spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

a multi grid technique for coupling fluid flow with porous media flow abstract in this paper we consider the coupling of fluid flow with porous media flow a multi grid finite element method for the coupled stokes darcy problem with the beavers joseph interface condition is proposed and discussed the optimal error estimates are obtained numerical experiment is given to verify the theoretical analysis and indicate the accuracy and efficiency of the multi grid method

direct minimization of the least squares spectral element functional part i direct solver this paper describes an equivalent but improved least squares formulation for the numerical approximation of the solution of partial differential equations instead of using variational analysis to impose the conditions for minimizing the residual the residuals are minimized directly thus leading to a method we will denote by direct minimization dm dm circumvents setting up the normal equations which consists of matrix matrix multiplications matrix matrix multiplications are expensive may lead to loss of accuracy and destroy the sparsity pattern present in the original system the condition number of the dm formulation is the square root of the condition number which would be obtained if variational analysis was employed an element by element procedure will be presented which allows for parallelization of dm a computational comparison between dm and the conventional least squares formulation based on variational analysis will be presented

least squares spectral collocation for discontinuous and singular perturbation problems a least squares spectral collocation scheme for discontinuous problems is proposed for the first derivative operator the domain is decomposed in subintervals where the jumps are imposed at the discontinuities equal order polynomials are used on all subdomains for the discretization spectral collocation with chebyshev polynomials is employed fast fourier transforms are now available the collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least squares the solution technique will only involve symmetric positive definite linear systems this approach is further extended to singular perturbation problems where least squares are used for stabilization by a suitable decomposition of the domain the boundary layer is well resolved numerical simulations confirm the high accuracy of our spectral least squares scheme

general relaxation methods for initial value problems with application to multistep schemes recently an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial value problems using runge kutta methods we generalize this approach to multistep methods including all general linear methods of order two or higher and many other classes of schemes we prove the existence of a valid relaxation parameter and high order accuracy of the resulting method in the context of general equations including but not limited to conservative or dissipative systems the theory is illustrated with several numerical examples

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

an efficient ensemble algorithm for numerical approximation of stochastic stokes darcy equations abstract we propose and analyze an efficient ensemble algorithm for fast computation of multiple realizations of the stochastic stokes darcy model with a random hydraulic conductivity tensor the algorithm results in a common coefficient matrix for all realizations at each time step making solving the linear systems much less expensive while maintaining comparable accuracy to traditional methods that compute each realization separately moreover it decouples the stokes darcy system into two smaller sub physics problems which reduces the size of the linear systems and allows parallel computation of the two sub physics problems we prove the ensemble method is long time stable and first order in time convergent under a time step condition and two parameter conditions numerical examples are presented to support the theoretical results and illustrate the application of the algorithm

linear conservation laws for odes abstract some physical systems described by odes have quantities that are conserved as the system evolves runge kutta formulas and linear multistep methods preserve linear conservation laws automatically a code for the integration of odes involves a number of algorithms in addition to the basic formula it is shown here that popular codes preserve linear conservation laws if they are used properly an application is made to differential algebraic equations of index 2 arising in the solution of the navier stokes equation by the method of lines

positive numerical integration methods for chemical kinetic systems abstract chemical kinetics conserves mass and renders nonnegative solutions a good numerical simulation would ideally produce mass balanced positive concentration vectors many time stepping methods are mass conservative however unconditional positivity restricts the order of a traditional method to one the projection method presented in this paper ensures mass conservation and positivity first a numerical approximation is computed with one step of a mass preserving traditional scheme if there are negative components the nearest vector in the reaction simplex is found by solving a quadratic optimization problem this vector is shown to better approximate the true solution a simpler version involves just one projection step and stabilizes the reaction simplex this technique works best when the underlying time stepping scheme favors positivity projected methods are more accurate than clipping and allow larger time steps for kinetic systems which are unstable outside the positive quadrant

