A Simplified Weak Galerkin Finite Element Method: Algorithm and Error Estimates
This is an incremental improvement for researchers using weak Galerkin methods, offering reduced computational cost while maintaining accuracy.
The authors developed a simplified weak Galerkin finite element method for convection-diffusion-reaction equations that uses only boundary degrees of freedom, reducing computational complexity. They proved stability and optimal order error estimates in H1 and L2 norms, with numerical verification including superconvergence on rectangular partitions.
In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations. The simplified weak Galerkin method utilizes only the degrees of freedom on the boundary of each element and, hence, has significantly reduced computational complexity over the regular weak Galerkin finite element method. A stability and some optimal order error estimates in the $H^1$ and $L^2$ norms are established for the corresponding numerical solutions. Numerical results are presented to verify the theory error estimates and a superconvergence phenomena on rectangular partitions.