An Analysis of the Weak Finite Element Method for Convection-Diffusion Equations
For researchers in numerical analysis, this provides a theoretical foundation and error analysis for weak finite element methods applied to convection-diffusion equations, though it is an incremental extension of existing methods.
The paper presents a weak finite element method for convection-diffusion equations, establishing optimal order error estimates in discrete H^1, L2, and L∞ norms, with H^1-superconvergence of order k+2 under certain conditions, validated by numerical examples.
We study the weak finite element method solving convection-diffusion equations. A weak finite element scheme is presented based on a spacial variational form. We established a weak embedding inequality that is very useful in the weak finite element analysis. The optimal order error estimates are derived in the discrete $H^1$-norm, the $L_2$-norm and the $L_\infty$-norm, respectively. In particular, the $H^1$-superconvergence of order $k+2$ is given under certain condition. Finally, numerical examples are provided to illustrate our theoretical analysis