a new class of higher order mixed finite volume methods for elliptic problems we introduce a new class of higher order mixed finite volume methods for elliptic problems we start from the usual way of changing the given equation into a mixed system using the darcy s law bf u cal k nabla p by integrating the system of equations with some judiciously chosen test spaces on each element we define new mixed finite volume methods of higher order we show that these new schemes are equivalent to the nonconforming finite element spaces used to define them the darcy velocity can be locally recovered from the solution of nonconforming finite element method hence our work opens a way to make higher order mixed method more practicable three dimensional extensions to parallelepiped elements are also presented this work can be viewed as a generalization of earlier works which were devoted to reducing the lowest order mixed finite element method to a corresponding nonconforming finite element method an optimal error analysis is carried out and numerical results are presented w

domain decomposition algorithms for mixed methods for second order elliptic problems domain decomposition algorithms for mixed finite element methods for linear second order elliptic problems in r 2 and r 3 are developed a convergence theory for two level and multilevel schwarz methods applied to the algorithms under consideration is given it is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems numerical experiments are presented to illustrate the present techniques

multigrid method and multilevel additive preconditioner for mixed element method for non self adjoint and indefinite problems in this paper a v cycle multigrid method is proposed for mixed element method for non self adjoint and indefinite second order elliptic problems and the uniform convergence of the v cycle multigrid method is proven under minimal regularity assumption meanwhile a multilevel additive preconditioner is given for these problems and an optimal convergence rate for preconditioned gmres method is obtained under minimal regularity assumption

preconditioned gmres methods for discretization equation of nonsymmetric and indefinite elliptic problem in this paper a class of preconditioners for gmres method are proposed for solving linear systems arising from discretization second order nonsymmetric and indefinite elliptic problems by finite element method the convergence of gmres method is verified and the rate of convergence is shown the generality of our result enables us to apply not only any known preconditioners designed for symmetric positive definite problems but also many other preconditioners based on original operator to nonsymmetric and indefinite problems without losing optimality

optimal l error estimates for nonconforming and mixed finite element methods of lowest order for second order linear elliptic problems it is proved that thep1 nonconforming finite element method has the samel asymptotic accuracy as thep1 conforming one this result is applied to derive optimall error estimates for both the displacement and the stress fields of the lowest order raviart thomas mixed finite element method and a superconvergence result at the barycenter of each element

error analysis of nonconforming and mixed fems for second order linear non selfadjoint and indefinite elliptic problems the state of the art proof of a global inf sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite second order linear elliptic pdes this paper therefore first analyses a nonconforming finite element discretization which converges owing to some a priori l 2 l2 error estimates even for reduced regularity on non convex polygonal domains an equivalence result of that nonconforming finite element scheme to the mixed finite element method mfem leads to the well posedness of the discrete solution and to a priori error estimates for the mfem the explicit residual based a posteriori error analysis allows some reliable and efficient error control and motivates some adaptive discretization which improves the empirical convergence rates in three computational benchmarks

characteristic nonconforming finite element methods for advection dominated diffusion problems in this paper characteristic nonconforming finite element methods are studied for timedependent advection dominated diffusion problems the diffusion term in these problems is discretized using nonconforming finite elements and the temporal differentiation and advection terms are treated by characteristic tracking schemes various nonconforming finite elements and characteristic tracking schemes are studied stability and convergence properties of the characteristic nonconforming methods are obtained unconditionally stable results and error estimates of optimal order are established

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

a parallel domain decomposition algorithm of mixed element equation for second order elliptic dirichlet boundary value problem a parallel domain decomposition algorithm of schwarz type is introduced to solve the mixed finite element equation for the dirichlet boundary value problem of a second order partial differential equation of elliptic type and convergence of this algorithm is analyzed

the spectrum of prime ideals in tensor triangulated categories we define the spectrum of a tensor triangulated category k as the set of so called prime ideals endowed with a suitable topology in this very generality the spectrum is the universal space in which one can define supports for objects of k this construction is functorial with respect to all tensor triangulated functors several elementary properties of schemes hold for such spaces e g the existence of generic points or some quasi compactness locally trivial morphisms are proved to be nilpotent we establish in complete generality a classification of thick ideal subcategories in terms of arbitrary unions of closed subsets with quasi compact complements thomason s theorem for schemes mutatis mutandis we also equip this spectrum with a sheaf of rings turning it into a locally ringed space we compute examples and show that our spectrum unifies the schemes of algebraic geometry and the support varieties of modular representation theory

