on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

maximum norm estimates for resolvents of elliptic finite element operators let be a convex domain with smooth boundary in rd it has been shown recently that the semigroup generated by the discrete laplacian for quasi uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum norm uniformly in the mesh width this implies a resolvent estimate of standard form in the maximum norm outside some sector in the right halfplane and conversely here we show directly that such a resolvent estimate holds outside any sector around the positive real axis with arbitrarily small angle this is useful in the study of fully discrete approximations based on a stable rational functions with small

on pfaffian random point fields we study pfaffian random point fields by using the moore dyson quaternion determinants first we give sufficient conditions that ensure that a self dual quaternion kernel defines a valid random point field and then we prove a clt for pfaffian point fields the proofs are based on a new quaternion extension of the cauchy binet determinantal identity in addition we derive the fredholm determinantal formulas for the pfaffian point fields which use the quaternion determinant

high order local rate of convergence by mesh refinement in the finite element method we seek approximations of the solution u of the neumann problem for the equation lu f in q with special emphasis on high order accuracy at a given point xo e f2 here q is a bounded domain in rn n 2 with smooth boundary and l is a second order uniformly elliptic differential operator with smooth coefficients an approximate solution uh is determined by the standard galerkin method in a space of continuous piecewise polynomials of degree at most r 1 on a partition ah xo a of u here h is a global mesh size parameter and a is the degree of a certain systematic refinement of the mesh around the given point x0 where larger a s mean finer mesh and a 0 corresponds to the quasi uniform case with no refinement it is proved that for suitable sufficiently large a s the high order error estimate u uh x0 o h2 2 holds a corresponding estimate with the same order of convergence is obtained for the first order derivatives of u uh these estimates are sharp in the sense that the required degree of refinement in each case is essentially the same as is needed for the local approximation to this order near xo for the estimates to hold it is sufficient that the exact solution u have derivatives to the rth order which are bounded close to xo and square integrable in the rest of u the proof of this uses high order negative norm estimates of u uh the number of elements in the considered partitions is of the same order as in the corresponding quasi uniform ones applications of the results to other types of boundary value problems are indicated 0 introduction let i2 be a bounded domain in rn n 2 with smooth boundary aq and consider the neumann problem to find u such that n a au n au 0 1 lu ax aija ea3 au f inn au n au 0 2 an i a a nj 0 on3ai2 a i j1 aj xi here l is assumed to be uniformly elliptic with smooth coefficients n nj and n denote the exterior normal and conormal to aq respectively assume also that the bilinear form n3v w n a v w e a a e a a w avw dx received september 22 1983 revised july 23 1984 1980 mathematics subject classification primary 65n30

pointwise error estimates of linear finite element method for neumann boundary value problems in a smooth domain pointwise error analysis of the linear finite element approximation for delta u u f in omega partial n u tau on partial omega where omega is a bounded smooth domain in mathbb r n is presented we establish o h 2 log h and o h error bounds in the l infty and w 1 infty norms respectively by adopting the technique of regularized green s functions combined with local h 1 and l 2 estimates in dyadic annuli since the computational domain omega h is only polyhedral one has to take into account non conformity of the approximation caused by the discrepancy omega h ne omega in particular the so called galerkin orthogonality relation utilized three times in the proof does not exactly hold and involves domain perturbation terms or boundary skin terms which need to be addressed carefully a numerical example is provided to confirm the theoretical result

uniform error estimates in the finite element method for a singularly perturbed reaction diffusion problem consider the problem e 2 u u f with homogeneous neumann boundary condition in a bounded smooth domain in r n the whole range 0 e 1 is treated the galerkin finite element method is used on a globally quasi uniform mesh of size h the mesh is fixed and independent of e a precise analysis of how the error at each point depends on h and e is is presented as an application first order error estimates in h which are uniform with respect to e are given

shape calculus and finite element method in smooth domains the use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary hence the approximation error consists of two parts the geometric part and the finite element part we propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain the finite element part of the error can be treated in the standard way to estimate the geometric part of the error we need quantitative estimates related to perturbation of the geometry we derive such estimates using the techniques developed for shape sensitivity analysis

an adaptive newton algorithm based on numerical inversion regularization as postconditioner we examine a class of approximate inversion processes satisfying estimates similar to those defined by finite element or truncated spectral approximations these are to be used as approximate right inverses for newton iteration methods when viewed at the operator level these approximations introduce a defect or loss of derivatives of order one or more regularization is introduced as a form of defect correction a superlinearly convergent approximate newton iteration is thereby obtained by using the numerical inversion adaptively i e with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect this adaptive choice makes possible ascribing an order to the convergent process and this is identified as essentially optimal for elliptic problems relative to complexity the design of the algorithm involves multi parameter selection thereby opening up interesting avenues for elliptic problems relative to complexity this applies also to the regularization which may be carried out in the fourier transform space and is band limited in the language of whittaker shannon sampling theory the norms employed in the analysis are of holder space type the iteration is an adaptation of nash moser interation and the complexity studies use vituskin s theory of information processing computational experience is described in the final section

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

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

cosine methods for nonlinear second order hyperbolic equations construction et analyse des methodes efficaces avec une precision d ordre eleve pour l approximation des solutions regulieres d une classe d equations hyperboliques non lineaires du second ordre application des discretisations de type galerkin et des schemas a deux pas estimations d erreurs d ordre optimal

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

optimal l error estimates for piecewise linear finite elements in 1 n 3 for 1 n 3 in this paper for two second order elliptic boundary value problems uniform convergence error estimates are derived using piecewise linear conforming and non conforming finite elements the estimates are valid for polyhedral convex regions rn with 1 n 3 the proof of the estimates is mainly based on sobolev s inequality by which the uniform estimates are reduced to estimates in the mean in this way it is possible to obtain the desired uniformo h2 estimate in the plane as a special case thus confirming theoretically natterer s numerical results

quasi optimal estimates for finite element approximations using orlicz norms on considere l approximation par des elements finis lineaires de la solution du probleme de dirichlet u f on etablit une relation entre l erreur dans la norme infinie et l erreur dans quelques espaces d orlicz on en deduit des estimations uniformes quasi optimales lorsque la solution ne possede des derivees secondes dans l espace d orlicz associe a la fonction exponentielle

optimal convergence for the finite element method in campanato spaces we prove a priori estimates and optimal error estimates for linear finite element approximations of elliptic systems in divergence form with continuous coefficients in campanato spaces the proofs rely on discrete analogues of the campanato inequalities for the solution of the system which locally measure the decay of the energy as an application of our results we derive w 1 p estimates and give a new proof of the well known w 1 results of rannacher and scott

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

on the discretization in time of semilinear parabolic equations with nonsmooth initial data single step discretization methods are considered for equations of the form u au f t u where a is a linear positive definite operator in a hubert space h it is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize even if the initial data v are only in h but also that in contrast to the situation in the linear homogeneous case higher order convergence is not possible in general without further assumptions on v

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

stability and accuracy of adapted finite element methods for singularly perturbed problems the stability and accuracy of a standard finite element method fem and a new streamline diffusion finite element method sdfem are studied in this paper for a one dimensional singularly perturbed connvection diffusion problem discretized on arbitrary grids both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity often in the form of boundary layers of the solution surprisingly the accuracy of the standard fem is shown to depend crucially on the uniformity of the grid away from the singularity in other words the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but in contrast it is robust with respect to perturbation of properly adapted grid inside the boundary layer motivated by this discovery a new sdfem is developed based on a special choice of the stabilization bubble function the new method is shown to have an optimal maximum norm stability and approximation property in the sense that u u n infty leq c inf v n in v n u v n infty where un is the sdfem approximation in linear finite element space vn of the exact solution u finally several optimal convergence results for the standard fem and the new sdfem are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed