A CHARACTERISTICS-MIXED FINITE-ELEMENT METHOD FOR ADVECTION-DOMINATEDTRANSPORT PROBLEMS

We define a new finite element method, called the characteristics-mixe d method, for approximating the solution to an advection-dominated tra nsport problem. The method is based on a space-time variational form o f the advection-diffusion equation. Our test functions are piecewise c onstant in space, and in time they approximately follow the characteri stics of the advective (i.e., hyperbolic) part of the equation. Thus t he scheme uses a characteristic approximation to handle advection in t ime. This is combined with a low-order mixed finite element spatial ap proximation of the equation. Boundary conditions are incorporated in a natural and mass conservative fashion. The scheme is completely local ly conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. A postprocessing step is includ ed in the scheme in which the approximation to the scalar unknown is i mproved by utilizing the approximate vector flux. This has the effect of improving the rate of convergence of the method. We show that it is optimally convergent to order one in time and at least suboptimally c onvergent to order 3/2 in space.