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

a multigrid algorithm for the lowest order raviart thomas mixed triangular finite element method an optimal order multigrid method for the lowest order raviart thomas mixed triangular finite element is developed the algorithm and the convergence analysis are based on the equivalence between raviart thomas mixed methods and certain nonconforming methods both the dirichlet and singular neumann boundary value problems for second order elliptic equations are discussed

least squares for second order elliptic problems abstract in this paper we introduce and analyze two least squares methods for second order elliptic differential equations with mixed boundary conditions these methods extend to problems which involve oblique derivative boundary conditions as well as nonsymmetric and indefinite problems as long as the original problem has a unique solution with the methods to be developed neumann and oblique boundary conditions are imposed weakly and thus avoid compatibility conditions on the finite element subspaces the resulting least squares approximations are unconditionally stable no conditions on the step size h and will be shown to converge at an optimal rate the first least squares method involves a discrete computable h 1 norm of the residual and stabilization terms consisting of the jumps at the interelement boundaries and a weighted elementwise l 2 norm of the residual over the finite elements this method is developed without the introduction of additional problem variables the second method involves the use of the flux as an additional unknown although this method is similar to the least squares method for first order systems introduced in 7 it differs in that discontinuous finite elements are allowed it is also more general in that it extends to the oblique boundary problem

spectrum of some triangulated categories abstract in this paper we compute the triangular spectrum as defined by p balmer of two classes of tensor triangulated categories which are quite common in algebraic geometry one of them is the derived category of g equivariant sheaves on a smooth scheme x for a finite group g the other class is the derived category of split superschemes

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

preconditioning of indefinite and almost singular finite element elliptic equations this paper deals with two ways of solving discretized finite element elliptic equations with indefinite and almost singular matrices such problems typically arise when applying the shifted inverse power iteration sipi method to the generalized eigenvalue problem a bf u lambda b bf u where a is defined from some discretized equation by a finite element self adjoint coercive second order elliptic operator and b is a mass matrix operator both methods explore two by two block partitioning of the given matrices one of the main matrix blocks corresponds to a coarse space or equivalently to coarse grid degrees of freedom and contains the main singularity of the original problem the other major block possibly indefinite is not as close to singular as the original matrix and is inverted by a preconditioned minimal residual minres method in the first method the second method which we use for comparison exploits preconditioned minres iterations for the reduced problem obtained by eliminating the block unknowns that correspond to the coarse discretization space here each iteration involves solving a coarse grid problem numerical examples are presented that demonstrate the various aspects of both approaches for solving generalized eigenvalue problems applied to second order finite element elliptic equations

on fourier multiplier transformations of banach valued functions we obtain analogues of the mihlin multiplier theorem and littlewood paley inequalities for functions with values in a suitable banach space b the requirement on b is that it have the unconditionality property for martingale difference sequences

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems a convergence estimate is established for nonsymmetric multigrid algorithms for finite element methods for nonselfadjoint and indefinite elliptic problems it is shown without further regularity assumption beyond h 1 necessary to define the weak form that the v cycle algorithm converges with rate bounded by 1 cj 1 for sufficiently small coarse mesh size h 1 where j is the number of levels in the multigrid algorithm

