matrix weighted besov spaces nazarov treil and volberg defined matrix ap weights and extended the theory of weighted norm inequalities on l p to the case of vector valued functions we develop some aspects of littlewood paley function space theory in the matrix weight setting in particular we introduce matrix weighted homogeneous besov spaces b q p w and matrix weighted sequence besov spaces b q p w as well as b q p a q where the a q are reducing operators for w under any of three different conditions on the weight w we prove the norm equivalences f b q p w s q q b q p w s q q b q p a q where s q q is the vector valued sequence of transform coefficients of f in the process we note and use an alternate more explicit characterization of the matrix ap class furthermore we introduce a weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on b q p w if w is doubling we also obtain the boundedness of almost diagonal operators on b q p w under any of the three conditions on w this leads to the boundedness of convolution and non convolution type calderon zygmund operators czos on b q p w in particular the hilbert transform we apply these results to wavelets to show that the above norm equivalence holds if the transform coefficients are replaced by the wavelet coefficients finally we construct inhomogeneous matrix weighted besov spaces b q p w and show that results corresponding to those above are true also for the inhomogeneous case

modification of the 8 node serendipity element this paper presents modification of the 8 node serendipity quadrilateral element which is widely used in finite element analysis first we give basic principle of constructing 8 d spaces including the cartesian quadratic polynomial space when the quadrilateral is of bilinear isoparametric shape then we derive a concrete example of such a space in which the isoparametric quadratic space is included as well explicit expressions of shape functions are given for the sake of practical use it is also shown that the usual 6 node quadratic triangle is obtained by applying the node degeneration technique to the present element comments are also given on the isoparametric use of the present element and on some existing elements related to ours moreover numerical results are included to show the effectiveness of our modification

newton multigrid least squares fem for the v v p formulation of the navier stokes equations we solve the v v p vorticity velocity pressure formulation of the stationary incompressible navier stokes equations based on the least squares finite element method for the discrete systems we use a conjugate gradient cg solver accelerated with a geometric multigrid preconditioner for the complete system in addition we employ a krylov space smoother inside of the multigrid which allows a parameter free smoothing combining this linear solver with the newton linearization we construct a very robust and efficient solver we use biquadratic finite elements to enhance the mass conservation of the least squares method for the inflow outflow problems and to obtain highly accurate results we demonstrate the advantages of using the higher order finite elements and the grid independent solver behavior through the solution of three stationary laminar flow problems of benchmarking character the comparisons show excellent agreement between our results and those of the galerkin mixed finite element method as well as available reference solutions

a locking free solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of lagrangian elements abstract this paper presents a new finite element solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of the classical bilinear or trilinear lagrangian elements it solves the primal variable displacement in the strain div formulation and can handle both displacement and traction boundary conditions it is a locking free solver based on conforming finite elements the solver has second order accuracy in displacement and first order accuracy in stress and dilation divergence of displacement as validated by theoretical analysis and illustrated by numerical experiments on benchmarks deal ii implementation is also discussed

optimality of the fully discrete filtered backprojection algorithm for tomographic inversion although the filtered backprojection algorithm fba has been the standard reconstruction algorithm in 2d computerized tomography for more than 30 years its convergence behavior is not completely settled so far relying on convergence results by rieder and faridani for the semi discrete fba siam j numer anal 41 3 869 892 2003 we show optimality of the fully discrete version for reconstructing sufficiently smooth density distributions further we introduce mfba a modified version of fba and prove its optimality under weaker smoothness requirements remarkably mfba may have a larger convergence order in the angular than in the lateral variable thus allowing optimal convergence in case of angular under sampling moreover mfba can be seen as a limit of the phantom view method introduced to increase angular resolution

fast iterative solvers for buoyancy driven flow problems we outline a new class of robust and efficient methods for solving the navier stokes equations with a boussinesq model for buoyancy driven flow we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the efficiency of the chosen preconditioning schemes with respect to the discretization parameters

gabor schauder bases and the balian low theorem the balian low theorem is a strong form of the uncertainty principle for gabor systems that form orthonormal or riesz bases for l2 r in this paper we investigate the balian low theorem in the setting of schauder bases we prove that new weak versions of the balian low theorem hold for gabor schauder bases but we constructively demonstrate that several variants of the blt can fail for gabor schauder bases that are not riesz bases we characterize a class of gabor schauder bases in terms of the zak transform and product a2 weights the riesz bases correspond to the special case of weights that are bounded away from zero and infinity