a direct solver for the least squares spectral collocation system on rectangular elements for the incompressible navier stokes equations a least squares spectral collocation scheme for the stokes and incompressible navier stokes equations is proposed the original domain is decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced in the least squares sense the collocation conditions and the interface conditions lead to overdetermined systems these systems are directly solved by qr decomposition of the underlying matrices by numerical simulations it is shown that the direct method leads to better results than the approach with normal equations furthermore it is shown that the condition numbers can be reduced by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero finally our scheme is successfully applied to the regularized and lid driven cavity flow problems

mass and momentum conservation of the least squares spectral collocation method for the navier stokes equations from the literature it is known that the least squares spectral element method lssem for the stationary stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum furthermore it is known that the least squares spectral collocation method lsscm leads to superior conservation of mass and momentum for the stationary stokes equations in the present paper we consider mass and momentum conservation of the lsscm for time dependent stokes and navier stokes equations we observe that the lsscm leads to improved conservation of mass and momentum for these problems furthermore the lsscm leads to the well known time dependent profiles for the velocity and the pressure profiles to obtain these results we use only a few elements each with high polynomial degree avoid normal equations for solving the overdetermined linear systems of equations and introduce the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with the least squares spectral collocation scheme to discretize the internal flow problems

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

constraint preserving integrators for general nonlinear higher index daes in the last few years there has been considerable research on numerical methods for differential algebraic equations daes f x x t 0 where f x is identically singular the index provides one measure of the singularity of a dae most of the numerical analysis literature on daes to date has dealt with daes with indices no larger than three even in this case the systems were often assumed to have a special structure recently a numerical method was proposed that could in principle be used to integrate general unstructured higher index solvable daes however that method did not preserve constraints this paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index daes

least squares finite element formulations for viscous incompressible and compressible fluid flows abstract we present least squares based finite element formulations for the numerical solution of viscous fluid flows governed by the navier stokes equations as an alternate approach to the well known weak form galerkin finite element formulations the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix we address the issue of norm equivalence of the least squares functional and its implications on the resulting finite element model in particular we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model the latter measured in terms of c k regularity across inter element boundaries we show through numerical examples that for the traditional c 0 basis such a compromise is possible when high p levels are used to span the finite element spaces when a low p level solution is desired guidelines are presented to obtain a reliable least squares collocation solution numerical examples are presented to demonstrate the high and low p level approaches these include incompressible flow past two circular cylinders in a side by side arrangement for gap sizes s d 2 0 and 0 85 incompressible flow past a square cylinder and subsonic transonic and supersonic compressible flow past a circular cylinder in addition we present a discontinuous least squares formulation where c k regularity across inter element boundaries is enforced in a weak sense through the least squares functional allowing for h and p type non conformities in the computational domain

stabilization of daes and invariant manifolds many methods have been proposed for the stabilization of higher index differential algebraic equations daes such methods often involve constraint differentiation and problem stabilization thus obtaining a stabilized index reduction a popular method is baumgarte stabilization but the choice of parameters to make it robust is unclear in practice here we explain why the baumgarte method may run into trouble we then show how to improve it we further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature our approach is to i consider stabilization of odes with invariants ii discretize the stabilizing term in a simple way generally different from the ode discretization and iii use orthogonal projections whenever possible the best methods thus obtained are related to methods of coordinate projection we discuss them and make concrete algorithmic suggestions

coupled generalized nonlinear stokes flow with flow through a porous medium in this article we analyze the flow of a fluid through a coupled stokes darcy domain the fluid in each domain is non newtonian modeled by the generalized nonlinear stokes equation in the free flow region and the generalized nonlinear darcy equation in the porous medium a flow rate is specified along the inflow portion of the free flow boundary we show existence and uniqueness of a variational solution to the problem we propose and analyze an approximation algorithm and establish a priori error estimates for the approximation

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

on the solution of coupled stokes darcy model with beavers joseph interface condition abstract in this paper we first get the uniqueness result of the possible solution to the steady state coupled stokes darcy model with beavers joseph interface condition for any physical parameters especially for any 0 then we show the existence of solutions for any 0 by using galerkin method furthermore we analyze the error of the corresponding coupled finite element scheme and derive the optimal error estimates

efficient energy preserving integrators for oscillatory hamiltonian systems in this paper we focus our attention on deriving and analyzing an efficient energy preserving formula for the system of nonlinear oscillatory or highly oscillatory second order differential equations q t mq t f q t where m is a symmetric positive semi definite matrix with m 1 and f q q u q is the negative gradient of a real valued function u q this system is a hamiltonian system with the hamiltonian h p q 1 2 p t p 1 2 q t mq u q the energy preserving formula exactly preserves the hamiltonian we analyze in detail the properties of the energy preserving formula and propose new efficient energy preserving integrators in the sense of numerical implementation the convergence analysis of the fixed point iteration is presented for the implicit integrators proposed in this paper it is shown that the convergence of implicit average vector field methods is dependent on m whereas the convergence of the new energy preserving integrators is independent of m the fermi pasta ulam problem and the sine gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well known average vector field methods in the scientific literature

numerical analysis of the navier stokes darcy coupling we consider a differential system based on the coupling of the navier stokes and darcy equations for modeling the interaction between surface and porous media flows we formulate the problem as an interface equation we analyze the associated nonlinear steklov poincare operators and we prove its well posedness we propose and analyze iterative methods to solve a conforming finite element approximation of the coupled problem

least squares spectral elements applied to the stokes problem least squares spectral element methods are based on two important and successful numerical methods spectral hp element methods and least squares finite element methods in this respect least squares spectral element methods seem very powerful since they combine the generality of finite element methods with the accuracy of the spectral methods and also have the theoretical and computational advantages of the least squares methods these features make the proposed method a competitive candidate for the solution of large scale problems arising in scientific computing in order to demonstrate its competitiveness the method has been applied to an analytical problem and the theoretical optimal and suboptimal a priori estimates have been confirmed for various boundary conditions moreover the exponential convergence rates typical for a spectral element discretization have also been confirmed the comparison with the classical galerkin spectral element method revealed that the least squares spectral element method is as accurate as the galerkin method for the smooth model problem

hp adaptive least squares spectral element method for hyperbolic partial differential equations this paper describes a hp adaptive spectral element formulation which is used to discretize the weak formulation obtained by minimizing the residuals in the l 2 norm the least squares error indicator will be briefly discussed refinement of the numerical approximation is based on an estimate of the regularity of the underlying exact solution if the underlying exact solution is sufficiently smooth polynomial enrichment is employed in areas with limited regularity h refinement is used for this purpose the sobolev regularity is estimated functionally and geometrically non conforming neighbouring elements are patched together using so called mortar elements results of this approach are compared to uniform h and p refinement for a linear advection equation

the analogue of grad div stabilization in dg methods for incompressible flows limiting behavior and extension to tensor product meshes abstract grad div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems for mitigating velocity errors that are sometimes called poor mass conservation such errors arise due to the relaxation of the divergence constraint in classical mixed methods and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures in this contribution an analogue of grad div stabilization for discontinuous galerkin methods is studied here the key is the penalization of the jumps of the normal velocities over facets of the triangulation which controls the measure valued part of the distributional divergence of the discrete velocity solution our contribution is twofold first we characterize the limit for arbitrarily large penalization parameters which shows that the stabilized nonconforming discontinuous galerkin methods remain robust and accurate in this limit second we extend these ideas to the case of non simplicial meshes here broken grad div stabilization must be used in addition to the normal velocity jump penalization in order to get the desired pressure robustness effect the analysis is performed for the stokes equations and more complex flows and crouzeix raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings

a second order pseudo transient method for steady state problems this article gives a second pseudo transient method for a special system of nonlinear equations which arises from chemical reaction rate equations this method uses a special second order rosenbrock method as the discrete difference scheme which satisfies a linear conservation law moreover it adaptively adjusts the time step in inverse proportion to an arithmetic mean of the current residual and the previous residual at every iteration step for a singular system of nonlinear equations under some standard assumptions local convergence of the new method is addressed finally some promise numerical results are also reported

