a priori and a posteriori analysis of finite volume discretizations of darcy s equations this paper is devoted to the numerical analysis of some finite volume discretizations of darcy s equations we propose two finite volume schemes on unstructured meshes and prove their equivalence with either conforming or nonconforming finite element discrete problems this leads to optimal a priori error estimates in view of mesh adaptivity we exhibit residual type error indicators and prove estimates which allow to compare them with the error in a very accurate way

the kharitonov theorem with degree drop the purpose of this paper is to present a proof of the kharitonov theorem based on bezoutians an interesting consequence of this proof is that it shows the validity of kharitonov s result in the presence of a degree drop

explicit and averaging a posteriori error estimates for adaptive finite volume methods local mesh refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady state convection diffusion reaction equations the paper establishes residual type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global h1 and l2 norms reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh refining algorithms over uniform mesh refining a discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper

a posteriori error estimations of some cell centered finite volume methods for diffusion convection reaction problems this paper presents an a posteriori residual error estimator for diffusion convection reaction problems approximated by some cell centered finite volume methods on isotropic or aniso tropic meshes in rd d 2 or 3 for that purpose we built a reconstructed approximation which is an appropriate interpolant of the finite volume solution the error is then the difference between the exact solution and this interpolant the residual error estimator is based on the jump of the normal derivative of the interpolant we then prove the equivalence between the energy norm of the error and the residual error estimator some numerical tests confirm our theoretical results

a posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions in this paper we shall derive a posteriori error estimates in the l 1 norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions this result is mainly based on some fundamental a priori error estimates published in a recent paper by c chainais hillairet the theoretical results are confirmed by numerical experiments

on supremum of a set in a multi dimensional space on donne une definition exacte du supremum d un ensemble dans l espace euclidien a n dimensions etendu et on montre qu elle est presque equivalente a celle proposee par kawasaki 1982 on etudie certaines proprietes de ce supremum ce qui pourra etre utile pour developper une theorie de dualite conjuguee en optimisation vectorielle

an edge theorem for polytopes of polynomials which can drop in degree abstract a new version of the edge theorem for robust stability of polynomials that can drop in degree is given the proof is of elementary nature and a counterexample is given to show that the original edge theorem of bartlett hollot and lin does not hold for the general case of arbitrary degree dropping

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

an explicit a posteriori error estimations for the cell functional minimization scheme of elliptic problems abstract this paper presents an explicit form of a posteriori error estimator for the cell functional minimization scheme of elliptic problems it is shown that this estimator is reliable with respect to an energy type error numerical experiments are conducted to demonstrate the performance of the error estimator in the adaptive mesh refinement algorithm

robust a posteriori error control and adaptivity for multiscale multinumerics and mortar coupling we consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains termed multinumerics coupled using the mortar technique the grids need not match along the interfaces we are also interested in the multiscale setting where the subdomains are partitioned by a mesh of size h whereas the interfaces are partitioned by a mesh of much coarser size h and where lower order polynomials are used in the subdomains and higher order polynomials are used on the mortar interface mesh we derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm our estimates are also locally efficient and one of them is robust with respect to the ratio h h under an assumption of sufficient regularity of the weak solution the present approach allows bounding separately and comparing mutually the subdomain and interface errors a subdomain interface adaptive refinement s

an a posteriori error estimate for a first kind integral equation in this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind the estimate is local and sharp for quasi uniform meshes and so improves earlier work of ours the mesh dependence of the constants is analyzed and shown to be weaker than expected from our previous work besides the galerkin boundary element method the collocation method and the qualocation method are considered a numerical example is given involving an adaptive feedback algorithm

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

coupling of the finite volume element method and the boundary element method an a priori convergence result the coupling of the finite volume element method and the boundary element method is an interesting approach to simulate a coupled system of a diffusion convection reaction process in an interior domain and a diffusion process in the corresponding unbounded exterior domain this discrete system maintains naturally local conservation and a possible weighted upwind scheme guarantees the stability of the discrete system also for convection dominated problems we show existence and uniqueness of the continuous system with appropriate transmission conditions on the coupling boundary provide a convergence and an a priori analysis in an energy semi norm and provide an existence and an uniqueness result for the discrete system all results are also valid for the upwind version numerical experiments show that our coupling is an efficient method for the numerical treatment of transmission problems which can also be convection dominated

a posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection diffusion equations this paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation c t nabla cdot u f c varepsilon delta c 0 the estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the l 1 norm in the situation where the diffusion parameter varepsilon is smaller or comparable to the mesh size numerical experiments underline the theoretical results

