Mixed schemes for quad-curl equations

Abstract

In this paper, amiable mixed schemes are presented for two variants of fourth order curl equations. Specifically, mixed formulations for the problems are constructed, which are wellposed in Babuška-Brezzi's sense and admit stable discretizations by finite element spaces of low smoothness and of low degree. The regularities of the mixed formulations and thus equivalently the primal problems are established, and some finite elements examples are given which can exploit the regularity of the solutions to an optimal extent.MHD, (∇×) 4 B is involved in the resistive system where B is the magnetic field as a primary variable [27], and in the inverse electromagnetic scattering theory, (∇×) 4 appears in computing the transmission eigenvalue [6]. Some more applications of (∇×) 4 can be found in the sequel works.2000 Mathematics Subject Classification. 65N30, 35Q60, 76E25, 76W05.

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

“…It is meaningful and urgent to design highly efficient and accurate numerical methods for quad-curl problems. By the way, singularities of the quad-curl problem were analyzed in [16,30].…”

“…In contrast to second-order curl problems, limited work has been done on numerical methods for quad-curl problems. Initially, numerical methods with various nonconformity/mix techniques, such as nonconforming finite element methods [32], discontinuous Galerkin methods [10], weak Galerkin methods [21], mixed finite element methods [19,20,14,25,31,30,23], and the Hodge decomposition method [4,3], were proposed to solve quad-curl problems as well as their related eigenvalue problems. Indeed, H(curl 2 )-conforming methods were unavailable for quad-curl problems until recently.…”

A Family of Curl-Curl Conforming Finite Elements on Tetrahedral Meshes

“…Another approach to deal with the quad-curl operator is to introduce an auxiliary variable and reduce the original problem to a second-order system [20]. Zhang proposed a different mixed scheme [24], which relaxes the regularity requirement in theoretical analysis.…”

A Quadratic C0 Interior Penalty Method for the Quad-Curl Problem

“…The most recent results on the regularity of the solution of (1.1)-(1.2) are as follows. On convex polyhedral domains, it was proved in [25] 3 , in the case where ∇ • f = 0. However, in [19] S. Nicaise showed that u / ∈ [H 2 (Ω)] 3 on general polyhedral domains.…”