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

a homotopy approach for stabilizing single input systems with control structure constraints a linear state feedback gain vector used in the control of a single input dynamical system may be constrained because of the way feedback is realized some examples of feedback realizations which impose constraints on the gain vector are static output feedback constant gain feedback for several operating points of a system and two controller feedback we consider a general class of problems of stabilization of single input dynamical systems with such structural constraints and give a numerical method to solve them each of these problems is cast into a problem of solving a system of equalities and inequalities in this formulation the coefficients of the quadratic and linear factors of the closed loop characteristic polynomial are the variables to solve the system of equalities and inequalities a continuous realization of the gradient projection method and a barrier method are used under the homotopy framework our method is illustrated with an example for each class of control structure constraint

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

conservation of mass and momentum of the least squares spectral collocation scheme for the stokes problem from the literature it is known that spectral least squares schemes perform poorly with respect to mass conservation and compensate this lack by a superior conservation of momentum this should be revised since the here presented new least squares spectral collocation scheme leads to an outstanding performance with respect to conservation of momentum and mass the reasons can be found in using only a few elements each with high polynomial degree avoiding normal equations for solving the overdetermined linear systems of equations and by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with our least squares spectral collocation scheme to discretize the internal flow problems

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints the equations of motion of mechanical multibody systems with algebraic constraints are of index 3 and therefore not directly solvable by standard ode or dae methods reducing the index by different

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

a two grid decoupling method for the mixed stokes darcy model in this paper we consider the mixed stokes darcy problem which describes a fluid flow coupled with a porous media we present a modified two grid method for decoupling this mixed model stability is proved and optimal error estimates are derived the numerical results show that the modified two grid method is effective and has the same accuracy as the coupling scheme when we choose h h 2

analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater surface water flows the most effective simulations of the multiphysics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes partitioned methods which solve the coupled problem by successively solving the subphysics problems have recently been studied for the stokes darcy coupling with convergence established over bounded time intervals with constants growing exponentially in t this report analyzes and tests two such partitioned noniterative domain decomposition methods for the fully evolutionary stokes darcy problem under a modest time step restriction of the form delta t leq c where c c physical parameters we prove unconditional long time over 0 leq t infty stability of both partitioned methods from this we derive an optimal error estimate that is uniform in time over 0 leq t infty

non negative solutions of odes this paper discusses procedures for enforcing non negativity in a range of codes for solving ordinary differential equations odes this codes implement both one step and multistep methods all of which use continuous extensions and have event finding capabilities examples are given

numerical analysis of two grad div stabilization methods for the time dependent stokes darcy model abstract we aim to present two algorithms for the non stationary stokes darcy model the first one is the standard grad div stabilization scheme the other one is a modular grad div based on the standard backward euler code which does not crash or slow down for large value grad div parameters both algorithms cannot only improve the efficiency and accuracy of calculation but also can improve mass conservation while the modular algorithm can be better we give a complete theoretical analysis of the stability and error estimations of the algorithms finally the theoretical results are verified by numerical experiments and the advantages of adding grad div stabilization terms are demonstrated

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

decoupled schemes for a non stationary mixed stokes darcy model we study numerical methods for solving a non stationary mixed stokes darcy problem that models coupled fluid flow and porous media flow a decoupling approach based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model error estimates are derived and numerical experiments are conducted to demonstrate the computational effectiveness of the decoupling approach

a strongly conservative finite element method for the coupling of stokes and darcy flow we consider a model of coupled free and porous media flow governed by stokes and darcy equations with the beavers joseph saffman interface condition this model is discretized using divergence conforming finite elements for the velocities in the whole domain discontinuous galerkin techniques and mixed methods are used in the stokes and darcy subdomains respectively this discretization is strongly conservative in h d i v w and we show convergence numerical results validate our findings and indicate optimal convergence orders

partitioned time stepping method for fully evolutionary stokes darcy flow with beavers joseph interface conditions in this report a partitioned time stepping algorithm for transient flow in a matrix domain coupled to a free flow in embedded conduits is analyzed the coupled flow is modeled by the fully evolutionary stokes darcy problem this method requires solving only one uncoupled stokes and darcy subphysics and subdomain per time step on the interface between the matrix and conduit beavers joseph interface conditions instead of the simplified beavers joseph saffman condition are imposed under a modest time step restriction of the form triangle t leq c where c c physical parameters we prove stability of the method we also derive error estimates numerical tests illustrate the validity of the theoretical results

design of software for odes we discuss the design of software that is easy to use for simple problems but still capable of solving complicated problems nowadays a design should accommodate related but fundamentally different tasks such as solving daes and ddes in addition to ivps and bvps for odes major issues are illustrated with the matlab ode suite and selected solvers written in fortran 90 and maple

on the convergence rate of grad div stabilized taylor hood to scott vogelius solutions for incompressible flow problems abstract it was recently proven in case et al 2010 2 that under mild restrictions grad div stabilized taylor hood solutions of navier stokes problems converge to the scott vogelius solution of that same problem however even though the analytical rate was only shown to be 1 2 where is the stabilization parameter the computational results suggest the rate may be improvable to 1 we prove herein the analytical rate is indeed 1 and extend the result to other incompressible flow problems including leray and mhd numerical results are given that verify the theory

dg approximation of coupled navier stokes and darcy equations by beaver joseph saffman interface condition in this work we couple the incompressible steady navier stokes equations with the darcy equations by means of the beaver joseph saffman s condition on the interface under suitable smallness conditions on the data we prove existence of a weak solution as well as some a priori estimates we establish local uniqueness when the data satisfy additional smallness restrictions then we propose a discontinuous galerkin scheme for discretizing the equations and do its numerical analysis

on the least squares method abstract it is shown that the theoretical basis of the general least squares method is the bounded inverse theorem this explains why the least squares finite element method lsfem within one mathematical computational framework without any special treatment can provide numerical solutions for all types of partial differential equations error estimates of lsfem for general and elliptic first order systems of partial differential equations are established as an example the incompressible stokes equations are considered the principal part of the stokes equations in the first order velocity pressure vorticity formulation consists of two div curl systems this knowledge is employed to derive all permissible combinations of non standard boundary conditions for the stokes equations and to show that under these conditions the stokes operator is bounded below in the full h1 sense and hence the corresponding lsfem is optimal the numerical results are also given to support this conclusion

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

spectral collocation schemes on the unit disc two spectral collocation schemes on the unit disc are presented the first one is based on the mapping of gordon and hall here the unit square is directly mapped onto the unit disc by means of an interpolation technique unlike other poisson solvers on the unit disc no polar coordinates are involved hence the usual problems with the singularity of polar coordinates are avoided this is also shown for more complex geometries the second method is based on a diameter approach where the collocation nodes are no more clustering in the center numerical results are presented which demonstrate the high accuracy of our new spectral collocation schemes

on the preservation of invariants by explicit runge kutta methods a new strategy to preserve invariants in the numerical integration of initial value problems with explicit runge kutta methods is presented it is proved that this technique retains the order of the original method has an easy and cheap implementation and can be used in adaptive runge kutta codes some numerical experiments with the classical code of dormand and prince dopri5 4 based on a pair of embedded methods with orders 5 and 4 are presented to show the behavior of the new method for several problems which possess invariants