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

can a finite element method perform arbitrarily badly in this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm and show that adaptive procedures cannot improve this slow convergence we also show that the l 2 norm and the nodal point errors converge arbitrarily slowly with the l 2 norm two cases need to be distinguished and the usual duality principle does not characterize the error completely the constructed elliptic problems are one dimensional

finite difference methods for the wide angle parabolic equation we consider a model initial and boundary value problem for the wide angle parabolic equation lur icu of underwater acoustics where l is a second order differential operator in the depth variable z with depth and range dependent coefficients we discretize the problem by the crank nicolson finite difference scheme and also by the forward euler method using nonuniform partitions both in depth and in range assuming that the problem admits a smooth solution and l is invertible for all r under the posed boundary and interface conditions we show stability of both schemes and derive error estimates

uniform convergence and schwarz method for the mortar projection nonconforming and mixed element methods for nonselfadjoint and indefinite problems abstract in this paper two mortar versions of the so called projection nonconforming and the mixed element methods are proposed respectively for nonselfadjoint and indefinite second order elliptic problems it is proven that the mortar mixed element method is equivalent to the mortar projection nonconforming element method based on this equivalence the existence uniqueness and uniform convergence of the solution for mortar mixed element method are shown only under minimal regularity assumption meanwhile the optimal error estimate is obtained under certain regularity assumption furthermore an additive schwarz preconditioning method is proposed for solving the discrete problem and the nearly optimal convergence rate for the preconditioned gmres method is proven under minimal regularity assumption finally the practical implementation of the method is adderssed and numerical experiments are presented

variational approximation of maxwell s equations in biperiodic structures consider a plane wave incident on a biperiodic diffractive structure the diffraction problem may be modeled by maxwell s equations with transparent boundary conditions and solved by a finite element method in this paper a variational approximation is studied the well posedness of the continuous and discretized problems is established in the following sense in the continuous case it is shown that the model problem attains a unique cal h 1 solution for all but possibly a discrete set of frequencies in the discrete case error estimates for the variational finite element approximation of the model problem with or without truncation of the nonlocal boundary operators are obtained

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

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems the effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied under similar assumption on the quadrature formula as for linear problems optimal error estimates in the l 2 and the h 1 norms are proved the numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh the analysis is valid for both simplicial and rectangular finite elements of arbitrary order numerical experiments corroborate the theoretical convergence rates

some new error estimates for ritz galerkin methods with minimal regularity assumptions new uniform error estimates are established for finite element approximations u h of solutions u of second order elliptic equations lu f using only the regularity assumption u 1 c f 1 using an aubin nitsche type duality argument we show for example that for arbitrary fixed e sufficiently small there exists an h 0 such that for 0 h h 0 u u h 0 e u u h 1 here s denotes the norm on the sobolev space h s other related results are established

convergence and quasi optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems recently an adaptive nonconforming finite element method anfem has been developed by carstensen and hoppe in numer math 103 251 266 2006 in this paper we extend the result to some nonsymmetric and indefinite problems the main tools in our analysis are a posteriori error estimators and a quasi orthogonality property in this case we need to overcome two main difficulties one stems from the nonconformity of the finite element space the other is how to handle the effect of a nonsymmetric and indefinite bilinear form an appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order crouzeix raviart element it is shown that the anfem is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops moreover quasi optimality in the sense of quasi optimal algorithmic complexity can be shown for the anfem the results of numerical experiments confirm the theoretical findings

multigrid algorithms for nonconforming and mixed methods for nonsymmetric and indefinite problems in this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for nonsymmetric and or indefinite elliptic problems we show that a simple cal v cycle multigrid iteration using conforming coarse grid corrections converges at a uniform rate provided that the coarsest level in the multilevel iteration is sufficiently fine but independent of the number of multigrid levels various types of smoothers for the nonsymmetric and indefinite problems are discussed extensive numerical results are presented

preconditioning nonconforming finite element methods this paper is concerned with the h sup 1 condition number and singular value distribution of preconditioned nonself adjoint discrete elliptic operators in particular a wide class of nonconforming finite element methods are investigated the main results give a precise interval on which the singular values cluster results of this kind have an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations

