriori error estimators for the stabilized p1 nonconforming approximation of the stokes problem involving the strain tensor this will be done by decomposing the numerical error in a proper way into conforming and nonconforming contributions the error estimator for the nonconforming error is obtained in the standard way and the implicit error estimator for the conforming error is derived by applying the equilibrated residual method a crucial part of this work is construction of approximate normal stresses on interelement boundaries which will serve as equilibrated neumann data for local stokes problems it turns out that such normal stresses can be simply computed by local weak residuals of the discrete system plus jumps of the velocity solution and that a stronger equilibration condition is satisfied to ensure solvability of local stokes problems we also derive a simple explicit error estimator based on the nonsymmetric tensor recovery of the normal stress error numerical results are provided to illustrate the performance of our error estimators

a priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations we study the mixed finite element approximation of the second order elliptic problem with gradient nonlinearities in two and three space dimensions existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates in l m omega are obtained also reliable and efficient a posteriori error estimators measured in the l m omega norm are derived numerical examples are provided to illustrate the performance of the proposed estimator

a local projection operator for quadrilateral finite elements this note studies the approximation error of a local projection operator on polynomials of total degree k defined on quadrilaterals among other applications this projection operator permits to derive easily error estimates for quadrature formulas

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

a discontinuous galerkin method to solve the eeg forward problem using the subtraction approach in order to perform electroencephalography eeg source reconstruction i e to localize the sources underlying a measured eeg the electric potential distribution at the electrodes generated by a dipolar current source in the brain has to be simulated which is the so called eeg forward problem to solve it accurately it is necessary to apply numerical methods that are able to take the individual geometry and conductivity distribution of the subject s head into account in this context the finite element fe method fem has shown high numerical accuracy with the possibility to model complex geometries and conductive features e g white matter conductivity anisotropy in this article we introduce and analyze the application of a discontinuous galerkin dg method an fem that includes features of the finite volume framework to the eeg forward problem the dg fem approach fulfills the conservation property of electric charge also in the discrete case making it attractive for a variety of applicat

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

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

discontinuous galerkin methods for the chemotaxis and haptotaxis models in this work first we formulate and compare three different discontinuous interior penalty galerkin methods for the 2d keller segel chemotaxis model keller segel chemotaxis model is the important starting step in the modeling of the real biological system we show in the numerical tests that two of the proposed methods fail to give accurate oscillation free solutions next we consider the application of the successful method for the keller segel model to the simulation of the more realistic and closely related haptotaxis model of tumor invasion into healthy tissues

superconvergence property of an over penalized discontinuous galerkin finite element gradient recovery method a polynomial preserving recovery method is introduced for over penalized symmetric interior penalty discontinuous galerkin solutions to a quasi linear elliptic problem as a post processing method the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns as well as general meshes satisfying condition by means of the averaging technique we prove the polynomial preserving recovery method for averaged solutions is superconvergent satisfying similar estimates as those for conforming finite element methods we deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator consequently the a posteriori error estimator based on the recovered gradient is asymptotically exact extensive numerical results consistent with our analysis are presented

interior penalty discontinuous approximations of convection diffusion problems with parabolic layers a nonsymmetric discontinuous galerkin finite element method with interior penalties is considered for two dimensional convection diffusion problems with regular and parabolic layers on an anisotropic shishkin type mesh with bilinear elements we prove error estimates uniformly in the perturbation parameter in an integral norm associated with this method on different types of interelement edges we derive the values of discontinuity penalization parameters numerical experiments complement the theoretical results

a structure preserving discontinuous galerkin scheme for the fisher kpp equation an implicit euler discontinuous galerkin scheme for the fisher kolmogorov petrovsky piscounov fisher kpp equation for population densities with no flux boundary conditions is suggested and analyzed using an exponential variable transformation the numerical scheme automatically preserves the positivity of the discrete solution a discrete entropy inequality is derived and the exponential time decay of the discrete density to the stable steady state in the l 1 norm is proved if the initial entropy is smaller than the measure of the domain the discrete solution is proved to converge in the l 2 norm to the unique strong solution to the time discrete fisher kpp equation as the mesh size tends to zero numerical experiments in one space dimension illustrate the theoretical results

a residual based error estimator for mortar finite element discretizations a residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed in particular we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using p1 approximations in each subdomain additionally a residual based error estimator for crouzeix raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation it is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the lagrange multiplier is taken into account

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

a priori error estimates of a combined mixed finite element and local discontinuous galerkin method for an incompressible miscible displacement problem abstract a numerical approximation for a kind of incompressible miscible displacement problems in high dimension in porous media is studied mixed finite element method is applied to the flow equation and the transport one is solved by the local discontinuous galerkin method ldg based on interpolation projection properties and the induction hypothesis a priori hp error estimates are obtained numerical results are presented which verify the theoretical results

large perturbations of elliptic operators and the solvability of the lp dirichlet problem in this article we investigate the solvability of the lp dirichlet problem for an elliptic operator in divergence form with bounded measurable coefficients to be specific assume that we have two such operators in the unit ball of rn suppose that we know that the lp dirichlet problem is solvable for the first operator we derive simple easy to check criteria on the difference between the coefficients of the two operators which imply that the second operator has its lp dirichlet problem solvable as well we emphasize that the value of p is the same for both operators and the criteria on the difference between the two coefficients allow for large perturbations the definition of the notion of large perturbation is given quite precisely

a priori error estimates for interior penalty discontinuous galerkin method applied to nonlinear sobolev equations abstract a discontinuous galerkin method with interior penalties is presented for nonlinear sobolev equations a semi discrete and a family of fully discrete time approximate scheme are formulated these schemes can be symmetric or nonsymmetric hp version error estimates are analyzed for these schemes just because of a damping term b u u t included in sobolev equation which is the distinct character different from parabolic equation special test functions are chosen to deal with this term finally a priori l h 1 error estimate is derived for the semi discrete time scheme and similarly l h 1 and l 2 h 1 for the fully discrete time schemes these results also indicate that spatial rates in h 1 and time truncation errors in l 2 are optimal

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

computational survey on a posteriori error estimators for nonconforming finite element methods for the poisson problem this paper compares different a posteriori error estimators for nonconforming first order crouzeix raviart finite element methods for simple second order partial differential equations all suggested error estimators yield a guaranteed upper bound of the discrete energy error up to oscillation terms with explicit constants novel equilibration techniques and an improved interpolation operator for the design of conforming approximations of the discrete nonconforming finite element solution perform very well in an error estimator competition with six benchmark examples

a multiscale discontinuous galerkin method for convection diffusion reaction problems we provide a general framework of multiscale discontinuous galerkin methods developed in buffa et al 2006 hughes et al 2006 for general second order partial differential equations we establish stability of the method and prove the error estimates

mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems we construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order pdes in the state variable e g pressure and its flux e g darcy velocity an important point is that no staggered grids or covolumes are used to stabilize the system only a single primary grid system is adopted and the degrees of freedom are imposed on the interfaces the approximate flux is sought in the lowest order raviart thomas space and the pressure field in the rotated q1 nonconforming space furthermore we demonstrate that the present finite volume method can be interpreted as a rotated q1 nonconforming finite element method for the pressure with a simple local recovery of flux numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method

an equilibrated a posteriori error estimator for the interior penalty discontinuous galerkin method interior penalty discontinuous galerkin ipdg methods for second order elliptic boundary value problems have been derived from a mixed variational formulation of the problem numerical flux functions across interelement boundaries play an important role in that theory residual type a posteriori error estimators for ipdg methods have been derived and analyzed by many authors including the convergence analysis of the resulting adaptive schemes typically the effectivity indices deteriorate with increasing polynomial order of the ipdg methods the situation is more favorable for a posteriori error estimators derived by means of the so called hypercircle method equilibrated fluxes are obtained by using a mixed method and an extension operator for brezzi douglas marini elements and this can be done in the same way for all the discontinuous galerkin methods that fit into a known unified framework the hypercircle method immediately provides the reliability of the estimator whereas its efficiency can be

new locally conservative finite element methods on a rectangular mesh a new family of locally conservative finite element methods for a rectangular mesh is introduced to solve second order elliptic equations our approach is composed of generating pde adapted local basis and solving a global matrix system arising from a flux continuity equation quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations next this approach is exploited to approximate stokes equations numerical results are presented for various examples including the lid driven cavity problem

the dirichlet problem for higher order equations in composition form the present paper commences the study of higher order differential equations in composition form specifically we consider the equation lu divb adiva u 0 where a and b are elliptic matrices with complex valued bounded measurable coefficients and a is an accretive function elliptic operators of this type naturally arise for instance via a pull back of the bilaplacian 2 from a lipschitz domain to the upper half space more generally this form is preserved under a lipschitz change of variables contrary to the case of divergence form fourth order differential equations we establish well posedness of the dirichlet problem for the equation lu 0 with boundary data in l2 and with optimal estimates in terms of nontangential maximal functions and square functions