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

estimations a posteriori d un schema de volumes finis pour un probleme non lineaire resumedans cet article nous developpons des estimations a posteriori de l erreur commise en approchant un probleme elliptique non lineaire par une methode de volumes finis mvf de type vertex centered les estimateurs obtenus mesurent le residu de l equation forte et l irregularite de la solution discrete qui se traduit par des sauts a travers les inter elelments une condition sur la fonction flux numerique sera imposee pour traiter la non conformite du probleme au cours de l evaluation de l estimation summary we derive a posteriori error estimates for vertex centered finite volume discretizations of a class of non linear elliptic pdes the error estimates are of residual type and incorporate residuals of the pde in its strong form on the control volumes and jumps of the pde s prinicipal part across the element boundaries the non conformity of the discretization is taken into account by a condition on the numerical flux function

a posteriori error estimate and adaptive mesh refinement for the cell centered finite volume method for elliptic boundary value problems we extend a result of nicaise siam j numer anal 43 2005 pp 1481 1503 for the a posteriori error estimation of the cell centered finite volume method for the numerical solution of elliptic problems having computed the piecewise constant finite volume solution u h we compute a morley type interpolant mathcal i u h for the exact solution u the energy error norm nabla mathcal t u mathcal i u h l 2 can be controlled efficiently and reliably by a residual based a posteriori error estimator eta the local contributions of eta are used to steer an adaptive mesh refining algorithm a model example serves the laplace equation in two dimensions with mixed dirichlet neumann boundary conditions

a posteriori error estimations of some cell centered finite volume methods this paper presents the natural framework to residual based a posteriori error estimation of some cell centered finite volume methods for the laplace equation in r d d 2 or 3 for that purpose we associate with the finite volume solution a reconstructed approximation which is a kind of morley interpolant the error is then the difference between the exact solution and this morley interpolant the residual error estimator is based on the jump of normal and tangential derivatives of the morley interpolant we then prove the equivalence between the discrete h1 seminorm of the error and the residual error estimator numerical tests confirm our theoretical results

a nonconforming a posteriori estimator for the coupling of cell centered finite volume and boundary element methods the coupling of the cell centered finite volume method and the boundary element method is an interesting approach to solve elliptic problems on an unbounded domain where local flux conservation is important based on the piecewise constant interior finite volume solution we define a morley type interpolant built on a nonconforming finite element together with the cauchy data of the exterior boundary element solution this allows us to define a residual based a posteriori error estimator with respect to an energy norm we prove reliability and efficiency of this estimator and use its local contributions to steer an adaptive mesh refining algorithm in two examples we illustrate the effectiveness of the new adaptive coupling method and compare it with the coupling approach with a conforming morley interpolant

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

from mixed finite elements to finite volumes for elliptic pdes in two and three dimensions the link between mixed finite element mfe and finite volume fv methods applied to elliptic partial differential equations has been investigated by many authors recently a fv formulation of the mixed approach has been developed this approach was restricted to 2d problems with a scalar for the parameter used to calculate fluxes from the state variable gradient this new approach is extended to 2d problems with a full parameter tensor and to 3d problems the objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy this is achieved by defining one new variable per element for the 2d case with full parameter tensor this new formulation exists for any kind of triangulation it allows the reduction of the number of unknowns to the number of elements instead of the number of edges no additional assumptions are required concerning the averaging of the parameter in hetero geneous domains for 3d problems we demonstrate that the new formulation cannot exist for a general 3d tetrahedral discretization unlike in the 2d problem however it does exist when the tetrahedrons are regular or deduced from rectangular parallelepipeds and allows reduction of the number of unknowns numerical experiments and comparisons between both formulations in 2d show the efficiency of the new formulation copyright 2003 john wiley sons ltd

stability of families of polynomials geometric considerations in coefficient space the stability robustness of both continuous and discrete time dynamical systems is tantamount to the stability of a whole family of characteristic polynomials the authors treat such families as sets in coefficient space and examine some of their geometric properties specifically they examine cases where the stability of the extreme points of a set implies the stability of the complete family they are also concerned with the construction and convexity of the stability region in coefficient space

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

on the coupled bem and fem for a nonlinear exterior dirichlet problem in r2 in this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior dirichlet problem in the plane specifically the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region and the laplace equation in the corresponding unbounded exterior region in addition to appropriate boundary and transmission conditions the main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation we provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence uniqueness and approximation results are established then we consider the case where the corresponding operator is strongly monotone and lipschitz continuous and derive asymptotic error estimates for a boundary finite element solution we prove the unique solvability of the discrete operator equations and based on a strang type abstract error estimate we show the strong convergence of the approximated solutions moreover under additional regularity assumptions on the solution of the continous operator equation the asymptotic rate of convergenceo h is obtained

