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

TL;DR

This paper introduces a weak finite element method for one-dimensional elliptic problems, demonstrating higher accuracy, local solvability, and superconvergence, with comprehensive error analysis and numerical validation.

Contribution

It develops a new weak finite element method for 1D elliptic problems with improved accuracy and local solvability, extending the weak Galerkin approach.

Findings

Higher accuracy in discrete norms

Local element-by-element solution process

Superconvergence phenomena observed

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.