A uniformly optimal-order error estimate for a bilinear finite element method for degenerate convection-diffusion equations

摘要

We prove an optimal-order error estimate in a degenerate-diffusion weighted energy norm for bilinear Galerkin finite element methods for two-dimensional time-dependent convection-diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal-order estimate of the Galerkin finite element method, in which the generic constants depend only on the Sobolev norms of the initial and right side data. Preliminary numerical experiments were conducted to verify these estimates numerically.