adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of biot s consolidation model abstract this paper is concerned with the analysis of coupled mixed finite element methods applied to the biot s consolidation model we consider two mixed formulations that use the stress tensor and darcy velocity as primary variables as well as the displacement and pressure the first formulation is with a symmetric stress tensor while the other enforces the symmetry of the stress weakly through the introduction of a lagrange multiplier the well posedness of the two formulations is shown through galerkin s method and suitable a priori estimates the two formulations are then discretized with the backward euler scheme in time and with two mixed finite elements in space we present next a general and unified a posteriori error analysis which is applicable for any flux and stress conforming discretization our estimates are based on h 1 conforming reconstruction of the pressure and a suitable h 1 d conforming reconstruction of the displacement both are continuous and piecewise affine in time these reconstructions are used to infer a guaranteed and fully computable upper bound on the energy type error measuring the differences between the exact and the approximate pressure and displacement the error components resulting from the spatial and the temporal discretization are distinguished they are then used to design an adaptive space time algorithm numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm

interpolatory quadrature formulas on the unit circle for chebyshev weight functions in this paper interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands a comparison is made with szego quadrature formulas finally an application to the interval 1 1 is also carried out

a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

connection between orthogonal polynomials on the unit circle and bounded interval in this paper we establish the connection between measures on a bounded interval and on the unit circle by a transformation related with the szego classical transformation we transform a measure on the interval 1 1 into a measure on 0 2 in the same form in which the orthogonality measure of the chebyshev polynomials of fourth kind becomes the lebesgue measure we relate the sequences of orthogonal polynomials with respect to both measures and we also relate the coefficients of the three term recurrence relation with the schur parameters when the measures belong to the szego class we study the asymptotic behavior of the orthogonal polynomials on the interval outside the support of the measure as well as inside we also transform the generalized polynomials and we study the orthogonality properties of the new polynomials obtaining new interesting results and finally we solve two inverse problems connected with the transformation studied

a posteriori error estimation based on potential and flux reconstruction for the heat equation we derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous galerkin various finite volume and mixed finite element methods in space and the backward euler scheme in time extensions to conforming and nonconforming finite element spatial discretizations are also outlined our estimates are based on a h 1 conforming reconstruction of the potential continuous and piecewise affine in time and a locally conservative mathbf h mathrm div conforming reconstruction of the flux piecewise constant in time they yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative local in time lower bounds are also derived for nonconforming methods on time varying meshes the lower bounds require a mild parabolic type constraint on the meshsize

diffusion in poro elastic media existence uniqueness and regularity theory is developed for a general initial boundary value problem for a system of partial differential equations which describes the biot consolidation model in poro elasticity as well as a coupled quasi static problem in thermoelasticity additional effects of secondary consolidation and pore fluid exposure on the boundary are included this quasi static system is resolved as an application of the theory of linear degenerate evolution equations in hilbert space and this leads to a precise description of the dynamics of the system

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

effects of approximations on the static and dynamic response of a cantilever with a tip mass abstract the problem of determining the equilibrium deflections of a beam with a tip mass and the frequency of infinitesimally small oscillations about the beam s equilibrium state e is analyzed the beam is able to experience flexure along two normal directions in space thus flexure in any direction and torsion numerical solutions of the full nonlinear beam static equilibrium equations are obtained by direct integration of a two point boundary value problem the results obtained are compared with an approximate perturbation expansion solution e for the equilibrium state e the frequencies associated with the small oscillations about the equilibrium state arc determind by linearizing the equations of motion about the equilibrium state and by using a transfer matrix technique on the resulting equations the effect on the calculated frequencies of using the approximate solution e obtained by a perturbation method instead of the more exact numerical solution e mentioned above in the linearized equations is assessed the use of an alternative way to determine an approximation for the natural frequencies of the system is also assessed for this small motions are assumed and the equations of motion are first expanded about the undeformed state of the beam the undeformed state is not a static equilibrium state if gravity is considered the resulting equations which contain only polynomial nonlinearities are then used to analyze the motion after the equilibrium solution to these equations e is determined by a perturbation expansion these same equations are then linearized about e and the natural frequencies are also determined by a transfer matrix technique it is shown that the approximation so obtained for the natural frequencies can be unsatisfactory for large values of the tip mass all the results obtained in this paper are compared with published unite elements und with experimental results

stable cell centered finite volume discretization for biot equations in this paper we discuss a new discretization for the biot equations the discretization treats the coupled system of deformation and flow directly as opposed to combining discretizations for the two separate subproblems the coupled discretization has the following key properties the combination of which is novel 1 the variables for the pressure and displacement are co located and are as sparse as possible e g one displacement vector and one scalar pressure per cell center 2 with locally computable restrictions on grid types the discretization is stable with respect to the limits of incompressible fluid and small time steps 3 no artificial stabilization term has been introduced furthermore due to the finite volume structure embedded in the discretization explicit local expressions for both momentum balancing forces and mass conservative fluid fluxes are available we prove stability of the proposed method with respect to all relevant limits together with consistency this proves conve

chebyshev series method for computing weighted quadrature formulas abstract in this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions the main goal is to evaluate accurately a definite integral whose mass is highly concentrated near some points the numerical implementation of this approach is based on the calculation of chebyshev series and some integration formulas which are exact for polynomials in terms of accuracy the proposed method can be compared with rational gauss quadrature formula

non linear flexural flexural torsional extensional dynamics of beams i formulation abstract the non linear differential equations of motion and boundary conditions for euler bernoulli beams able to experience flexure along two principal directions and thus flexure in any direction in space torsion and extension are formulated the beam s material is assumed to be hookean but its properties may vary along its span the nonlinearities present in the differential equations include contributions from the curvature expression and from inertia terms a set of differential equations with polynomial nonlinearities to cubic order suitable for a perturbation analysis of the motion is also developed and the validity of the inextensional approximation is assessed the equations developed here reduce to those for an inextensional beam in part ii of this paper a specific example of application is analyzed and the results obtained are compared with those available in the literature where several non linear terms have been neglected a priori

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

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

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

dynamic lateral torsional post buckling of a beam mass system theory abstract in this paper the theory of a cosserat point is used as a numerical model of a nonlinear elastic beam mass system and simulations of dynamic lateral torsional buckling are compared with results of experiments the cosserat equations are solved using the newmark time integration scheme and an analytical expression for the tangent stiffness of the resulting newton raphson iteration procedure is developed also the effects of material damping aerodynamic drag and gravity have been included the simulations reproduce the experimental result that two different nonlinear modes of vibration occur at the same excitation amplitude and frequency one mode torsion i is a post buckling mode associated with out of plane motion of the beam mass system which is dominated by oscillating torsion of the beam the second mode bending ii is dominated by a nonlinear second bending mode in the weak bending plane the simulations of torsion i and bending ii and another mode torsion ii are in reasonably good quantitative agreement with the experimental results however the model is not able to accurately simulate the response of torsion iii which was similar to torsion i and ii but with a larger amplitude of vibration comparison is also made with the commercial code ansys

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

adaptive asynchronous time stepping stopping criteria and a posteriori error estimates for fixed stress iterative schemes for coupled poromechanics problems abstract in this paper we develop adaptive iterative coupling schemes for the biot system modeling coupled poromechanics problems we particularly consider the space time formulation of the fixed stress iterative scheme in which we first solve the problem of flow over the whole space time interval then exploiting the space time information for solving the mechanics two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in space and the backward euler scheme in time therefrom adaptive fixed stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions these ingredients are used to derive a posteriori error estimates for the fixed stress algorithms distinguishing the different error components namely the spatial discretization the temporal discretization and the fixed stress iteration components precisely at the iteration k 1 of the adaptive algorithm we prove that our estimate gives a guaranteed and fully computable upper bound on the energy type error measuring the difference between the exact and approximate pressure and displacement these error components are efficiently used to design adaptive asynchronous time stepping and adaptive stopping criteria for the fixed stress algorithms numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms

stress and flux reconstruction in biots poro elasticity problem with application to a posteriori error analysis we derive equilibrated reconstructions of the darcy velocity and of the total stress tensor for biots poro elasticity problem both reconstructions are obtained from mixed finite element solutions of local neumann problems posed over patches of elements around mesh vertices the darcy velocity is reconstructed using raviartthomas finite elements and the stress tensor using arnoldwinther finite elements so that the reconstructed stress tensor is symmetric both reconstructions have continuous normal component across mesh interfaces using these reconstructions we derive a posteriori error estimators for biots poro elasticity problem and we devise an adaptive spacetime algorithm driven by these estimators the algorithm is illustrated on test cases with analytical solution on the quarter five spot problem and on an industrial test case simulating the excavation of two galleries

a multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs abstract in coupled flow and poromechanics phenomena representing hydrocarbon production or co 2 sequestration in deep subsurface reservoirs the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs the typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock the former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs in order to address these challenges we augment the fixed stress split iterative scheme with upscaling and downscaling operators to enable modeling flow and mechanics on overlapping nonmatching hexahedral grids flow is solved on a finer mesh using a multipoint flux mixed finite element method and mechanics is solved on a coarse mesh using a conforming galerkin method the multiscale operators are constructed using a procedure that involves singular value decompositions a surface intersections algorithm and delaunay triangulations we numerically demonstrate the convergence of the augmented scheme using the classical mandel s problem solution

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

finite volume methods for elasticity with weak symmetry we introduce a new cell centered finite volume discretization for elasticity with weakly enforced symmetry of the stress tensor the method is motivated by the need for robust discretization methods for deformation and flow in porous media and falls in the category of multi point stress approximations mpsa by enforcing symmetry weakly the resulting method has additional flexibility beyond previous mpsa methods this allows for a construction of a method which is applicable to all grid types and in particular the method amends a crucial shortcoming in previous mpsa methods for simplex grids by formulating the method as a discrete variational problem we prove convergence of the new method for a wide range of problems with conditions that can be verified at the time of discretization we present the first set of comprehensive numerical tests for the mpsa methods in three dimensions covering cartesian and simplex grids with both heterogeneous and nearly incompressible media the tests show that the new method consistently is second order convergent in displacement despite being lowest order with a rate that mostly is between 1 and 2 for stresses the results further show that the new method is more robust and computationally cheaper than previous mpsa methods

special linear combinations of orthogonal polynomials abstract the paper lists a number of problems that motivate consideration of special linear combinations of polynomials orthogonal with the weight p x on the interval a b we study properties of the polynomials as well as the necessary and sufficient conditions for their orthogonality the special linear combinations of chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations

nonlinear dynamics of a flexible beam in a central gravitational field i equations of motion abstract the complete nonlinear differential equations governing the nonlinear motions of a beam able to undergo bending and pitching in space are formulated in this paper the formulation is based on a variational principle and accounts for all the nonlinearities due to deformation and gravity gradient effects the nonlinearities due to deformation arise due to geometric effects which consist of nonlinear curvature and nonlinear inertia terms expanded equations governing the nonlinear perturbed motion about an equilibrium are also developed for the case when the beam is in circular orbit such equations are suited for a perturbation analysis of the motion and nonlinearities up to cubic order in a bookkeeping parameter are retained in them nonlinear motions involving interactions between bending and pitching of the beam are investigated in part ii of this work using the equations developed here

guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the biot problem abstract the paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi static linear biot equations the system is decoupled by employing the fixed stress split scheme which leads to an iteratively solved semi discrete system the error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations the estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method they are fully computable do not contain mesh dependent constants and provide reliable global estimates of the error measured in the energy norm moreover they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures

a posteriori error estimations of a coupled mixed and standard galerkin method for second order operators in this paper we consider a discretization method proposed by wieners and wohlmuth the coupling of mixed and conforming finite element discretizations in domain decomposition methods vol 10 contemporary mathematics vol 218 american mathematical society providence ri 1998 pp 547 554 see also r d lazarov j pasciak p s vassilevski iterative solution of a coupled mixed and standard galerkin discretization method for elliptic problems numer linear algebra appl 8 2001 13 31 for second order operators which is a coupling between a mixed method in a subdomain and a standard galerkin method in the remaining part of the domain we perform an a posteriori error analysis of residual type of this method by combining some arguments from a posteriori error analysis of galerkin methods and mixed methods the reliability and efficiency of the estimator are proved some numerical tests are presented and confirm the theoretical error bounds

adaptive regularization linearization and discretization and a posteriori error control for the two phase stefan problem we consider in this paper the time dependent two phase stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward euler temporal discretizations regularization of the enthalpy temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual as well as on the l2 l2 error of the temperature and the l2 h1 error of the enthalpy moreover they allow to distinguish the space time regularization and linearization error components an adaptive algorithm is proposed which ensures computational savings through the online choice of a sufficient regularization parameter a stopping criterion for the linearization iterations local space mesh refinement time step adjustment and equilibration of the spatial and temporal errors we also prove the efficiency of our estimate our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization as an example we apply it to the vertex centered finite volume finite element with mass lumping and quadrature and newton methods numerical results illustrate the effiectiveness of our estimates and the performance of the adaptive algorithm

adaptive iterative multiscale finite volume method the multiscale finite volume msfv method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients it has been shown for a wide range of test cases that the msfv results are in close agreement with those obtained with a classical computationally expensive technique the method however fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions recently a convergent iterative msfv i msfv method was developed to enhance the quality of the multiscale results by improving the localization conditions although the i msfv method proved to be efficient for most practical problems it is still favorable to improve the localization condition adaptively i e only for a sub domain where the original msfv localization conditions are not acceptable e g near shale layers and long coherent structures with high permeability contrasts in this paper a space time adaptive i msfv ai msfv method is introduced it is shown how to improve the msfv results adaptively in space and simulation time the fine scale smoother which is necessary for convergence of the i msfv method is also applied locally finally for multiphase flow problems two criteria are investigated for adaptively updating the msfv interpolation functions 1 a criterion based on the total mobility change for the transient coefficients and 2 a criterion based on the pressure equation residual for the accuracy of the results for various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub domains and not everywhere the findings show that the error introduced in the msfv framework can be controlled and improved very efficiently with very little additional computational cost compared to the original non iterative msfv method

a posteriori error estimates for the mortar mixed finite element method several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived a residual based estimator provides optimal upper and lower bounds for the pressure error an efficient and reliable estimator for the velocity and mortar pressure error is also derived which is based on solving local element problems in a higher order space the interface flux jump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

gaussian quadrature formulae on the unit circle let be a probability measure on 0 2 in this paper we shall be concerned with the estimation of integrals of the form i f 1 2 02 f ei d for this purpose we will construct quadrature formulae which are exact in a certain linear subspace of laurent polynomials the zeros of szego polynomials are chosen as nodes of the corresponding quadratures we will study this quadrature formula in terms of error expressions and convergence as well as its relation with certain two point pade approximants for the herglotz riesz transform of furthermore a comparison with the so called szego quadrature formulae is presented through some illustrative numerical examples

dispersion analysis of plane wave discontinuous galerkin methods summary the plane wave dg pwdg method for the helmholtz equation was introduced and analyzed in gittelson c hiptmair r and perugia i plane wave discontinuous galerkin methods analysis of the h version math model numer anal 43 2009 297 331 as a generalization of the so called ultra weak variational formulation see o cessenat and b despres application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz equation siam j numer anal 35 1998 pp 255 299 the method relies on trefftz type local trial spaces spanned by plane waves of different directions and links cells of the mesh through numerical fluxes in the spirit of dg methods we conduct a partly empirical dispersion analysis of the method in a discrete translation invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction we find agreement of the wave numbers for directions represented in the local trial spaces for other directions the pwdg methods turn out to incur both phase and amplitude errors this manifests itself as a pollution effect haunting the h version of the method our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves copyright 2014 john wiley sons ltd

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

fully computable error bounds for discontinuous galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes we obtain fully computable a posteriori error bounds on the broken energy seminorm and discontinuous galerkin norm dg norm of the error in first order symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg finite element approximations of a linear second order elliptic problem on meshes containing an arbitrary number of levels of hanging nodes and comprised of triangular elements the estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and dg norm of the error these estimators are also shown to provide a lower bound for the broken energy seminorm and dg norm of the error up to a constant and higher order data oscillation terms we also obtain an explicit computable bound for the value of the interior penalty parameter needed to ensure the existence of the discontinuous galerkin finite element approximation for all versions of the method

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

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

error indicators for the mortar finite element discretization of the laplace equation the mortar technique turns out to be well adapted to handle mesh adaptivity in finite elements since it allows for working with nonnecessarily compatible discretizations on the elements of a nonconforming partition of the initial domain the aim of this paper is to extend the numerical analysis of residual error indicators to this type of methods for a model problem and to check their efficiency thanks to some numerical experiments

a comprehensive analysis of the dynamics of a helicopter rotor blade a comprehensive and versatile analysis of the response of a helicopter rotor blade is presented in this paper first the full nonlinear partial differential equations that govern the motion of the blade taking into account the geometrical nonlinearities that arise due to deformation are presented the equilibrium solution exhibited by the system is then determined using a relatively simple model for the aerodynamic forces common in the rotorcraft dynamics literature the equilibrium solution is determined by numerical integration of a nonlinear two point boundary value problem the blade s equilibrium is then perturbed and the stability of infinitesimally small perturbations is analyzed in detail the analysis presented here yields essentially exact results for the equilibrium solution and for the eigenvalues and eigenfunctions associated with the infinitesimally small perturbations about the equilibrium

a posteriori error estimates for a discontinuous galerkin method applied to elliptic problems log number r74 abstract a posteriori error estimates for locally mass conservative methods for subsurface flow are presented these methods are based on discontinuous approximation spaces and referred to as discontinuous galerkin methods in the case where penalty terms are added to the bilinear form one obtains the nonsymmetric interior penalty galerkin methods in a previous work we proved optimal rates of convergence of the methods applied to elliptic problems here h adaptivity is investigated for flow problems in 2d we derive global explicit estimators of the error in the l 2 norm and we numerically investigate an implicit indicator of the error in the energy norm model problems with discontinuous coefficients are considered

a nonconforming high order method for the biot problem on general meshes in this work we introduce a novel algorithm for the biot problem based on a hybrid high order discretization of the mechanics and a symmetric weighted interior penalty discretization of the flow the method has several assets including in particular the support of general polyhedral meshes and arbitrary space approximation order our analysis delivers stability and error estimates that hold also when the specific storage coefficient vanishes and shows that the constants have only a mild dependence on the heterogeneity of the permeability coefficient numerical tests demonstrating the performance of the method are provided

a posteriori error estimates for biot system using enriched galerkin for flow abstract we analyze the biot system solved with a fixed stress split enriched galerkin eg discretization for the flow equation and galerkin for the mechanics equation residual based a posteriori error estimates are established with both lower and upper bounds these theoretical results are confirmed by numerical experiments performed with the mandel s problem the efficiency of these a posteriori error estimators to guide dynamic mesh refinement is demonstrated with a prototype unconventional reservoir model containing a fracture network we further propose a novel stopping criterion for the fixed stress iterations using the error indicators to balance the fixed stress split error with the discretization errors the new stopping criterion does not require hyperparameter tuning and demonstrates efficiency and accuracy in numerical experiments

a finite element framework for some mimetic finite difference discretizations in this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified nedelec raviart thomas finite element methods for model problems in h curl and h div this provides a simple and transparent way to analyze such mimetic finite difference discretizations using the well known results from finite element theory the finite element framework that we develop is also crucial for the design of efficient multigrid methods for mimetic finite difference discretizations since it allows us to use canonical inter grid transfer operators arising from the finite element framework we provide special local fourier analysis and numerical results to demonstrate the efficiency of such multigrid methods

monolithic and splitting solution schemes for fully coupled quasi static thermo poroelasticity with nonlinear convective transport abstract this paper concerns monolithic and splitting based iterative procedures for the coupled nonlinear thermo poroelasticity model problem the thermo poroelastic model problem we consider is formulated as a three field system of pde s consisting of an energy balance equation a mass balance equation and a momentum balance equation where the primary variables are temperature fluid pressure and elastic displacement due to the presence of a nonlinear convective transport term in the energy balance equation it is convenient to have access to both the pressure and temperature gradients hence we introduce these as two additional variables and extend the original three field model to a five field model for the numerical solution of this five field formulation we compare six approaches that differ by how we treat the coupling decoupling between the flow and from heat and from the mechanics suitable for varying coupling strength between the three physical processes the approaches have in common a simultaneous application of the so called l scheme which works both to stabilize iterative splitting as well as to linearize nonlinear problems and can be seen as a generalization of the undrained and fixed stress split algorithms more precisely the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion we provide a convergence proof for the derived algorithms and demonstrate their performance through several numerical examples investigating different strengths of the coupling between the different processes

space time finite element approximation of the biot poroelasticity system with iterative coupling abstract we analyze an optimized artificial fixed stress iterative scheme for a space time finite element approximation of the biot system modeling fluid flow in deformable porous media the iteration is based on a prescribed constant artificial volumetric mean total stress in the first half step the optimization comes through the adaptation of a numerical stabilization or tuning parameter and aims at an acceleration of the iterations the separated subproblems of fluid flow written as a mixed first order in space system and mechanical deformation are discretized by space time finite element methods of arbitrary order continuous and discontinuous galerkin discretizations of the time variable are encountered the convergence of the iterative schemes is proved for the continuous and fully discrete case the choice of the optimization parameter is identified in the proofs of convergence of the iterations the analyses are illustrated and confirmed by numerical experiments

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials in the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients as an application we obtain quadrature rules for polynomial modifications of the bernstein measures on 1 1 having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients

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

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

parameter robust discretization and preconditioning of biot s consolidation model biot s consolidation model in poroelasticity has a number of applications in science medicine and engineering the model depends on various parameters and in practical applications these parameters range over several orders of magnitude a current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations the purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete biot systems the approach taken here is to consider the stability of the problem in nonstandard or weighted hilbert spaces and employ the operator preconditioning approach we derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement the parameters of interest are small time step sizes large bulk and shear moduli and small hydraulic conductivity

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

a posteriori error estimates including algebraic error and stopping criteria for iterative solvers for the finite volume discretization of a second order elliptic model problem we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system we show that the algebraic error can be bounded by constructing an equilibrated raviart thomas nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector next claiming that the discretization error and the algebraic one should be in balance we construct stopping criteria for iterative algebraic solvers an attention is paid in particular to the conjugate gradient method which minimizes the energy norm of the algebraic error using this convenient balance we also prove the efficiency of our a posteriori estimates i e we show that they also represent a lower bound up to a generic constant for the overall energy error a local version of this result is also stated this makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

stability and convergence of sequential methods for coupled flow and geomechanics drained and undrained splits abstract we perform a stability and convergence analysis of sequential methods for coupled flow and geomechanics in which the mechanics sub problem is solved first we consider slow deformations so that inertia is negligible and the mechanical problem is governed by an elliptic equation we use biot s self consistent theory to obtain the classical parabolic type flow problem we use a generalized midpoint rule parameter between 0 and 1 time discretization and consider two classical sequential methods the drained and undrained splits the von neumann method provides sharp stability estimates for the linear poroelasticity problem the drained split with backward euler time discretization 1 is conditionally stable and its stability depends only on the coupling strength and it is independent of time step size the drained split with the midpoint rule 0 5 is unconditionally unstable the mixed time discretization with 1 0 for mechanics and 0 5 for flow has the same stability properties as the backward euler scheme the von neumann method indicates that the undrained split is unconditionally stable when 0 5 we extend the stability analysis to the nonlinear regime poro elastoplasticity via the energy method it is well known that the drained split does not inherit the contractivity property of the continuum problem thereby precluding unconditional stability for the undrained split we show that it is b stable therefore unconditionally stable at the algorithmic level when 0 5 we also analyze convergence of the drained and undrained splits and derive the a priori error estimates from matrix algebra and spectral analysis we show that the drained split with a fixed number of iterations is not convergent even when it is stable the undrained split with a fixed number of iterations is convergent for a compressible system i e finite biot modulus for a nearly incompressible system i e very large biot modulus the undrained split loses first order accuracy and becomes non convergent in time we also study the rate of convergence of both splits when they are used in a fully iterated sequential scheme when the medium permeability is high or the time step size is large which corresponds to a high diffusion of pressure the error amplification of the drained split is lower and therefore converges faster than the undrained split the situation is reversed in the case of low permeability and small time step size we provide numerical experiments supporting all the stability and convergence estimates of the drained and undrained splits in the linear and nonlinear regimes we also show that our spatial discretization finite volumes for flow and finite elements for mechanics removes the well documented spurious instability in consolidation problems at early times

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

on the fixed stress split scheme as smoother in multigrid methods for coupling flow and geomechanics the fixed stress split method has been widely used as solution method in the coupling of flow and geomechanics in this work we analyze the behavior of an inexact version of this algorithm as smoother within a geometric multigrid method in order to obtain an efficient monolithic solver for biot s problem this solver combines the advantages of being a fully coupled method with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm moreover the fixed stress split smoother is based on the physics of the problem and therefore all parameters involved in the relaxation are based on the physical properties of the medium and are given a priori a local fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained the proposed geometric multigrid algorithm is used to solve several tests in semi structured triangular grids in order to show the good behavior of the method and its practical utility

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational results

optimization of the multigrid convergence rate on semi structured meshes by local fourier analysis in this paper a local fourier analysis for multigrid methods on tetrahedral grids is presented different smoothers for the discretization of the laplace operator by linear finite elements on such grids are analyzed a four color smoother is presented as an efficient choice for regular tetrahedral grids whereas line and plane relaxations are needed for poorly shaped tetrahedra a novel partitioning of the fourier space is proposed to analyze the four color smoother numerical test calculations validate the theoretical predictions a multigrid method is constructed in a block wise form by using different smoothers and different numbers of pre and post smoothing steps in each tetrahedron of the coarsest grid of the domain some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

quadrature formulas associated with rational modifications of the chebyshev weight functions making use of the connection between quadrature formulas on the unit circle and the interval 1 1 recently established in 1 explicit expressions for the nodes and the weights of the gaussian formulas associated with rational modifications of the chebyshev weight functions are given some illustrative numerical examples are also presented

adaptive variational multiscale methods based on a posteriori error estimation energy norm estimates for elliptic problems we develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution the fine scale is approximated by a sum of solutions to decoupled localized problems which are solved numerically on a fine grid partition of a patch of coarse grid elements the sizes of the patches of elements may be increased to control the error caused by localization we derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters the coarse grid mesh size the fine grid mesh size and the sizes of the patches based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters finally we show how the method works in practice by presenting various numerical examples

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

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

mixed finite element methods for linear elasticity with weakly imposed symmetry in this paper we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima tions to both stresses and displacements the methods are based on a modified form of the hellinger reissner variational principle that only weakly imposes the symmetry condition on the stresses although this approach has been previously used by a number of authors a key new ingredient here is a constructive derivation of the elasticity complex starting from the de rham complex by mimicking this construction in the discrete case we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de rham complex these elements appear to be simpler than the ones previously derived for example we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field

a residual based error estimator for mortar finite element discretizations a residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed in particular we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using p1 approximations in each subdomain additionally a residual based error estimator for crouzeix raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation it is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the lagrange multiplier is taken into account

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

convergence analysis of multirate fixed stress split iterative schemes for coupling flow with geomechanics abstract we consider multirate iterative schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium the multirate iterative coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we adapt the fixed stress split algorithm that decouples the flow and mechanics equations for the multirate case and perform an iteration between the two problems until convergence we provide a fully discrete scheme that uses backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics our analysis is based on studying the equations satisfied by the difference of iterates and using banach contraction argument to prove that the corresponding scheme is a fixed point contraction the analysis provides the value of an adjustable coefficient used in the proposed iterative coupling algorithms furthermore we show that the converged quantities satisfy the variational weak form for the coupled discrete system

a partially parallel in time fixed stress splitting method for biot s consolidation model in this work we study the parallel in time iterative solution of coupled flow and geomechanics in porous media modelled by a two field formulation of biot s equations in particular we propose a new version of the fixed stress splitting method which has been widely used as solution method of these problems this new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization the method is partially parallel in time we present a rigorous convergence analysis of the method and numerical experiments to demonstrate the robust behaviour of the algorithm

fourier analysis for multigrid methods on triangular grids in this paper a local fourier analysis technique for multigrid methods on triangular grids is presented the analysis is based on an expression of the fourier transform in new coordinate systems both in space variables and in frequency variables associated with reciprocal bases this tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids different smoothers for the discrete laplace operator obtained with linear finite elements are analyzed a new three color smoother has been studied and has proven to be the best choice for near equilateral triangles it is also shown that the block line smoothers are more appropriate for irregular triangles numerical test calculations validate the theoretical predictions

exact de rham sequences of spaces defined on macro elements in two and three spatial dimensions this paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure the main requirement is that the agglomerates share faces that have closed boundaries composed of 1 d edges the spaces resulting from the agglomerated elements are subspaces of the original de rham sequence of h 1 conforming h mathbf curl conforming h mathbf div conforming and piecewise constant spaces associated with an unstructured fine mesh the procedure can be recursively applied so that a sequence of nested de rham complexes can be constructed as an illustration we generate coarser spaces from the sequence corresponding to the lowest order nedelec spaces lowest order raviart thomas spaces and for piecewise linear h 1 conforming spaces all in three dimensions the resulting v cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case

mortar finite element discretization for the flow in a nonhomogeneous porous medium abstract darcy s equations model the flow of a viscous incompressible fluid in a rigid porous medium one of the parameters of the system depends on the permeability of the medium and when this medium is not homogeneous the variations of the parameter could be very high to handle this phenomenon we propose a discretization of the model that relies on the mortar finite element method indeed the idea is to construct a decomposition of the domain such that the permeability is constant on each element of the partition and to use independent meshes on the different subdomains we perform the a priori and a posteriori analysis of this discretization and present some numerical experiments which are in good coherency with the results of the analysis

on numerical improvement of the first kind gauss chebyshev quadrature rules one of the integration methods of the equality type is gauss chebyshev quadrature rule which is in the following form 11f x 1 x 2dx pn k 1nfcos 2k 1 p2n 2 p2 2 n 2n f 2 n h 1 h 1 according to gauss quadrature rules the precision degree of above formula is the highest i e 2n 1 hence it is not possible to increase the precision degree of gauss chebyshev integration formulas anymore in this way we present a matrix proof for this matter but on the other hand we claim that we can improve the above formula numerically to do this we consider the integral bounds as two unknown variables this causes to numerically be extended the monomial space f x x j from j 0 1 2n 1 to j 0 1 2n 1 this means that we have two monomials more than gauss chebyshev integration method in other words we give an approximate formula as abf x 1 x 2dx i 1nw if x i in which a b and w 1 w 2 w n and x 1 x 2 x n are all unknowns and the formula is almost exact for the monomial basis f x x j j 0 1 2n 1 some important examples are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual gauss chebyshev nodes and weights let us add that in this part we have also some wonderful 2 point formulas that are comparable with 71 point formulas of gauss chebyshev quadrature rules in average