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

interpolation error estimates of a modified 8 node serendipity finite element interpolation error estimates for a modified 8 node serendipity finite element are derived in both regular and degenerate cases the latter of which includes the case when the element is of triangular shape for u in w 3 p k defined over a quadrilateral k the error for the interpolant pi k u is estimated as u pi k u w alpha p k le ch 3 alpha k u w 3 p k alpha 0 1 where 1 le p le infty in the regular case and 1 le p 3 in the degenerate case respectively thus the obtained error estimate in the degenerate case is of the same quality as in the regular case at least for 1 le p 3 results for some related elements are also given

wavelets and the angle between past and future the main result the riesz projectionp or equivalently hilbert transformt is bounded in the weighted spacel2 w wherewis a matrix valued weight if and only ifsupi 1 i iw 1 2 1 i iw 1 1 2 wheresupremumis taken over all intervalsi motivation for this problem comes from stationary processes riesz projection is bounded means the angle between past and future of a stationary process with spectral measurewis nonzero in the scalar case the result is the well known hunt muckenhoupt wheeden theorem the main step in our proof is to show that a vector haar system forms an unconditional basis inl2 w as a byproduct of our approach we get some new results about bases of wavelets in weighted spaces in both scalar and vector valued cases

quadratic serendipity finite elements on polygons using generalized barycentric coordinates we introduce a nite element construction for use on the class of convex planar polygons and show it obtains a quadratic error convergence estimate on a convex n gon satisfying simple geometric criteria our construction produces 2n basis functions associated in a lagrange like fashion to each vertex and each edge midpoint by transforming and combining a set of n n 1 2 basis functions known to obtain quadratic convergence the technique broadens the scope of the so called serendipity elements previously studied only for quadrilateral and regular hexahedral meshes by employing the theory of generalized barycentric coordinates uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges numerical evidence is provided on a trapezoidal quadrilateral mesh previously not amenable to serendipity constructions and applications to adaptive meshing are discussed

lowest order virtual element approximation of magnetostatic problems abstract we give here a simplified presentation of the lowest order serendipity virtual element method and show its use for the numerical solution of linear magneto static problems in three dimensions the method can be applied to very general decompositions of the computational domain as is natural for virtual element methods and uses as unknowns the constant tangential component of the magnetic field h on each edge and the vertex values of the lagrange multiplier p used to enforce the solenoidality of the magnetic induction b h in this respect the method can be seen as the natural generalization of the lowest order edge finite element method the so called first kind nedelec elements to polyhedra of almost arbitrary shape and as we show on some numerical examples it exhibits very good accuracy for being a lowest order element and excellent robustness with respect to distortions

trigonometric bases for matrix weighted lp spaces abstract we give a complete characterization of 2 periodic matrix weights w for which the vector valued trigonometric system forms a schauder basis for the matrix weighted space l p t w then trigonometric quasi greedy bases for l p t w are considered quasi greedy bases are systems for which the simple thresholding approximation algorithm converges in norm it is proved that such a trigonometric basis can be quasi greedy only for p 2 and whenever the system forms a quasi greedy basis the basis must actually be a riesz basis

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

the approximate inverse in action with an application to computerized tomography the approximate inverse is a scheme to obtain stable numerical inversion formulae for linear operator equations of the first kind yet in some applications the computation of a crucial ingredient the reconstruction kernel is time consuming and instable it may even happen that the kernel does not exist for a particular semidiscrete system to cure this dilemma we propose and analyze a technique that is based on a singular value decomposition of the underlying operator the results are applied to the reconstruction problem in 2d computerized tomography where they enable the design of reconstruction filters and lead to a novel error analysis of the filtered backprojection algorithm

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

ultraconvergence of zz patch recovery at mesh symmetry points the ultraconvergence property of the zienkiewicz zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even order finite elements the result is valid at all rectangular mesh symmetry points different smoothing strategies are discussed and numerical examples are demonstrated

a monolithic fem multigrid solver for non isothermal incompressible flow on general meshes we present special numerical simulation methods for non isothermal incompressible viscous fluids which are based on lbb stable fem discretization techniques together with monolithic multigrid solvers for time discretization we apply the fully implicit crank nicolson scheme of 2nd order accuracy while we utilize the high order q 2 p 1 finite element pair for discretization in space which can be applied on general meshes together with local grid refinement strategies including hanging nodes to treat the nonlinearities in each time step as well as for direct steady approaches the resulting discrete systems are solved via a newton method based on divided differences to calculate explicitly the jacobian matrices in each nonlinear step the coupled linear subproblems are solved simultaneously for all quantities by means of a monolithic multigrid method with local multilevel pressure schur complement smoothers of vanka type for validation and evaluation of the presented methodology we perform the mit benchmark 2001 m a christon p m gresho s b sutton computational predictability of natural convection flows in enclosures in first mit conference on computational fluid and solid mechanics vol 40 elsevier 2001 pp 1465 1468 of natural convection flow in enclosures to compare our results with respect to accuracy and efficiency additionally we simulate problems with temperature and shear dependent viscosity and analyze the effect of an additional dissipation term inside the energy equation moreover we discuss how these fem multigrid techniques can be extended to monolithic approaches for viscoelastic flow problems

analysis of some low order quadrilateral reissner mindlin plate elements four quadrilateral elements for the reissner mindlin plate model are considered the elements are the stabilized mitc4 element of lyly stenberg and vihinen 1933 the min4 element of tessler and hughes 1983 the q4bl element of zienkiewicz et al 1993 and the fmin4 element of kikuchi and ishii 1999 for all elements except the q4bl element a unifying variational formulation is introduced and optimal h 1 and l 2 error bounds uniform in the plate thickness are proven moreover we propose a modified q4bl element and show that it admits the optimal h 1 and l 2 error bounds uniform in the plate thickness in particular we study the convergence behavior of all elements regarding the mesh distortion

framework for the a posteriori error analysis of nonconforming finite elements this paper establishes a unified framework for the a posteriori error analysis of a large class of nonconforming finite element methods the theory assures reliability and efficiency of explicit residual error estimates up to data oscillations under the conditions h1 h2 and applies to several nonconforming finite elements the crouzeix raviart triangle element the han parallelogram element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element of hu and shi the p 1 element on parallelograms due to park and sheen and the dssy parallelogram element the theory is extended to include 1 irregular meshes with at most one hanging node per edge

the semidiscrete filtered backprojection algorithm is optimal for tomographic inversion the filtered backprojection algorithm is probably the most often used reconstruction algorithm in two dimensional computerized tomography for a semidiscrete version in the parallel scanning geometry we prove optimal l2 convergence rates for density distributions in sobolev spaces additionally we show l2 convergence without rates when the density distribution is only in l2 the key to success is a new representation of the filtered backprojection which enables us to apply techniques from approximation theory our analysis provides further a modification of the shepp logan reconstruction filter with an improved convergence behavior numerical experiments in the fully discrete setting reproduce the theoretical predictions

hilbert spaces of vector valued functions generated by quadratic forms abstract we study hilbert spaces l 2 e g where e r d is a measurable set e 0 and for almost every t e the matrix g t see 3 is a hermitian positive definite matrix we find necessary and sufficient conditions for which the projection operators t k f f k e k 1 k n are bounded the obtained results allow us to translate various questions in the spaces l 2 e g to weighted norm inequalities with weights which are the diagonal elements of the matrix g t in section 3 we study the properties of the system m t e j 1 j n m n in the space l 2 e g where m m 1 is a complete orthonormal system defined on a measurable set e r we concentrate our study on two classical systems the haar and the trigonometric systems simultaneous approximations of n elements f 1 f n of some banach spaces x 1 x n with respect to a system which is a basis in any of those spaces are studied

numerical solution of stokes equations with pressure and filtration boundary conditions a numerical algorithm is presented for two dimensional stokes equations plane and axisymmetric case with pressure and filtration boundary conditions the numerical procedure is based on a divergence free finite element method and is applicable to multiply connected domains comparisons between two types of finite elements are performed in order to choose the better one the numerical method is tested on flows with known numerical and analytical solutions and on different grids stress boundary conditions are briefly discussed and compared with pressure ones the influence of a closely placed outlet on the accuracy of numerical results is studied numerical examples are presented including flow past a system of bodies in a channel flows in branched channels with or without particles and in channels of arbitrary shape with filtrating walls

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements