a characterization of hybridized mixed methods for second order elliptic problems in this paper we give a new characterization of the approximate solution given by hybridized mixed methods for second order self adjoint elliptic problems we apply this characterization to obtain an explicit formula for the entries of the matrix equation for the lagrange multiplier unknowns resulting from hybridization we also obtain necessary and sufficient conditions under which the multipliers of the raviart thomas and the brezzi douglas marini methods of similar order are identical

discrete spline methods for solving two point fractional bagley torvik equation a new discrete spline method is developed to solve the two point fractional bagley torvik equation the method is based on discrete spline function and a nonstandard grunwald letnikov difference nsgd and the weighted and shifted grunwald letnikov difference wsgd operators to approximate the fractional derivative bounds for grunwald letnikov weights are considered convergence analysis is discussed and a class of second order and third order methods are obtained illustrative examples are presented to validate the practical usefulness of the methods

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

explicit methods for fractional differential equations and their stability properties the use of explicit methods in the numerical treatment of differential equations of fractional order is an area not yet widely investigated in this paper stability properties of some multistep methods of explicit type are investigated and new methods with larger intervals of stability are proposed some numerical experiments are presented in order to validate theoretical results

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

a three dimensional mathematical model of the human masticatory system predicting maximum possible bite forces abstract a three dimensional mathematical model of the human masticatory system containing 16 muscle forces and two joint reaction forces is described the model allows simulation of static bite forces and concomitant joint reaction forces for various bite point locations and mandibular positions the system parameters for the model were obtained from a cadaver head maximum possible bite forces were computed using optimization techniques the optimization criterion we used was the minimizing of the relative activity of the most active muscle the model predicts that at each specific bite point bite forces can be generated in a wide range of directions and that the magnitude of the maximum bite force depends on its direction the relationship between bite force direction and its maximum magnitude depends on bite point location and mandibular position in general the direction of the largest possible bite force does not coincide with the direction perpendicular to the occlusal plane

superconvergence of the numerical traces of discontinuous galerkin and hybridized methods for convection diffusion problems in one space dimension in this paper we uncover and study a new superconvergence property of a large class of finite element methods for one dimensional convection diffusion problems this class includes discontinuous galerkin methods defined in terms of numerical traces discontinuous petrov galerkin methods and hybridized mixed methods we prove that the so called numerical traces of both variables superconverge at all the nodes of the mesh provided that the traces are conservative that is provided they are single valued in particular for is local discontinuous galerkin method we show that the superconvergence is order 2p 1 when polynomials of degree at most p are used extensive numerical results verifying our theoretical results are displayed

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

application and validation of a three dimensional mathematical model of the human masticatory system in vivo abstract a previously described three dimensional mathematical model of the human masticatory system predicting maximum possible bite forces in all directions and the recruitment patterns of the masticatory muscles necessary to generate these forces was validated in in vivo experiments the morphological input parameters to the model for individual subjects were collected using mri scanning of the jaw system experimental measurements included recording of maximum voluntary bite force magnitude and direction and surface emg from the temporalis and masseter muscles for bite forces with an angle of 0 10 and 20 relative to the normal to the occlusal plane the predicted maximum possible bite forces were between 0 9 and 1 2 times the measured ones and the average ratio of measured to predicted maximum bite force was close to unity the average measured and predicted muscle recruitment patterns showed no striking differences nevertheless some systematic differences dependent on the bite force direction were found between the predicted and the measured maximum possible bite forces in a second series of simulations the influence of the direction of the joint reaction forces on these errors was studied the results suggest that they were caused primarily by an improper determination of the joint force directions

an optimal galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere the kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically conducting liquid one of its main uses is as a proof of concept tool to test hypotheses about self exciting dynamo action indeed it provided the very first quantitative evidence for the possibility of the geodynamo despite its utility due to the requirement of resolving fine structures historically numerical work has proven difficult and reported solutions were often plagued by poor convergence in this paper we demonstrate the numerical superiority of a galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere after adopting a poloidal toroidal decomposition and expanding in spherical harmonics we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials each basis function is constructed from a terse sum of one sided jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior but is everywhere infinitely differentiable including at the origin this galerkin method exhibits more rapid convergence for a given problem size than any other scheme hitherto reported as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature in the axisymmetric flows we consider in this paper at a magnetic reynolds number of o 100 a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions alternatively 4 significant figures requires 20 radial functions the terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics we exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of bachtiar et al 4 thereby verifying a counter example to the zel dovich anti dynamo theorem in a spherical geometry

a discontinuous petrov galerkin method with lagrangian multipliers for second order elliptic problems we present a discontinuous petrov galerkin dpg method for the finite element discretization of second order elliptic boundary value problems the novel approach emanates from a one element weak formulation of the differential problem this procedure which is typical of discontinuous galerkin dg methods is based on introducing variables defined in the interior and on the boundary of the element the interface variables are suitable lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative thus providing the proper connection between neighboring elements the internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole lagrangian multipliers a stability and convergence analysis of the novel formulation is carried out and its connection with mixed hybrid and dg methods is explored numerical tests on several benchmark problems are included to validate the convergence performance and the flux conservation properties of the dpg method

multigrid for the mortar finite element method a multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed these problems are posed on domains partitioned into subdomains each of which is independently triangulated in a multilevel fashion the multilevel mortar finite element spaces based on such triangulations which need not align across subdomain interfaces are in general not nested suitable grid transfer operators and smoothers are developed which lead to a variable v cycle preconditioner resulting in a uniformly preconditioned algebraic system computational results illustrating the theory are also presented

error analysis of variable degree mixed methods for elliptic problems via hybridization a new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree raviart thomas method for second order elliptic problems the approach gives error estimates for the lagrange multipliers without using error estimates for the other variables error estimates for the primal and flux variables then follow from those for the lagrange multipliers in contrast traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the lagrange multipliers the new approach not only gives new error estimates for the new variable degree raviart thomas method but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates the error analysis is achieved by using a variational characterization of the lagrange multipliers wherein the other unknowns do not appear this approach can be applied to other hybridized mixed methods as well

spectral approximation on the unit ball spectral approximation by polynomials on the unit ball is studied in the frame of the sobolev spaces w s p mathbb b d 1 p infty the main results give sharp estimates on the order of approximation by polynomials in the sobolev spaces and explicit construction of approximating polynomials one major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the sobolev space w 2 s mathbb b d as an application a direct and efficient spectral galerkin method based on our orthogonal polynomials is proposed for the second and fourth order elliptic equations on the unit ball its optimal error estimates are explicitly derived for both procedures in the sobolev spaces and finally numerical examples are presented to illustrate the theoretic results

the hellan herrmann johnson method some new error estimates and postprocessing analyse du comportement de la methode mixte de hellan herrmann johnson pour la resolution du probleme biharmonique 2 f etablissement d un resultat de superconvergence

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

stability regions for linear fractional differential systems and their discretizations this paper concerns with basic stability properties of linear autonomous fractional differential and difference systems involving derivative operators of the riemann liouville type we derive stability regions for special discretizations of the studied fractional differential systems including a precise description of their asymptotics our analysis particularly shows that discretizations based on backward differences can retain the key qualitative properties of underlying fractional differential systems in addition we introduce the backward discrete laplace transform and employ some of its properties as the main proof tool

analytical solution of the bagley torvik equation by adomian decomposition method abstract the fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials motion of a large thin plate in a newtonian fluid creep and relaxation functions for viscoelastic materials the pi d controller for the control of dynamical systems etc phenomena in electromagnetics acoustics viscoelasticity electrochemistry and material science are also described by differential equations of fractional order the solution of the differential equation containing fractional derivative is much involved instead of application of the existing methods an attempt has been made in the present analysis to obtain the solution of bagley torvik equation r l bagley p j torvik on the appearance of the fractional derivative in the behavior of real materials asme j appl mech 51 1984 294 298 i podlubny fractional differential equations academic press san diego ca usa 1999 by the relatively new adomian decomposition method the results obtained by this method are then graphically represented and then compared with those available in the work of podlubny i podlubny fractional differential equations academic press san diego ca usa 1999 a good agreement of the results is observed

an application of one sided jacobi polynomials for spectral modeling of vector fields in polar coordinates a spectral tau method is proposed for solving vector field equations defined in polar coordinates the method employs one sided jacobi polynomials as radial expansion functions and fourier exponentials as azimuthal expansion functions all the regularity requirements of the vector field at the origin and the physical boundary conditions at a circumferential boundary are exactly satisfied by adjusting the additional tau coefficients of the radial expansion polynomials of the highest order the proposed method is applied to linear and nonlinear dispersive time evolution equations of hyperbolic type describing internal kelvin and poincare waves in a shallow stratified lake on a rotating plane

