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

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

a low order nonconforming method for linear elasticity on general meshes in this work we construct a low order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions the method is obtained by hacking the hybrid high order method of di pietro and ern cmame 2015 283 1 21 that requires the use of polynomials of degree k ge 1 for stability specifically we show that coercivity can be recovered for k 0 by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces this term plays a key role in the fulfillment of a discrete korn inequality on broken polynomial spaces for which a novel proof valid for general polyhedral meshes is provided locking free error estimates are derived for both the energy and the l 2 norms of the error that are shown to convergence for smooth solutions as h and h 2 respectively here h denotes the meshsize a thorough numerical validation on a complete panel of two and three dimensional test cases is provided

quasi optimal convergence rate of an adaptive discontinuous galerkin method we analyze an adaptive discontinuous finite element method adfem for symmetric second order linear elliptic operators the method is formulated on nonconforming meshes made of simplices or quadrilaterals with any polynomial degree and in any dimension geq2 we prove that the adfem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops we design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent the geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the adfem we show that the adfem and the afem on nonconforming meshes yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

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

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

hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods a weak continuity condition at the interfaces is enforced by means of lagrange multipliers the two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical two level splitting the first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the lagrange multiplier it is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted l2 norm of the jumps across the interfaces

a staggered dg method of minimal dimension for the stokes equations on general meshes abstract in this paper a locally conservative lowest order staggered discontinuous galerkin method is developed for the stokes equations the proposed method allows rough grids and is based on the partition of the domain into arbitrary shapes of quadrilaterals or polygons which makes the method highly desirable for practical applications a priori error analysis covering low regularity is demonstrated a new postprocessing scheme for the velocity earning faster convergence is constructed next adaptive mesh refinement is highly appreciated on quadrilateral and polygonal meshes since hanging nodes are allowed therefore we propose two guaranteed type error estimators in l 2 error of stress and energy error of the postprocessed velocity respectively numerical experiments confirm our theoretical findings and illustrate the flexibility of the proposed method and accuracy of the guaranteed upper bounds

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

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

algebraic multilevel iteration method for lowest order raviart thomas space and applications an optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems that have their weak formulation in the space h div is presented the algorithm is developed for the discrete problem obtained by using the lowest order raviart thomas space the method is theoretically analyzed and supporting numerical examples are presented furthermore as a particular application the algorithm is used for the solution of the discrete minimization problem which arises in the functional type a posteriori error estimates for the discontinuous galerkin approximation of elliptic problems copyright 2011 john wiley sons ltd

a multilevel preconditioner for the interior penalty discontinuous galerkin method in this article we present a multilevel preconditioner for interior penalty discontinuous galerkin discretizations of second order elliptic boundary value problems that gives rise to uniformly bounded condition numbers without any additional regularity assumptions on the solution the underlying triangulations are assumed only to be shape regular but may have hanging nodes subject to certain mild grading conditions a key role is played by certain decompositions of the discontinuous trial space into a conforming subspace and a nonconforming subspace that is controlled by the jumps across the edges

an improved a priori error analysis of nitsche s method for robin boundary conditions in a previous paper juntunen and stenberg in math comput 78 1353 1374 2009 we have extended nitsche s method in abhandlungen aus dem mathematischen seminar der universitat hamburg 1970 71 for the poisson equation with general robin boundary conditions the analysis required that the solution is in h s with s 3 2 here we give an improved error analysis using a technique proposed by gudi math comput 79 2169 2189 2010

discontinuous finite element methods for a bi wave equation modeling d wave superconductors this paper concerns discontinuous nite element approximations of a fourth order bi wave equation arising as a simplied ginzburg landau type model for d wave superconductors in the absence of an applied magnetic eld in the rst half of the paper we construct a variant of the morley nite element method which was originally developed for approximating the fourth order biharmonic equation for the bi wave equation it is proved that unlike the biharmonic equation it is necessary to impose a mesh constraint and to include certain penalty terms in the method to guarantee convergence nearly optimal order o by a factor j lnhj error estimates in the energy norm and in the h 1 norm are established for the proposed morley type nonconforming method in the second half of the paper we develop a symmetric interior penalty discontinuous galerkin method for the bi wave equation using general meshes and prove optimal order error estimates in the energy norm finally numerical experiments are provided to gauge the eciency of the proposed methods and to validate the theoretical error bounds

