Improvements of the Mizukami–Hughes method for convection–diffusion equations

We consider the Mizukami–Hughes method for the numerical solution of scalar two-dimensional steady convection–diffusion equations using conforming triangular piecewise linear finite elements. We propose several modifications of this method to eliminate its shortcomings. The improved method still satisfies the discrete maximum principle and gives very accurate discrete solutions in convection-dominated regime, which is illustrated by several numerical experiments. In addition, we show how the Mizukami–Hughes method can be applied to convection–diffusion–reaction equations and to three-dimensional problems.