A nonconforming finite element method for fourth order curl equations in ℝ³

Abstract

In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both ( ∇ × ) 2 (\nabla \times )^2 and ( ∇ × ) 4 (\nabla \times )^4 operators. The element has a very small number of degrees of freedom, and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.