on kharitonov type theorems for real polynomials when is degree drop admissible a root locus approach is adopted to show that extension of kharitonov type theorem for real interval polynomial with degree drop is not only applicable to the open left half plane but also other regions such as the left shifted open left half plane and the left sector

a posteriori error estimator for finite volume methods in this paper first we discuss a technique to compare finite volume method and some well known finite element methods namely the dual mixed methods and nonconforming primal methods for elliptic equations these both equivalences are exploited to give us a posteriori error estimator for finite volume methods this estimator is explicitly given easy to compute and asymptotically exact without any regularity of the solution in unstructured grids

goal oriented a posteriori error estimation for finite volume methods a general framework for goal oriented a posteriori error estimation for finite volume methods is developed the framework does not rely on recasting finite volume methods as special cases of finite element methods but instead directly determines error estimators from the discretized finite volume equations thus the framework can be applied to arbitrary finite volume methods it also provides the proper functional settings to address well posedness issues for the primal and adjoint problems numerical results are presented to illustrate the validity and effectiveness of the a posteriori error estimates and their applicability to adaptive mesh refinement

a posteriori error estimation for the discrete duality finite volume discretization of the laplace equation an efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the laplace equation on very general two dimensional meshes the main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

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

a posteriori error analysis of a cell centered finite volume method for semilinear elliptic problems in this paper we conduct a goal oriented a posteriori analysis for the error in a quantity of interest computed from a cell centered finite volume scheme for a semilinear elliptic problem the a posteriori error analysis is based on variational analysis residual errors and the adjoint problem to carry out the analysis we use an equivalence between the cell centered finite volume scheme and a mixed finite element method with special choice of quadrature

robust a posteriori error estimation for the nonconforming fortin soulie finite element approximation we obtain a computable a posteriori error bound on the broken energy norm of the error in the fortin soulie finite element approximation of a linear second order elliptic problem with variable permeability this bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms the estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error

the validity of johnson nedelec s bem fem coupling on polygonal interfaces in this short article we prove that the classical one equation or johnson nedelec coupling of finite and boundary elements can be applied with a lipschitz coupling interface because of the way it was originally approached from the analytical standpoint this bem fem scheme required smooth boundaries and hence produced a consistency error in the finite element part with a variational argument we prove that this requirement is not needed and that stability holds for all pairs of discrete space as it inherits the underlying ellipticity of the problem

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

a dual graph norm refinement indicator for finite volume approximations of the euler equations we generalise and apply a refinement indicator of the type originally designed by mackenzie suli and warnecke in 15 and 16 for linear friedrichs systems to the euler equations of inviscid compressible fluid flow the euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in 15 can be applied we discuss the details of the implementation of the refinement indicator into the dlr tau code which is based on a finite volume method of box type on an unstructured grid and present numerical results

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

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

conjugate duality in vector optimization in this paper we develop a conjugate duality theory in vector optimiza tion conjugate duality was fully developed in scalar optimization by rockafellar 3 and provides a unified framework for the duality theory the author and sawaragi extended it to the case of multiobjective optimization by introducing some new concepts such as conjugate maps and subgradients for vector valued set valued maps 4 6 their results are based on the efficiency pareto maximality kawasaki refined their results and obtained a reciprocal duality by introducing the concept of supremum set in an extended euclidean space 1 2 his supremum is based on the weak efficiency weak pareto maximality of the closure of a set in an extended sense the author s newly defined the supremum of a set in the extended multi dimensional euclidean space on the basis of weak ehiciency through several discussions on supremum the definition satisfies some desirable fundamental properties it can be extended to the case of general partially ordered linear topological spaces in a straightforward manner based on this definition of supremum some useful concepts such as conjugate maps and subgradients are introduced for vector valued set valued maps in this paper these concepts enable us to develop the conjugate duality in vector optimization although the results obtained are quite similar to the earlier works by the author and sawaragi 4 or kawasaki 2 our new approach makes the proofs much easier and more understandable and provides a better version of conjugate duality

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

robust a posteriori error estimation for nonconforming finite element approximation the equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media the estimator is shown to provide a computable upper bound on the error and up to a constant depending only on the geometry provides two sided bounds on the error the robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly

kharitonov s theorem extension to interval polynomials which can drop in degree a nyquist approach this paper studies the hurwitz stability of interval polynomials which can drop in degree from a nyquist point of view these families make it possible to deal with system families taking into account different dynamic behaviors in the modeling of a plant the behavior of the interval polynomials is studied and it is proven that kharitonov s theorem can be extended

averaging techniques for the effective numerical solution of symm s integral equation of the first kind averaging techniques for finite element error control occasionally called zz estimators for the gradient recovery enjoy a high popularity in engineering because of their striking simplicity and universality one does not even require a pde to apply the nonexpensive post processing routines recently averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method this paper establishes a class of averaging error estimators for boundary integral methods symm s integral equation of the first kind with a nonlocal single layer integral operator serves as a model equation studied both theoretically and numerically we introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order the higher order terms depend on the regularity of the exact solution several numerical experiments illustrate the theoretical results and show that the normally unknown error is sharply estimated by the proposed estimators i e error and estimators almost coincide

a posteriori error estimates and adaptive mesh refinement for the coupling of the finite volume method and the boundary element method we consider the coupling of the finite volume method and the boundary element method of erath siam j numer anal 50 2012 pp 574 594 in two and three dimensions this method can be used for example to approximate a solution of the transport of a concentration in a fluid where no boundary conditions are available we derive residual based a posteriori estimates also for an upwind version these upper and lower bounds measure the error in an energy semi norm between the exact solution and the numerical solution the upper bound is robust in the sense that it does not depend on the variation of the model data the lower bound however depends additionally on the local peclet number the local contributions of the a posteriori estimates are used to steer an adaptive mesh refining algorithm this strategy turns out to be very suitable for the numerical treatment of transmission problems which have singularities or boundary internal layers several numerical examples illustrate the effectivenes

a duality theorem in multiobjective nonlinear programming we give a duality theorem in multiobjective nonlinear programming problems by means of conjugate set valued functions which were introduced in kawasaki h 1981 conjugate relations and weak subdifferentials of relations math oper res 6 593 607 the duality theorem is reflexive in the sense analogous to ekeland i r temam 1976 convex analysis and variational problems north holland amsterdam

conjugate relations and weak subdifferentials of relations the duality theory for convex programs is based on a concept conjugate of functions and the set of solutions is characterized by subdifferential of a function the purpose of this paper is to extend the concepts conjugate and subdifferential to those of set valued functions relations and to investigate their properties

a posteriori error estimates for boundary element methods this paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples namely symm s integral equation an integral equation with a hypersingular operator and a boundary integral equation for a transmission problem based on these estimates an analog of eriksson and johnson s adaptive finite element method is proposed for the h version of the galerkin boundary element method for integral equations of the first kind the efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities the construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations and we refer to the pioneering work of babuska and miller 3 and eriksson and johnson 10 11 whereas the main features ofadaptivity for finite element methods now seem to be visible and the door is open to implementation 15 comparably little is known for boundary element methods for integral equations see e g 1 13 18 19 24 in this paper a new adaptive version of the galerkin discretization for the boundary element method is presented based on a posteriori error estimates a general framework for these a posteriori error estimates is derived in 2 and three examples are discussed in 3 5 involving the dirichlet problem the neumann problem for a closed and an open surface and a transmission problem for the laplacian leading to integral equations with strongly elliptic pseudodifferential operators even for smooth data the lack of regularity of the solution near corners of a polygonal domain q leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used in practical problems such information is missing e g when we have singular or nearly singular data and the main problem is how to balance a graded mesh refinement towards singularities and a global

continuous interior penalty hp finite element methods for advection and advection diffusion equations a continuous interior penalty hp finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced error estimates are obtained for first order and advection dominated transport operators the analysis relies on three technical results that are of independent interest an hp inverse trace inequality a local discontinuous to continuous hp interpolation result and hp error estimates for continuous l2 orthogonal projections

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

a new conservative numerical scheme for flow problems on unstructured grids and unbounded domains abstract we consider the prototype of a flow and transport problem in two dimensions in particular in many fluid dynamics problems the boundary conditions may be unknown or the domain may be unbounded furthermore only numerical schemes which conserve mass and are stable with respect to dominating convection are appropriate to solve these problems therefore we develop a new discrete scheme to address these issues more precisely we couple the cell centered finite volume method and the boundary element method and introduce suitable transmission conditions also semi robust a posteriori estimates of residual type are developed which allow us to use an adaptive mesh refinement algorithm to treat efficiently problems which have singularities or boundary internal layers several examples illustrate the effectiveness of our new adaptive coupling method

technical communique on kharitonov s theorem without invariant degree assumption the original statement of kharitonov s theorem requires that all polynomials in the family have the same degree it has been shown recently that such an assumption is unnecessary here we show that the validity of the stronger statement follows from a classical proof combined with additional elementary considerations

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

a robust nonconforming h 2 element finite element methods for some elliptic fourth order singular perturbation problems are discussed we show that if such problems are discretized by the nonconforming morley method in a regime close to second order elliptic equations then the error deteriorates in fact a counterexample is given to show that the morley method diverges for the reduced second order equation as an alternative to the morley element we propose to use a nonconforming h 2 element which is h 1 conforming we show that the new finite element method converges in the energy norm uniformly in the perturbation parameter

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

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients