subgrid stabilized defect correction method for a steady state natural convection problem in this paper we consider a subgrid stabilized defect correction method for the steady state natural convection problem we state the stabilities and error results for the defect step and the first correction step of the defect correction method the derived theoretical results are supported by several numerical examples

a finite element dynamical nonlinear subscale approximation for the low mach number flow equations in this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows the physical model is described by the low mach number equations which are obtained as a limit of the compressible navier stokes equations in the small mach number regime in contrast to the commonly used boussinesq approximation this model permits to take volumetric deformation into account although the former is more general than the latter both systems have similar mathematical structure and their numerical approximation can suffer from the same type of instabilities we propose a stabilized finite element approximation based on the variational multiscale method in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed modeling the subscale and taking its effect on the coarse scale problem into account results in a stable formulation the quality of the final approximation accuracy efficiency depends on the particular model the distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms the first ingredient permits to obtain an improved time discretization scheme higher accuracy better stability no restrictions on the time step size the second ingredient permits to prove global conservation properties it also allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view numerical tests show that nonlinear and dynamic subscales give more accurate solutions than classical stabilized methods

new subgrid artificial viscosity galerkin methods for the navier stokes equations abstract we study subgrid artificial viscosity methods for approximating solutions to the navier stokes equations two methods are introduced that add viscous stabilization via an artificial viscosity then remove it only on a coarse mesh these methods can be considered as conforming mixed methods the first for velocity and vorticity and the second for velocity and its gradient the former incorporating a naturally arising grad div stabilization term in this paper we rigorously study the first scheme analytically showing that it is unconditionally stable and optimally convergent as well as both schemes computationally numerical experiments demonstrate the advantages of both of these methods

a finite element variational multiscale method for incompressible flows based on two local gauss integrations in this article we present a finite element variational multiscale vms method for incompressible flows based on two local gauss integrations and compare it with common vms method which is defined by a low order finite element space l h on the same grid as x h for the velocity deformation tensor and a stabilization parameter a the best algorithmic feature of our method is using two local gauss integrations to replace projection operator we theoretically discuss the relationship between our method and common vms method for the taylor hood elements and show that the nonlinear system derived from our method by finite element discretization is much smaller than that of common vms method computationally additionally we present numerical simulations to demonstrate the effectiveness storage computational complexity of our method finally we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method

a projection based stabilized finite element method for steady state natural convection problem abstract we formulate a projection based stabilization finite element technique for solving steady state natural convection problems in particular we consider heat transport through combined solid and fluid media this stabilization does not act on the large flow structures based on the projection stabilization idea finite element error analysis of the problem is investigated and optimal errors for the velocity temperature and pressure are established we also present some numerical tests which both verify the theoretical predictions and demonstrate the method s promise

subgrid stabilized defect correction methods for the navier stokes equations we consider the synthesis of a recent subgrid stabilization method with defect correction methods the combination is particularly efficient and combines the best algorithmic features of each we prove convergence of the method for a fixed number of corrections as the mesh size goes to zero and derive parameter scalings from the analysis we also present some numerical tests which both verify the theoretical predictions and illustrate the method s promise

a new stabilized finite element method for the transient navier stokes equations abstract this paper is concerned with the development and analysis of a new stabilized finite element method based on two local gauss integrations for the two dimensional transient navier stokes equations by using the lowest equal order pair of finite elements this new stabilized finite element method has some prominent features parameter free avoiding higher order derivatives or edge based data structures and stabilization being completely local at the element level an optimal error estimate for approximate velocity and pressure is obtained by applying the technique of the galerkin finite element method under certain regularity assumptions on the solution compared with other stabilized methods using the same pair of mixed finite elements for the two dimensional transient navier stokes equations through a series of numerical experiments it is shown that this new stabilized method has better stability and accuracy results

a least squares galerkin petrov nonconforming mixed finite element method for the stationary conduction convection problem abstract in this paper a least squares galerkin petrov nonconforming mixed finite element method lsgpnmfm is proposed and analyzed for the stationary conduction convection problem we use p 2 nonconforming as approximation space for the velocity the linear element for the pressure space and the quadratic element for the temperature space the mixed finite element spaces x h and m h need not satisfy inf sup condition the existence uniqueness and convergence of the discrete solution are presented and error estimates of optimal order are derived in the case of sufficient viscosity

a second order characteristics finite element scheme for natural convection problems in this paper a second order characteristics finite element scheme is applied to the numerical solution of natural convection problems firstly after recalling the mathematical model a second order time discretization of the material time derivative is introduced next fully discretized schemes are proposed by using finite element methods numerical results for the two dimensional problem of buoyancy driven flow in a square cavity with differentially heated side walls are given and compared with a reference solution

a connection between subgrid scale eddy viscosity and mixed methods we consider a new mixed method related to the evss method in computational viscoelasticity for the convection dominated convection diffusion equation in which stabilization is added then removed through the extra mixed variables this consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in u h by suitable choice of the mixed spaces a method of guermond is recovered exactly we show that for a different natural choice a new method results with global error estimates similar to both guermond s method and the streamline diffusion supg method