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented

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

the approximation of vector functions and their derivatives on the sphere a vector basis is defined for vector functions that are given on the sphere it is shown that the basis is orthogonal that it is complete and that the convergence properties of the spectral representation are determined by the smoothness of the vector function in cartesian coordinates the method for analyzing a vector function in terms of this basis uses the methods for analyzing scalar functions in terms of surface harmonics a vector function that is smooth in cartesian coordinates and nonzero at the pole will be discontinuous in spherical coordinates this is a result of the discontinuous spherical coordinate system rather than the vector function itself this in turn induces singularities in certain surface derivatives on the sphere even though the function is bounded and differentiable in cartesian coordinates many of the individual terms in a partial differential equation on the sphere are unbounded at the poles however the cancellation between these terms is sufficient to yield a bounded resu

approximate solution of the bagley torvik equation by hybridizable discontinuous galerkin methods in this paper we introduce a hybridizable discontinuous galerkin method for numerically solving a boundary value problem associated with the bagley torvik equation that arises in the study of the motion of a plate immersed in a newtonian fluid one of the main features of these methods is that they are efficiently implementable since it is possible to eliminate all internal degrees of freedom and obtain a global linear system that only involves unknowns at the element interfaces we display the results of a series of numerical experiments to ascertain the performance of the method

features of human jaw design which maximize the bite force a linear programming model operating in a three dimensional coordinate system was written to study the effect on unilateral maximum bite forces mbfs of small changes in the design of human jaws under some conditions the mbf was larger on a more anterior tooth than a more posterior tooth for a given bite direction the mbf was increased i as the bite force was tilted forward ii as the jaw was opened if its direction was kept perpendicular to the occlusal plane of the lower jaw iii by placing teeth nearer the midline iv by raising the height of the teeth for forward bite forces and v by tilting forward the articular surface of the condyle equivalent to creating an articular eminence increasing the distance between the condyles spread the load more evenly between the two joints the articular eminence was the only design feature which seemed to be clearly related to increasing the mbf

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

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

the grunwald letnikov method for fractional differential equations this paper is devoted to the numerical treatment of fractional differential equations based on the grunwald letnikov definition of fractional derivatives finite difference schemes for the approximation of the solution are discussed the main properties of these explicit and implicit methods concerning the stability the convergence and the error behavior are studied related to linear test equations the asymptotic stability and the absolute stability of these methods are proved error representations and estimates for the truncation propagation and global error are derived numerical experiments are given

an analysis of the embedded discontinuous galerkin method for second order elliptic problems the embedded discontinuous galerkin methods are obtained from hybridizable discontinuous galerkin methods by a simple change of the space of the hybrid unknown in this paper we consider embedded methods for second order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous functions this change results in a significantly smaller stiffness matrix whose size and sparsity structure coincides with those of the stiffness matrix of the statically condensed continuous galerkin method it is shown that this computational advantage has to be balanced against the fact that the approximate solutions for the scalar variable and its flux lose each a full order of convergence indeed we prove that if polynomials of degree k ge1 are used for the original hybridizable discontinuous galerkin method its approximations to the scalar variable and its flux converge with order k 2 and k 1 respectively whereas those of the corresponding embedded discontinuous galerkin method converge with orders k 1 and k respectively only we also provide numerical results comparing the relative efficiency of the methods

analysis of a multiscale discontinuous galerkin method for convection diffusion problems we study a multiscale discontinuous galerkin method introduced in t j r hughes g scovazzi p bochev and a buffa comput meth appl mech engrg 195 2006 pp 2761 2787 that reduces the computational complexity of the discontinuous galerkin method seemingly without adversely affecting the quality of results for a stabilized variant we are able to obtain the same error estimates for the convection diffusion equation as for the usual discontinuous galerkin method we assess the stability of the unstabilized case numerically and find that the inf sup constant is positive bounded uniformly away from zero and very similar to that for the usual discontinuous galerkin method

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

on riemann and caputo fractional differences in this paper we define left and right caputo fractional sums and differences study some of their properties and then relate them to riemann liouville ones studied before by miller k s and ross b atici f m and eloe p w abdeljawad t and baleanu d and a few others also the discrete version of the q operator is used to relate the left and right caputo fractional differences a caputo fractional difference equation is solved the solution proposes discrete versions of mittag leffler functions

navier stokes spectral solver in a sphere the paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere a variational formulation of the navier stokes equations is adopted using a fractional step time discretization with the classical second order backward difference scheme combined with explicit extrapolation of the nonlinear term the resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors the spectral matrices for radial operators are characterized by a minimal sparsity diagonal stiffness and tridiagonal mass matrix closed form expressions of their nonzero elements are provided here for the first time showing that the condition number of the relevant matrices grows as the second power of the truncation order a new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the coriolis force in a rotating frame but requires a minimal coupling between the modal velocity components in the fourier space the numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution by construction these results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by kida and nakayama where the velocity field is represented in terms of poloidal and toroidal functions

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

hybridized globally divergence free ldg methods part i the stokes problem we devise and analyze a new local discontinuous galerkin ldg method for the stokes equations of incompressible fluid flow this optimally convergent method is obtained by using an ldg method to discretize a vorticity velocity formulation of the stokes equations and by applying a new hybridization to the resulting discretization one of the main features of the hybridized method is that it provides a globally divergence free approximate velocity without having to construct globally divergence free finite dimensional spaces only elementwise divergence free basis functions are used another important feature is that it has significantly less degrees of freedom than all other ldg methods in the current literature in particular the approximation to the pressure is only defined on the faces of the elements on the other hand we show that as expected the condition number of the schur complement matrix for this approximate pressure is of order h 2 in the mesh size h finally we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates

a hybridizable discontinuous galerkin method for fractional diffusion problems we study the use of the hybridizable discontinuous galerkin hdg method for numerically solving fractional diffusion equations of order alpha with 1 alpha 0 1 0 for exact time marching we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular thus if for each time t in 0 t t 0 t the approximations are taken to be piecewise polynomials of degree k ge 0 k 0 on the spatial domain varomega the approximations to u u in the l infty bigr 0 t l 2 varomega bigr l 0 t l2 norm and to nabla u u in the l infty bigr 0 t mathbf l 2 varomega bigr l 0 t l2 norm are proven to converge with the rate h k 1 hk 1 where h h is the maximum diameter of the elements of the mesh moreover for k ge 1 k 1 and quasi uniform meshes we obtain a superconvergence result which allows us to compute in an elementwise manner a new approximation for u u converging with a rate of sqrt log t h 2 alpha 1 h k 2 log th 2 1 hk 2

exact and discretized stability of the bagley torvik equation abstract the paper discusses stability and asymptotic properties of a two term linear fractional differential equation involving the bagley torvik equation as the particular case these properties are analysed for the exact as well as numerical solutions obtained from the grunwald letnikov discretization of the studied differential equation as the main results precise descriptions of the exact and discretized stability regions are presented including the decay rate of the solutions these results enable us among others to observe similarities and distinctions between the asymptotic behaviour of the classical and fractional damping models

a hybridizable discontinuous galerkin method for a class of fractional boundary value problems abstract in this paper we present a hybridizable discontinuous galerkin hdg method for solving a class of fractional boundary value problems involving caputo derivatives the hdg methods have the computational advantage of eliminating all internal degrees of freedom and the only globally coupled unknowns are those at the element interfaces furthermore the global stiffness matrix is tridiagonal symmetric and positive definite internal degrees of freedom are recovered at an element by element postprocessing step we carry out a series of numerical experiments to ascertain the performance of the proposed method

efficient spectral galerkin methods iv spherical geometries fast spectral galerkin algorithms are developed for elliptic equations on the sphere the algorithms are based on a double fourier expansion and have quasi optimal optimal up to a logarithmic term computational complexity numerical experiments indicate that they are significantly more efficient and or accurate when compared with the algorithms based on spherical harmonics and on finite difference extensions to problems in spherical layers and to vector equations are also discussed

on systems of linear fractional differential equations with constant coefficients abstract this paper deals with the study of linear systems of fractional differential equations such as the following system 1 y a x y b x where y is the riemann liouville or the caputo fractional derivative of order 0 2 a x a 11 x a 1 n x a n 1 x a nn x b x b 1 x b n x are matrices of known real functions in a way analogous to the usual case we show how a generalized matrix exponential function and certain fractional green function in connection with the mittag leffler type functions would allow us to obtain an explicit representation of the general solution to the system 1 when a is a constant matrix

a local post processing technique for improving the accuracy in mixed finite element approximations a general simple post processing technique is given in this paper under certain conditions if an algorithm approximating a function u also provides a good approximation for its gradient and for some locally defined projection pu then by this method a better approximation for u can be easily obtained essentially the leading term of the error between u and its original approximation can be computed it is shown how the method may be applied to various mixed type finite element methods

locally conservative fluxes for the continuous galerkin method the standard continuous galerkin cg finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations a much needed quantity in many applications we show how to overcome this shortcoming by using a two step postprocessing the first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous galerkin methods this computation is nonlocal in that it requires the solution of a symmetric positive definite system but the system is well conditioned independently of mesh size so it can be solved at asymptotically optimal cost the second step is a local element by element postprocessing of the cg solution incorporating the result of the first step this leads to a conservative flux approximation with continuous normal components this postprocessing applies for the cg method in its standard form or for a hybridized version of it we present the hybridized version since it allows easy handling of variable degree polynomials and hanging nodes furthermore we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the raviart thomas mixed method of corresponding degree

a projection based error analysis of hdg methods for timoshenko beams in this paper we give the first a priori error analysis of the hybridizable discontinuous galerkin hdg methods for timoshenko beams the analysis is based on the use of a projection especially designed to fit the structure of the numerical traces of the hdg method this property allows to prove in a very concise manner that the projection of the errors is bounded in terms of the distance between the exact solution and its projection the study of the influence of the stabilization function on the approximation is then reduced to the study of how they affect the approximation properties of the projection in a single element surprisingly and unlike any other discontinuous galerkin method this can be done without assuming any positivity property of the stabilization function of the hdg method we apply this approach to hdg methods using polynomials of degree k 0 in all the unknowns and show that the projection of the error in each of them superconverges with order k 2 when k 1 and converges with order 1 for k 0 as a result we show that the hdg methods converge with optimal order k 1 for all the unknowns and that they are free from shear locking finally we show that all the numerical traces converge with order 2k 1 numerical experiments validating these results are shown

spectral solvers for spherical elliptic problems a new family of direct spectral solvers for the 3d helmholtz equation in a spherical gap and inside a sphere for nonaxisymmetric problems is presented a variational formulation no collocation is adopted based on the fourier expansion and the associated legendre functions to represent the angular dependence over the sphere and using basis functions generated by legendre or jacobi polynomials to represent the radial structure of the solution in the present method boundary conditions on the polar axis and at the sphere center are not required and never mentioned by construction the spectral solution of the vector dirichlet problem is also considered by employing a transformation that uncouples the spherical components of the fourier modes and that is implemented here for the first time the condition numbers of the matrices involved in the scalar solvers are computed and the spectral convergence of all the proposed solution algorithms is verified by numerical tests

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

hierarchical mixed hybridized methods for elliptic problems abstract in this article we propose a computational procedure for the efficient implementation of dual mixed hybridized methods of arbitrary degree the procedure relies on the decomposition of the finite element spaces into a vertical p type hierarchy consisting of a lower order part and a defect correction coupled with an additional horizontal decomposition of the defect correction space for the vector variable based on the helmholtz principle an appropriate definition of the basis function set allows us to obtain a systematic substructuring of the block matrix system this property in turn naturally gives rise to an efficient implementation of the procedure through an approximate fixed point block iteration exploiting the equivalence between the principle of defect correction and the variational multiscale modeling theory we also devise and numerically validate a hierarchical a posteriori error estimator for dual mixed methods in hybridized form

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