Applied Math Seminar-Gradcurl-Conforming Finite Elements Based on De Rham Complexes for the Fourth-Order Curl Problems

Time: Friday, Oct. 21st. 3:30-4:30 pm

Place: Zoom

Zoom link: https://gwu-edu.zoom.us/j/95630895609

Speaker: Qian Zhang, Michigan Technological University

Title: Gradcurl-Conforming Finite Elements Based on De Rham Complexes for the Fourth-Order Curl Problems

Abstract: The fourth-order curl operator appears in various models, such as electromagnetic interior transmission eigenvalue problems, magnetohydrodynamics in hot plasmas, and couple stress theory in linear elasticity. The key to discretizing these problems is to discretize the fourth-order curl operator. In this talk, I will present the conforming finite element method for a simplified fourth-order curl model. Discretizing the quad-curl equations using smoother elements (such as H^2-conforming elements) would lead to wrong solutions. Specific finite elements need to be designed for the fourth-order curl operator. However, constructing such elements is a challenging task because of the continuity required by the curlcurl-conformity and the naturally divergence-free property of the curl operator. In this presentation, we provide the construction of the curlcurl-conforming elements in both 2D and 3D based on the de Rham complex. In 2D, the lowest-order grad curl-conforming element has only 6 and 8 degrees of freedom on a triangle and a rectangle, respectively. In 3D, we relate the fourth-order curl problem to fluid mechanics and a de Rham complex with higher regularity.  The lowest-order element has only 18 degrees of freedom on a tetrahedron. As a by-product, we construct a family of stable and mass-preserving finite element pairs for solving the Navier-Stokes equations.