uniform convergence and preconditioning methods for projection nonconforming and mixed methods for nonselfadjoint and indefinite problems abstract the purpose of this paper is to prove the existence uniqueness and uniform convergence of the solutions of so called projection nonconforming and mixed element methods and the equivalence between projection nonconforming element method and mixed element method with nonquasi uniform partition for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption meanwhile the optimal error estimate of the solution of mixed element method is obtained under h 2 smoothness hypothesis for nonselfadjoint and indefinite elliptic problems without h 2 regularity finally the discrete compactness result for nonconforming element space with nonquasi uniform partition is proven and some preconditioning methods for projection nonconforming and mixed element methods with nonquasi uniform partition are given it is proven that the h 1 condition number of preconditioned operator is uniformly bounded and its singular values cluster in a relatively small finite interval

justification of the saturation assumption the saturation assumption is widely used in computational science and engineering usually without any rigorous theoretical justification and even despite of counterexamples for some coarse meshes known in the mathematical literature on the other hand there is overwhelming numerical evidence at least in an asymptotic regime for the validity of the saturation in the generalized form the assumption states for any 0 varepsilon le 1 0 1 that begin aligned u hat u 2 le 1 varepsilon c u u 2 varepsilon mathrm osc 2 f mathcal n end aligned u u 2 1 c u u 2 osc2 f n for the exact solution u and the first order conforming finite element solution u resp hat u u of the poisson model problem with respect to a regular triangulation mathcal t t resp hat mathcal t t and its uniform refinement hat mathcal t t within the class mathbb t t of admissible triangulations the point is that the patch oriented oscillations mathrm osc f mathcal n osc f n vanish for constant right hand sides f equiv 1 f 1 and may be of higher order for smooth f while the strong reduction factor 1 varepsilon c 1 1 c 1 involves some universal constant c which exclusively depends on the set of admissible triangulations and so on the initial triangulation only this paper proves the inequality sa for the energy norms of the errors for any admissible triangulation mathcal t t in mathbb t t up to computable pathological situations characterized by failing the weak saturation test ws this computational test ws for some triangulation mathcal t t states that the solutions u and hat u u do not coincide for the constant right hand side f equiv 1 f 1 the set of possible counterexamples is characterized as mathcal t t with no interior node or exactly one interior node which is the vertex of all triangles and hat mathcal t t is a particular uniform bisec3 refinement in particular the strong saturation assumption holds for all triangulations with more than one degree of freedom the weak saturation test ws is only required for zero or one degree of freedom and gives a definite outcome with o 1 operations the only counterexamples known so far are regular n polygons the paper also discusses a generalization to linear elliptic second order pdes with small convection to prove that saturation is somehow generic and fails only in very particular situations characterised by ws

an optimal order multigrid method for p1 nonconforming finite elements etude d une methode multigrille d ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l aide d elements finis non conformes

uniform convergence and preconditioning method for mortar mixed element method for nonselfadjoint and indefinite problems abstract in this paper a mortar version of the modified mixed element method is proposed for nonselfadjoint and indefinite second order elliptic problems it is proven that the modified mortar mixed element method is equivalent to a modified mortar nonconforming element method based on this equivalence the existence uniqueness and uniform convergence of the solution for the modified mortar mixed element method are shown under minimal regularity assumption meanwhile an optimal error estimate for the modified mortar mixed element method is obtained under h2 smoothness assumption furthermore an additive schwarz preconditioning method for solving the discrete problem is provided and the nearly optimal convergence rate for the preconditioned gmres method is proven under the minimal regularity assumption

error analysis of the dtn fem for the scattering problem in acoustics via fourier analysis in this paper we are concerned with the error analysis for the nite element solution of the two dimensional exterior neumann boundary value problem in acoustics in particular we establish an explicit priori error estimates in h 1 and l 2 norms including both the eect of the truncation of the dtn mapping and that of the numerical discretization to apply the nite element method fem to the exterior problem the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a dirichlet to neumann dtn mapping represented in terms of the fourier expansion series we discuss essential features of the corresponding variational equation and its modication due to the truncation of the dtn mapping in appropriate function spaces numerical tests are presented to validate our theoretical results

