Superconvergence in finite element method by smoothing

TL;DR

This paper introduces a smoothing-based postprocessing technique to achieve superconvergence in finite element methods, applicable to various equations and discretizations.

Contribution

It develops an algebraic, easy-to-implement smoothing approach that enhances finite element solutions with proven superconvergence properties.

Findings

Superconvergence is achieved through a few smoothing iterations.

The method is effective for Poisson, Maxwell, biharmonic, and Helmholtz equations.

Applicable to high-order and 3D discretizations.

Abstract

This paper develops a smoothing-based postprocessing method for superconvergence in finite element methods. The method applies a few smoothing iterations, such as damped Jacobi, Gauss-Seidel, or conjugate gradient, with initial guess being the current finite element solution embedded in an enriched finite element space. The resulting procedure is algebraic, easy to implement, and applicable to high-order and three-dimensional discretizations. For symmetric and positive-definite problems, we prove superconvergence of the smoothed solutions under additive and multiplicative smoothers. Effectiveness of the proposed method is demonstrated by numerical experiments for the Poisson, Maxwell, biharmonic and Helmholtz equations.