A weak finite element method for elliptic problems in one space dimension

Abstract

We present a weak finite element method for elliptic problems in one space dimension. Our analysis shows that this method has more advantages than the known weak Galerkin method proposed for multi-dimensional problems, for example, it has higher accuracy and the derived discrete equations can be solved locally, element by element. We derive the optimal error estimates in the discrete $H^1$-norm, the $L_2$-norm and $L_\infty$-norm, respectively. Moreover, some superconvergence results are also given. Finally, numerical examples are provided to illustrate our theoretical analysis.