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

uniform convergence and schwarz method for the mortar element method for non selfadjoint and indefinite problems in this paper the mortar element method for non selfadjoint and indefinite second order elliptic problems is studied only under minimal regularity assumption the existence uniqueness and uniform convergence of the solution for the mortar element method are proven furthermore an additive schwarz preconditioning method is proposed and nearly optimal convergence rate for the preconditioned gmres method is shown under minimal regularity assumption

preconditioning projection nonconforming element method for the lowest order raviart thomas mixed triangular element method this paper is concerned with the h 1 condition number and the b h singular value distribution of preconditioned operators b h 1a h 0 h 1 where a h is the projection nonconforming element discretization of the nonself adjoint and possibly indefinite second order elliptic operator a b h is the nonconforming element discretization of the self adjoint and positive definite second order elliptic operator b the projection nonconforming element method is equivalent to the lowest order raviart thomas mixed triangular element method for second order elliptic problems it is proved that the h 1 condition number of operators b h 1a h 0 h 1 are uniformly bounded and the b h singular values cluster in a positive finite interval finally a simple v cycle multigrid implementation of b h 1 is given

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

analysis of the spectrum of a cartesian perfectly matched layer pml approximation to acoustic scattering problems in this paper we study the spectrum of the operator which results when the perfectly matched layer pml is applied in cartesian geometry to the laplacian on an unbounded domain this is often thought of as a complex change of variables or complex stretching the reason that such an operator is of interest is that it can be used to provide a very effective domain truncation approach for approximating acoustic scattering problems posed on unbounded domains stretching associated with polar or spherical geometry lead to constant coefficient operators outside of a bounded transition layer and so even though they are on unbounded domains they and their numerical approximations can be analyzed by more standard compact perturbation arguments in contrast operators associated with cartesian stretching are non constant in unbounded regions and hence cannot be analyzed via a compact perturbation approach alternatively to show that the scattering problem pml operator associated with cartesian geometry is stable for real nonzero wave numbers we show that the essential spectrum of the higher order part only intersects the real axis at the origin this enables us to conclude stability of the pml scattering problem from a uniqueness result given in a subsequent publication

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

uniform convergence of finite volume element method with crouzeix raviart element for non self adjoint and indefinite elliptic problems in this paper we consider the finite volume element method based on the crouzeix raviart element and prove the existence uniqueness and uniform convergence of the finite volume element approximations for the non self adjoint and indefinite elliptic problems under minimal elliptic regularity assumption

three scale finite element discretizations for quantum eigenvalue problems based on globally and locally coupled discretizations some three scale finite element schemes are proposed in this paper for a class of quantum eigenvalue problems it is shown that the solution of a quantum eigenvalue problem on a fine grid may be reduced to the solution of an eigenvalue problem on a relatively coarse grid and the solutions of linear algebraic systems on a globally mesoscopic grid and the locally fine grid and the resulting solution is still very satisfactory

on the accuracy of the mortar finite volume element method with lagrange multipliers abstract in this paper the mortar finite volume element mfve method with lagrange multipliers is considered for second order elliptic boundary value problems we prove l 2 error estimates which are affected by regularities in both the exact solution and the source term meanwhile the uniform convergence of the solution is obtained for the mfve method under minimal regularity assumption

a local global principle for the telescope conjecture abstract we prove an etale local global principle for the telescope conjecture and use it to show that the telescope conjecture holds for derived categories of azumaya algebras on noetherian schemes as well as for many classifying stacks and gerbes this specializes to give another proof of the fact that the telescope conjecture holds for noetherian schemes

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

local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems in this paper we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems two types of local smoothers are constructed one is based on the original nonsymmetric problems and the other is defined in terms of the associated symmetric problems it is shown that the local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems are optimal which means that the convergence rates of the local multilevel methods are independent of mesh sizes and mesh levels provided that the coarsest grid is sufficiently fine numerical experiments are reported to confirm our theory

numerical analysis of diffraction by periodic structures tm polarization the diffraction of a time harmonic wave in tm polarization by some periodic inhomogeneous material is studied in this paper the diffraction problem is modeled by a generalized helmholtz equation with transparent boundary conditions existence and uniqueness are proved for the model problem uniform error estimates for the finite element approximation of the solution are established error estimates are also obtained when the truncation of the nonlocal boundary operators takes place

