A Uniformly Optimal-Order Estimate for Bilinear Finite Element Method for Transient Advection-Diffusion Equations

Abstract

We prove an optimal-order error estimate in a weighted energy norm for bilinear Galerkin finite element method for two-dimensional time-dependent advection-diffusion equations by the means of integral identities or expansions, in the sense that the generic constants in the estimates depend only on certain Sobolev norms of the true solution but not on the scaling parameter $\varepsilon$. These estimates, combined with a priori stability estimates of the governing partial differential equations, yield an "$\varepsilon$-uniform estimate of the bilinear Galerkin finite element method, in which the generic constants depend only on the Sobolev norms of the initial and right data but not on the scaling parameter $\varepsilon$.