H($\text{curl}^2$)-conforming quadrilateral spectral element method for quad-curl problems

TL;DR

This paper introduces a novel $H(\text{curl}^2)$-conforming quadrilateral spectral element method for solving quad-curl problems, utilizing generalized Jacobi polynomials and hierarchical construction for efficient and accurate solutions.

Contribution

It develops the first explicit construction of $H(\text{curl}^2)$-conforming spectral elements on arbitrary convex quadrilaterals with minimal degrees of freedom.

Findings

Method achieves high accuracy with only 8 degrees of freedom per element.

Numerical results confirm the method's effectiveness and efficiency.

Applicable to solving quad-curl equations and eigenvalue problems.

Abstract

In this paper, we propose an $H(\text{curl}^2)$-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. $H(\text{curl}^2)$-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is astonishing that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our $H(\text{curl}^2)$-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.