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

presheaves of triangulated categories and reconstruction of schemes to any triangulated category with tensor product k otimes we associate a topological space rm spc k otimes by means of thick subcategories of k a la hopkins neeman thomason moreover to each open subset u of this space spc k otimes we associate a triangulated category cal k u producing what could be thought of as a presheaf of triangulated categories applying this to the derived category k otimes rm d rm perf x otimes l of perfect complexes on a noetherian scheme x the topological space rm spc k otimes turns out to be the underlying topological space of x moreover for each open u subset x the category cal k u is naturally equivalent to rm d rm perf u as an application we give a method to reconstruct any reduced noetherian scheme x from its derived category of perfect complexes rm d rm perf x considering the latter as a tensor triangulated category with otimes l

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

analysis of a finite element method for the drift diffusion semiconductor device equations the multidimensional case an explicit finite element method for numerically solving the drift diffusion semiconductor device equations in two space dimensions is analyzed the method is based on the use of a mixed finite element method for the approximation of the electric field and a discontinuous upwinding finite element method for the approximation of the electron and hole concentrations the mixed method gives an approximate electric field in the precise form needed by the discontinuous method which is trivially conservative and fully parallelizable it is proven that the method produces uniformly bounded concentrations and electric fields and that it converges to the exact solution provided there is a convergent subsequence of the electron concentrations numerical simulations are presented that display the performance of the method and indicate the behavior of the solution

convergence analysis of the schwarz algorithm and multilevel decomposition iterative methods ii nonselfadjoint and indefinite elliptic problems convergence estimate for the schwarz alternating algorithm applied to nonselfadjoint and indefinite elliptic problems is established for a certain overlapping domain decomposition multilevel iterative algorithms are proposed and analyzed for the finite element method for the same problem with locally refined grids the author shows that the algorithm based on the hierarchical basis decomposition converges with rate bounded by 1 cj 2 where j is the number of levels the author also analyzes the algorithm based on the full multigrid decomposition and shows that the algorithm converges with rate 1 cj 1 for problems of any dimension all the results are established without regularity assumption for the differential equation

iterative schemes for nonsymmetric and indefinite elliptic boundary value problems the purpose of this paper is twofold the first is to describe some simple and robust iterative schemes for nonsymmetric and indefinite elliptic boundary value problems the schemes are based in the sobolev space h omega and require minimal hypotheses the second is to develop algorithms utilizing a coarse grid approximation this leads to iteration matrices whose eigenvalues lie in the right half of the complex plane in fact for symmetric indefinite problems the iteration is reduced to a well conditioned symmetric positive definite system which can be solved by conjugate gradient interation applications of the general theory as well as numerical examples are given 20 refs 8 tabs

convergence and domain decomposition algorithm for nonconforming and mixed methods for nonselfadjoint and indefinite problems in this paper the existence uniqueness and convergence of the solutions of nonconforming and mixed methods with non quasi uniform partition are proven for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption moreover uniform convergence of the solution of nonconforming method is obtained meanwhile the domain decomposition algorithm which suits non quasi uniform partition is proposed for nonconforming and mixed methods

analysis of a cartesian pml approximation to acoustic scattering problems in r2 abstract in this paper we consider a cartesian pml approximation to solutions of acoustic scattering problems on an unbounded domain in r 2 the perfectly matched layer pml technique in a curvilinear coordinate system has been researched for acoustic scattering applications both in theory and computation our goal will be to extend the results of spherical cylindrical pml to pml in cartesian coordinates that is the well posedness of cartesian pml approximation on both the unbounded and truncated domains the exponential convergence of approximate solutions as a function of domain size is also shown we note that once the stability and convergence of the continuous truncated problem has been achieved the analysis of the resulting finite element approximations is then classical finally the results of numerical computations illustrating the theory and efficiency of the cartesian pml approach will be given