Adaptivity and Error Estimation for the Finite Element Method Applied to Convection Diffusion Problems

Abstract

A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto “upwinded” polynomials. The error is measured in the $L_p$ mesh dependent norm. The method is shown to be quasi-optimal, provided that the input data is piecewise smooth—a reasonable assumption in practice. A posteriors error estimates are derived having the property that the effectivity index $\theta = $ (error estimate/true error) converges to one as the maximum mesh size goes to zero. These error estimates are composed of locally computable error indicators, providing for an adaptive mesh refinement strategy. Numerical results show that $\theta $ is nearly one even on coarse meshes, and optimal rates of convergence are attained by the adaptive procedure. The robustness of the algorithm is tested on a nonlinear turning point problem modeling flow through an expanding duct.