A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

MIXED FINITE ELEMENT METHOD FOR THE CONVECTION-DOMINATED DIFFUSION PROBLEMS WITH SMALL PARAMETER ε

In this paper a mixed finite element method for the convection dominated diffusion problems with small parameter ε is presented,the effect of the parameter ε on the approximation error is considered and a sufficient condition for optimal error estimates is derived.The paper also shows that under some conditions,the standard finite element method only gives a bounded solution,however the mixed finite element method gives a convergent one. Received March 1,1997. 1991 MR Subject Classification: 65N30,65M15.

The Variations of Dominant Convection Modes over Asia, Indian Ocean, and Western Pacific Ocean during the Summers of 1997-2004

The NOAA daily outgoing longwave radiation （OLR） and the Global Precipitation Climatology Project （GPCP）daily precipitation data are used to study the variation of dominant convection modes and their relationships over Asia, the Indian Ocean, and the western Pacific Ocean during the summers from 1997 to 2004. Major findings are as follows： （1） Regression analysis with the OLR indicates the convective variations over Asian monsoon region are more closely associated with the convective activities over the western subtropical Pacific （WSP） than with those over the northern tropical Indian Ocean （NTIO）. （2） The EOF analysis of OLR indicates the first mode （EOF1） exhibits the out-of-phase variations between eastern China and India, and between eastern China and the WSP. The OLR EOF1 primarily exhibits seasonal and even longer-term variations. （3） The OLR EOF2 mostly displays in-phase convective variations over India, the Bay of Bengal, and southeastern China. A wavelet analysis reveals intraseasonal variation （ISV） features in 2000, 2001, 2002, and 2004. However, the effective ISV does not take place in every year and it seems to occur only when the centers of an east-west oriented dipole reach enough intensity over the tropical Indian and western Pacific Oceans. （4） The spatial patterns of OLR EOF3 are more complicated than those of EOF1 and EOF2, and an effective ISV is noted from 1999 to 2004. The OLR EOF3 implies there is added complexity of the OLR pattern when the effective ISV occurs. （5） The correlation analysis suggests the precipitation over India is more closely associated with the ISV, seasonal variations, and even longer-term variations than precipitation occurring over eastern China.

GENERALIZED UPWIND SCHEME WITH FRACTIONAL STEPS FOR 3-D PROBLEM OF CONVECTION DOMINATING GROUNDWATER TRANSPORT

A generalized upwind scheme with fractional steps for 3-D mathematical models of convection dominating groundwater quality is suggested. The mass transport equation is split into a convection equation and a dispersive equation. The generalized upwind scheme is used to solve the convection equation and the finite element method is used to compute the dispersive equation.These procedures which not only overcome the phenomenon of the negative concentration and numerical dispersion appear frequently with normal FEM or FDM to solve models of convection dominating groundwater transport but also avoid the step for computing each node velocity give a more suitable method to calculate the concentrations of the well points.

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L^2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.

LOCAL ERROR ESTIMATES FOR METHODS OF CHARACTERISTICS INCORPORATING STREAMLINE DIFFUSION

Allen and Liu (1995) introduced a new method for a time-dependent convection dominated diffusion problem, which combines the modified method of characteristics and method of streamline diffusion. But they ignored the fact that the accuracy of time discretization decays at half an order when the characteristic line goes out of the domain. In present paper, the author shows that, as a remedy, a simple lumped scheme yields a full accuracy approximation. Forthermore, some local error bounds independent of the small viscosity axe derived for this scheme outside the boundary layers.

Fast Numerical Simulation of Two-Phase Transport Model in the Cathode of a Polymer Electrolyte Fuel Cell

In this paper,we apply streamline-diffusion and Galerkin-least-squares fi-nite element methods for 2D steady-state two-phase model in the cathode of polymer electrolyte fuel cell(PEFC)that contains a gas channel and a gas diffusion layer(GDL).This two-phase PEFC model is typically modeled by a modified Navier-Stokes equation for the mass and momentum,with Darcy’s drag as an additional source term in momentum for flows through GDL,and a discontinuous and degenerate convectiondiffusion equation for water concentration.Based on the mixed finite element method for the modified Navier-Stokes equation and standard finite element method for water equation,we design streamline-diffusion and Galerkin-least-squares to overcome the dominant convection arising from the gas channel.Meanwhile,we employ Kirchhoff transformation to deal with the discontinuous and degenerate diffusivity in water concentration.Numerical experiments demonstrate that our finite element methods,together with these numerical techniques,are able to get accurate physical solutions with fast convergence.