Superconvergence in finite element method by smoothing
For computational scientists using finite element methods, this provides an easy-to-implement, algebraic postprocessing technique to improve solution accuracy, though it is an incremental improvement over existing superconvergence methods.
The paper introduces a smoothing-based postprocessing method that achieves superconvergence in finite element methods by applying a few smoothing iterations (e.g., damped Jacobi, Gauss-Seidel, or conjugate gradient) with an enriched initial guess. Numerical experiments on Poisson, Maxwell, biharmonic, and Helmholtz equations demonstrate effectiveness.
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.