a priori and computable a posteriori error estimates for an hdg method for the coercive maxwell equations abstract in this paper we present and analyze a hybridizable discontinuous galerkin hdg method for a mixed curl curl formulation of the steady state coercive maxwell equations with a discrete sobolev embedding type estimates for the discontinuous polynomials we provide a priori error estimates for the electric field and the lagrange multiplier in the energy norm with the smooth or minimal regularity assumption on the exact solution we have optimal convergence rate for the electric field and the lagrange multiplier in the energy norm the a priori error estimate for the electric field in the l 2 norm is also obtained by the duality argument and the approximation is also optimal for the electric field in the l 2 norm moreover by employing suitable helmholtz decompositions of the error together with the upper bound estimate for the lagrange multiplier we provide a computable residual based a posteriori error estimator which is derived based on the error measured in terms of a mesh dependent energy norm the efficiency of the a posteriori error estimator is also established three dimensional numerical results testing the performance of the a priori and a posteriori error estimates for the maxwell equations are given

a combined finite element method for elliptic problems posted in domains with rough boundaries abstract a combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes while good quality meshes often over refine unnecessarily the interior of the domain the basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size h h for other portions of the domain to reduce some unnecessary computational effort the transmission conditions across the fine coarse mesh interface are treated by the penalty technique the key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form which avoids the affection of the ratio h h in the error estimate we prove a quasi optimal convergence in terms of elements since there is no whole h 2 regularity in the domain with rough boundaries numerical results are provided for elliptic equations in domains with non oscillating or oscillating boundaries to illustrate the theoretical results

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

convergence of adaptive discontinuous galerkin approximations of second order elliptic problems a residual type a posteriori error estimator is introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem with dirichlet neumann type boundary conditions an adaptive algorithm using this estimator together with specific marking and refinement strategies is constructed and shown to achieve any specified error level in the energy norm in a finite number of cycles the convergence rate is in effect linear with a guaranteed error reduction at every cycle results of numerical experiments are presented

discontinuous galerkin finite element convergence for incompressible miscible displacement problems of low regularity in this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous galerkin method under minimal regularity assumptions the main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem in order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries the careful construction of a reflexive subspace of the space of bounded variation which compactly embeds into l 2 omega and of a lifting operator which is compatible with the nonlinear diffusion coefficient are required an equivalent skew symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method numerical experiments underline the robustness of the proposed algorithm

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

a priori and a posteriori error analysis of a staggered discontinuous galerkin method for convection dominant diffusion equations abstract an upwind staggered discontinuous galerkin upwind sdg method for convection dominant diffusion problems is developed optimal a priori error estimates can be achieved for both the scalar and vector functions approximated by the method to efficiently capture the layer problems we propose a robust a posteriori error estimator for upwind sdg method measured in a natural norm and a semi norm associated with the convective derivative the semi norm can be neglected when the mesh peclet number is sufficiently small the key is to bound the conforming contribution by the dual norm of the residual and the robustness arises from the incorporation of the semi norm associated with the convective derivative finally various numerical examples are tested to illustrate the performance of the robust a posteriori error estimator the results indicate that optimal convergence rates can be achieved and the singularity can be well captured by the proposed error estimator

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

hp fem for a stabilized three field formulation of the biharmonic problem abstract in this paper a three field formulation for the biharmonic equation is considered that consists of the potential u the flux and a lagrange multiplier the use of stabilization techniques circumvents the discrete inf sup condition and thus enables the application of arbitrary conforming finite element spaces for the three fields a priori error estimates as well as reliable and efficient a posteriori error estimates are derived the proof of the latter is based on an implicit never to be computed h 2 reconstruction of the discrete potential several numerical experiments underline the theoretical results and show the existence of discretization spaces for which the a posteriori error estimate is p robust with an efficiency index tending towards the optimal value of one the numerical experiments indicate that the proposed method is applicable even if the theoretically required minimal regularity of the lagrange multiplier for the a priori error estimate is not fulfilled in particular h and h p adaptivity recover optimal algebraic and exponential converge rates respectively

a framework for obtaining guaranteed error bounds for finite element approximations we give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming non conforming mixed and discontinuous finite element schemes whilst the details of the actual estimator are rather different for each particular scheme there is nonetheless a common underlying structure at work in all cases we aim to illustrate this structure by treating conforming non conforming and discontinuous finite element schemes in a single framework in taking a rather general viewpoint some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but in return we hope to allow the reader to see the wood despite the trees

remarks on convergence of trigonometric series with special varying coefficients abstract we give an essential generalization of telyakovskii s results on the convergence of trigonometric series with rarely changing coefficients in the norm of l 1

