Uniform Estimates Of Characteristics-Mixed Finite Methods For Time-Dependent Convection-Diffusion Problems

In this paper, we prove uniform optimal-order error estimates for characteristics-mixed finite element methods for two-dimensional convection-dominated diffusion equations. The convection-dominated diffusion equation is strictly parabolic, how-ever, in many applicationsεis small enough, thus advection dominates diffusion, the equation is nearly hyperbolic in nature. The concentration often develops sharp fronts that are nearly shocks. It is well known that the traditional parabolic dis-cretization schemes applied to the problem do not work well, it tends to introduce into the solution an excessive amount of numerical diffusion near the fronts. In order to overcome the defects of the traditional methods and purse a high per-formance of an algorithm, people have developed the characteristics-mixed finite element method. This kind of methods is divided into two classes, one is the com-bination of characteristics method and mixed finite element method proposed in [9], the other is the modified method of characteristics combined with mixed finite element method proposed in [1]. A great many numerical experiences show that these methods are practically effective. They not only ensure the high stability of the method in approximating the sharp fronts and decrease the numerical diffusion, but also possess the optimal L2-crror approximation for the unknown function and the vector flux. However, it is worthy to point out that these error estimates were proved via the introduction of the mixed elliptic projcction whose approximation property depends on the inverse of thc small parameter,thus thc constants in these error estimates also dcpend onε-1.Whenεtends to zero,the finito clement solution will b1ow up as the lower bound of the diffusion approaches to zero.In this paper,we still adopt the characteristies-mixed finite element schemes for convection-dominated diffusion equations with a periodic boundary condition. For derivingε-uniform estimates,the mixed elliptic projcctions are rcplaced by the Raviart-Thomas projection and the interpolation operators in the error derivation, therefore the uniform error estimates for the unknown funetion and its flux are derived.The generic constants in the error estimates do not explicitly depend onε, but depend linearly on certain Sobolev norms of the true solution.Moreover,in the derivation of the first characteristics-mixed method,we obtain the optimally uniform L2-error estimates when the grid ratio h/△t≤1.Furthermore,combining the estimates with the stability estimates of the true solution,we prove the constants depend only on the initial and the right side data.Meanwhile,we derive the optimally uniform L2-error estimates for the local balance characteristics-mixed finite element method when the grid ratio h3/2/△t≤1.A large number of numerical experiments are presented to confirm our theoretical findings.