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

numerical experiments for the arnold winther mixed finite elements for the stokes problem the stress velocity formulation of the stationary stokes problem allows an arnold winther mixed finite element formulation with some superconvergent reconstruction of the velocity this local postprocessing gives rise to two reliable a posteriori error estimators which recover optimal convergence order for the stress error estimates the theoretical results are investigated in numerical benchmark examples

improved zz a posteriori error estimators for diffusion problems conforming linear elements abstract in cai and zhang 2009 we introduced and analyzed an improved zienkiewicz zhu zz estimator for the conforming linear finite element approximation to elliptic interface problems the estimator is based on the piecewise constant flux recovery in the h div conforming finite element space this paper extends the results of cai and zhang 2009 to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear h div space

the non symmetric nitsche method for the parameter free imposition of weak boundary and coupling conditions in immersed finite elements abstract we explore the use of the non symmetric nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh in contrast to symmetric nitsche methods it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters we first review the available mathematical background recollecting relevant aspects of the method from a numerical analysis viewpoint we then compare accuracy and convergence of symmetric and non symmetric nitsche methods for a laplace problem a kirchhoff plate and in 3d elasticity our numerical experiments confirm that the non symmetric method leads to reduced accuracy in the l 2 error but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux bending moments or stress based on our numerical evidence the non symmetric nitsche method is a viable alternative for problems with diffusion type operators in particular when the accuracy of derivative quantities is of primary interest

stencil reduction algorithms for the local discontinuous galerkin method the problem of reducing the stencil of the local discontinuous galerkin method applied to second order differential operator is discussed heuristic algorithms to minimize the total number of non zero blocks of the reduced stiffness matrix are presented and tested on a wide variety of unstructured and structured grids in 2d and 3d copyright 2009 john wiley sons ltd

on l2 error estimate for nonsymmetric interior penalty galerkin approximation to linear elliptic problems with nonhomogeneous dirichlet data in this paper we present improved a priori error estimates for a nonsymmetric interior penalty galerkin method nipg with super penalty for the problem email protected f in w and u g on email protected using piecewise polynomials of degree less than or equal to r our new l 2 error estimate is of order h r r 1 2 when email protected h r 1 2 email protected and is optimal i e of order h r r 1 when email protected h r 1 email protected where h denotes the mesh size numerical experiments are presented to illustrate the theoretical results

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

hp adaptation driven by polynomial degree robust a posteriori error estimates for elliptic problems we devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions the adaptation is based on equilibrated flux estimates these estimates are presented here for inhomogeneous dirichlet and neumann boundary conditions for spatially varying polynomial degree and for mixed rectangular triangular grids possibly containing hanging nodes they deliver a global error upper bound with constant one and up to data oscillation error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound we numerically asses the estimates and several hp adaptive strategies using the interior penalty discontinuous galerkin method asymptotic exactness is observed for all the symmetric nonsymmetric odd degrees and incomplete variants on non nested unstructured triangular grids for a smooth solution and uniform refinement exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement

recovery based error estimators for interface problems mixed and nonconforming finite elements in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 we introduced and analyzed a recovery based a posteriori error estimator for conforming linear finite element approximation to interface problems it was shown theoretically that the estimator is robust with respect to the size of jumps provided that the distribution of coefficients is locally monotone numerical examples showed that this condition is unnecessary this paper extends the idea in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to mixed and nonconforming finite element methods for developing and analyzing robust estimators numerical results on test problems are also presented moreover an a priori error estimate is obtained when the underlying problem has low regularity

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

analysis of a nonsymmetric discontinuous galerkin method for elliptic problems stability and energy error estimates in this paper we analyze a nonsymmetric discontinuous galerkin method for elliptic problems proposed by oden babuska and baumann our main results are a complete inf sup stability analysis and as a consequence error estimates in a mesh dependent energy norm allowing variable meshsize and order of polynomials the analysis is carried out in two spatial dimensions on an unstructured triangulation

efficient computable error bounds for discontinuous galerkin approximations of elliptic problems we present guaranteed and computable both sided error bounds for the discontinuous galerkin dg approximations of elliptic problems these estimates are derived in the full dg norm on purely functional grounds by the analysis of the respective differential problem and thus are applicable to any qualified dg approximation based on the triangle inequality the underlying approach has the following steps for a given dg approximation 1 computing a conforming approximation in the energy space using the oswald interpolation operator and 2 application of the existing functional a posteriori error estimates to the conforming approximation various numerical examples with varying difficulty in computing the error bounds from simple problems of polynomial type analytic solution to problems with analytic solution having sharp peaks or problems with jumps in the coefficients of the partial differential equation operator are presented which confirm the efficiency and the robustness of the estimates