error indicators for incompressible darcy flow problems using enhanced velocity mixed finite element method abstract local mesh adaptivity serves as a practical tool in numerical simulations to accurately capture features of interest while reducing computational time and memory requirements in this work we suggest a refinement strategy based on pressure and flux error estimates for numerical simulation of an incompressible single phase flow and transport process in the subsurface porous media we derive a posteriori error estimates for an enhanced velocity mixed finite element method evmfem as a spatial domain decomposition approach we note that the flux errors play an important role in coupled flow and transport systems later demonstrated using numerical experiments a comparison between explicit residual based error estimators and an implicit error estimator based upon the post processing proposed by arbogast and chen 1995 shows that the latter performs better a residual based error estimator for pressure was found to be both computationally efficient while sufficiently indicating the large error subdomains numerical studies are also presented that confirm our theoretical derivations while demonstrating the advantages of post processing in detecting velocity errors

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

adaptive numerical solution of a discontinuous galerkin method for a helmholtz problem in low frequency regime we develop an a posteriori error analysis for helmholtz problem using the local discontinuous galerkin ldg for short approach for the sake of completeness we give a description of the main a priori results of this method indeed under some assumptions on regularity of the solution of an adjoint problem we prove that a the corresponding indefinite discrete scheme is well posed b the approach is convergent with the expected convergence rates as long as the meshsize h is small enough we give precise information on how small h has to be in terms of the size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the laplacian after that we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number we finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work we derive an a posteriori error estimator that is reliable and locally quasi efficient the efficiency and reliability constants have been characterized with respect to the physical parameter we include numerical examples that validate our theoretical results recognizing singularities and inner layers for moderate values of

an a posteriori error estimate for the variable degree raviart thomas method we propose a new a posteriori error analysis of the variable degree hybridized version of the raviart thomas method for second order elliptic problems on conforming meshes made of simplexes we establish both the reliability and eciency of the estimator for the l2 norm of the error of the ux we also nd the explicit dependence of the estimator on the order of the local spaces k 0 the only constants that are not explicitly computed are those depending on the shape regularity of the simplexes in particular the constant of the local eciency in equality is proven to behave like k 2 3 2 however we present numer ical experiments suggesting that such constant is actually independent of k

continuous interior penalty finite element method for oseen s equations in this paper we present an extension of the continuous interior penalty method of douglas and dupont interior penalty procedures for elliptic and parabolic galerkin methods in computing methods in applied sciences lecture notes in phys 58 springer verlag berlin 1976 pp 207 216 to oseen s equations the method consists of a stabilized galerkin formulation using equal order interpolation for pressure and velocity to counter instabilities due to the pressure velocity coupling or due to a high local reynolds number we add a stabilization term giving l2 control of the jump of the gradient over element faces edges in two dimensions to the standard galerkin formulation boundary conditions are imposed in a weak sense using a consistent penalty formulation due to nitsche we prove energy type a priori error estimates independent of the local reynolds number and give some numerical examples recovering the theoretical results

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

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

adaptive discontinuous galerkin methods on surfaces we present a dual weighted residual based a posteriori error estimate for a discontinuous galerkin approximation of a surface partial differential equation we restrict our analysis to a linear second order elliptic problem posed on hypersurfaces in mathbb r 3 r3 which are implicitly represented as level sets of smooth functions we show that the error in the energy norm may be split into a residual part and a higher order geometric part upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator we also present a novel geometric driven refinement strategy for pdes on surfaces which considerably improves the performance of the method on complex surfaces

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

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

hp adaptive discontinuous galerkin methods for non stationary convection diffusion problems an a posteriori error estimator for the error in the l2 h1 l1 l2 type norm for an interior penalty discontinuous galerkin dg spatial discretisation and backward euler temporal discretisation of linear non stationary convection diusion initial boundary value problems is derived allowing for anisotropic elements the proposed error estimator is used to drive an hp space time adaptive algorithm wherein directional mesh renement is employed to give rise to highly anisotropic elements able to accurately capture layers the performance of the hp space time adaptive algorithm is assessed via a number of standard test problems characterised by sharp and or moving layers

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

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

a delta regularization finite element method for a double curl problem with divergence free constraint to deal with the divergence free constraint in a double curl problem rm curl mu 1 rm curl u f and rm div varepsilon u 0 in omega where mu and varepsilon represent the physical properties of the materials occupying omega we develop a delta regularization method rm curl mu 1 rm curl u delta delta varepsilon u delta f to completely ignore the divergence free constraint rm div varepsilon u 0 we show that u delta converges to u in h rm curl omega norm as delta rightarrow 0 the edge finite element method is then analyzed for solving u delta with the finite element solution u delta h a quasioptimal error bound in the h rm curl omega norm is obtained between u and u delta h including a uniform with respect to delta stability of u delta h in the h rm curl omega norm all the theoretical analysis is done in a general setting where mu and varepsilon may be discontinuous an

