Error analysis of a decoupled finite element method for quad-curl problems

TL;DR

This paper analyzes a decoupled finite element method for quad-curl problems, demonstrating improved convergence rates and adaptive strategies for singular solutions, supported by theoretical and numerical validation.

Contribution

It introduces a simplified element construction for quad-curl problems and develops an adaptive error estimator with proven convergence.

Findings

Quadratic convergence for curl approximation in convex domains

First-order convergence in energy norm

Effective adaptive method for singular solutions

Abstract

Finite element approximation to a decoupled formulation for the quad--curl problem is studied in this paper. The difficulty of constructing elements with certain conformity to the quad--curl problems has been greatly reduced. For convex domains, where the regularity assumption holds for Stokes equation, the approximation to the curl of the true solution has quadratic order of convergence and first order for the energy norm. If the solution shows singularity, an a posterior error estimator is developed and a separate marking adaptive finite element procedure is proposed, together with its convergence proved. Both the a priori and a posteriori error analysis are supported by the numerical examples.