An adaptive superconvergent finite element method based on local residual minimization

TL;DR

This paper presents an adaptive finite element method that combines superconvergent postprocessing with residual minimization for efficient and accurate solutions of PDEs involving diffusion, supported by theoretical analysis and numerical experiments.

Contribution

It introduces a novel adaptive superconvergent finite element method based on local residual minimization and develops new a posteriori error estimators for mixed formulations.

Findings

Numerical experiments confirm the sharpness of the error estimators.

The proposed method achieves superconvergence and efficiency in solving PDEs.

Residual minimization on local postprocessing schemes reduces computational effort.

Abstract

We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal variable with an adaptive finite element method via residual minimization. Such a residual minimization procedure is performed on a local postprocessing scheme, commonly used in the context of mixed finite element methods. Given the local nature of that approach, the underlying saddle point problems associated with residual minimizations can be solved with minimal computational effort. We propose and study a posteriori error estimators, including the built-in residual representative associated with residual minimization schemes; and an improved estimator which adds, on the one hand, a residual term quantifying the mismatch between discrete fluxes and, on the other hand, the interelement jumps of the postprocessed solution. We present numerical experiments in two dimensions using Brezzi-Douglas-Marini elements as input for our methodology. The experiments perfectly fit our key theoretical findings and suggest that our estimates are sharp.