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

a posteriori error estimator based on gradient recovery by averaging for discontinuous galerkin methods we consider some anisotropic and piecewise constant diffusion problems in domains of r 2 approximated by a discontinuous galerkin method with polynomials of any fixed degree we propose an a posteriori error estimator based on gradient recovery by averaging it is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability the lower bound is also established moreover these additional terms are negligible when the recovered gradient is superconvergent the reliability and efficiency of the proposed estimator is confirmed by some numerical tests

discontinuous galerkin finite element approximation of nonlinear second order elliptic and hyperbolic systems we develop the convergence analysis of discontinuous galerkin finite element approximations to symmetric second order quasi linear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in mathbb r d subject to mixed dirichlet neumann boundary conditions optimal order asymptotic bounds are derived on the discretization error in each case without requiring the global lipschitz continuity or uniform monotonicity of the stress tensor instead only local smoothness and a garding inequality are used in the analysis

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

an adaptive stabilized conforming finite element method via residual minimization on dual discontinuous galerkin norms abstract we design and analyze a new adaptive stabilized finite element method we construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf sup stability we formulate this residual minimization as a stable saddle point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement numerical results on an advection reaction model problem show competitive error reduction rates when compared to discontinuous galerkin methods on uniformly refined meshes and smooth solutions moreover the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

energy norm a posteriori error estimation for discontinuous galerkin methods in this paper we present a residual based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous galerkin approximations of elliptic problems the theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

robust a posteriori error estimates for conforming and nonconforming finite element methods for convection diffusion problems a posteriori error estimation is carried out within a unified framework for various conforming and nonconforming finite element methods for convection diffusion problems our main contribution is finding an appropriate norm to measure the error which incorporates a discrete energy norm a discrete dual semi norm of the convective derivative and jumps of the approximate solution over element faces edges in two dimensions the error estimator is shown to be robust with respect to the peclet number in the sense of the modified norm based on a general error decomposition we show that the key ingredient of error estimation is the estimation on the consistency error related to the particular numerical scheme and the remaining terms can be bounded in a unified way the numerical results are presented to illustrate the robustness and practical performance of the estimator in an adaptive refinement strategy

analysis of backward euler extended finite element discretization of parabolic problems with moving interfaces abstract numerical experiments performed by several authors repeatedly confirmed that the extended finite element method combined with classical time stepping schemes such as backward euler or crank nicolson provides optimal rates of convergence when applied to problems with moving interfaces in spite of the rapid spread of this discretization approach a complete space and time error analysis is not available yet the objective of this work is to contribute to closing the gap between observations and rigorous analysis more precisely we preform a thorough space and time error analysis of different variants of the extended finite element method combined with backward euler time advancing scheme when applied to problems with moving interfaces our results prove that the discretization error measured in the natural norms decreases with the expected rate when space and time discretization steps are refined we also show that the space and time approximation properties are fully decoupled as a consequence high order approximation schemes could be developed and analyzed within the theoretical framework proposed here numerical experiments are finally addressed for the verification of the algorithms besides the interest of providing rigorous error bounds we believe that a general theoretical framework is extremely helpful as a guide for further developing and refining extended finite element methods for free interface problems

a posteriori error estimation for discontinuous galerkin finite element approximation it is shown that the interelement discontinuities in a discontinuous galerkin finite element approximation are subordinate to the error measured in the broken h 1 seminorm one consequence is that the dg norm of the error is equivalent to the broken energy seminorm computable a posteriori error bounds are obtained for the error measured in both the dg norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms the estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error

energy norm error estimates for averaged discontinuous galerkin methods a mathematical analysis is presented for a class of interior penalty ip discontinuous galerkin approximations of elliptic boundary value problems in the framework of the present theory one can derive some overpenalized ip bilinear forms avoiding the notion of fluxes and artificial penalty terms the main idea is to start from bilinear forms for the local average of discontinuous approximations which are rewritten using the theory of distributions it is pointed out that a class of overpenalized ip bilinear forms can be obtained using a perturbation of these also error estimations can be derived between the local averages of the discontinuous approximations and the analytic solution in the h1 seminorm using the local averages the analysis is performed in a conforming framework without any assumption on extra smoothness for the solution of the original boundary value problem

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

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

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

weak galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of weak galerkin wg finite element methods based on the second order elliptic interface problems this estimate can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator has been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

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

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