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection abstract we consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation free forcing in the momentum equation we provide analysis that suggests for such problems obtaining accurate solutions necessitates either the use of pointwise divergence free finite elements such as scott vogelius or heavy grad div stabilization of weakly divergence free elements the theory is demonstrated in numerical experiments for a benchmark natural convection problem where large irrotational forcing occurs with high rayleigh numbers

a stabilized finite element method based on two local gauss integrations for the stokes equations this paper considers a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for the stokes equations approximated by the lowest equal order finite element pairs i e the p 1 p 1 and q 1 q 1 pairs this method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf sup condition optimal error estimates are obtained by applying the standard galerkin technique finally the numerical illustrations agree completely with the theoretical expectations

error analysis of a fully discrete finite element variational multiscale method for the natural convection problem abstract for the natural convection problem we propose a new projection based finite element variational multiscale method by defining the stabilization terms via two local gauss integrations at the element level based on the implicit backward euler and implicit crank nicolson schemes for temporal discretization and stabilized mixed finite element spatial discretization we establish two numerical schemes for the natural convection problem unconditional stabilities of the two numerical schemes are proved we derive error bounds of the fully discrete solution which are first and second order in time respectively the optimal error estimates in space could be achieved for velocity and temperature in the h 1 semi norm and for pressure in the l 2 norm with the proper choosing of stabilized parameters however the error estimates in space are suboptimal for velocity and temperature in the l 2 norm the derived theoretical results are supported by two numerical examples

a defect correction method for the incompressible navier stokes equations a defect correction method for the incompressible navier stokes equation with a high reynolds number is considered in the defect step the artificial viscosity parameter s is added to the reynolds number as a stability factor and the residual is taken care of in the correction step h 1 and l 2 error estimations are derived for the one step defect correction method and the results of some numerical experiments are presented these results show that for the driven cavity two defect correction steps antidiffuse the artificial viscosity approximation nearly optimally this combination gives on a very coarse mesh results indistinguishable from a benchmark very fine mesh calculation

a finite element variational multiscale method for the navier stokes equations this paper presents a variational multiscale method vms for the incompressible navier stokes equations which is defined by a large scale space lh for the velocity deformation tensor and a turbulent viscosity nu t the connection of this method to the standard formulation of a vms is explained the conditions on lh under which the vms can be implemented easily and efficiently into an existing finite element code for solving the navier stokes equations are studied numerical tests with the smagorinsky large eddy simulation model for nu t are presented

the time viscosity splitting method for the boussinesq problem abstract in this article we consider the time viscosity splitting method for the boussinesq problem by using this technique the considered problem is decoupled into two subproblems and each subproblem is solved more easily than the original one in the first step two linear elliptic problems with semi implicit schemes for the convection terms are solved the advantage of using such numerical technique is that a linear system with the constant coefficient matrices is obtained and then the computation becomes easy in the second step a system consists of a stokes problem and a linear elliptic problem is solved the main results of this article include that 1 establish the stability of numerical solutions in the time viscosity splitting method 2 provide the convergence results of strongly second order for the velocity and temperature and strongly first order for the pressure finally some numerical results are provided to display the performances of the developed numerical algorithms

hilbert valued perturbed subgradient algorithms we propose a hilbert valued perturbed subgradient algorithm with stochastic noises and we provide a convergence proof for this algorithm under classical assumptions on the descent direction and new assumptions on the stochastic noises instead of requiring the stochastic noises to correspond to martingale increments we only require these noises to be asymptotically so furthermore the variance of these noises is allowed to grow infinitely under the control of a decreasing sequence linked with the subgradient stepsizes this algorithm can be used to solve stochastic closed loop control problems without any a priori discretization of the uncertainty such as linear decision rules or tree representations it can also be used as a way to perform stochastic dynamic programming without state space discretization or a priori functional bases i e approximate dynamic programming both problems arise frequently for example in power systems scheduling or option pricing this article focuses on the theorical foundations of the algorithm and gives references to detailed practical experimentations

the decoupled crank nicolson adams bashforth scheme for the boussinesq equations with nonsmooth initial data abstract in this paper the decoupled crank nicolson adams bashforth scheme for the boussinesq equations is considered with nonsmooth initial data our numerical scheme is based on the implicit crank nicolson scheme for the linear terms and the explicit adams bashforth scheme for the nonlinear terms for the temporal discretization standard galerkin finite element method is used to the spatial discretization in order to improve the computational efficiency the decoupled method is introduced as a consequence the original problem is split into two linear subproblems and these subproblems can be solved in parallel we verify that our numerical scheme is almost unconditionally stable for the nonsmooth initial data u0 0 with the divergence free condition furthermore under some stability conditions we show that the error estimates for velocity and temperature in l2 norm is of the order o h 2 t 3 2 in h1 norm is of the order o h 2 t and the error estimate for pressure in a certain norm is of the order o h 2 t finally some numerical examples are provided to verify the established theoretical findings and test the performances of the developed numerical scheme