Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations

Abstract

Abstract. We propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Hodge decomposition for divergencefree vector fields. In this approach an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P 1 finite element method.

A Weak Galerkin Finite Element Method for the Maxwell Equations

“…To define the weak curl, we require weak functions v = {v 0 , v b } such that v 0 ∈ [L 2 (K )] 3 and v b × n ∈ [L 2 (∂ K )] 3 …”

“…and a second stabilization term 3 . For simplicity, one may take Q b φ as the standard L 2 projection of the boundary value φ on each boundary segment.…”

“…The Maxwell equations have been studied extensively in literature by using various numerical methodologies including H (curl; )-conforming edge element approaches [1,9,10,12,13] and discontinuous Galerkin methods [2,3,6,7,14,15]. Particularly in [8], a mixed DG formulation for the problem (1.7)-(1.10) was introduced and analyzed.…”

“…Particularly in [8], a mixed DG formulation for the problem (1.7)-(1.10) was introduced and analyzed. In this DG formulation, both u and p are approximated by piecewise [P k (T )] 3 and P k (T ) functions if T is a tetrahedron and by piecewise [Q k (T )] 3 and Q k (T ) if T is a parallelepiped, where P k (T ) denotes the set of polynomials of total degree k and Q k (T ) the set of polynomials of degree k in each variable.…”

“…In essence, this procedure shall introduce a discrete curl operator, which shall be combined with the discrete weak gradient as introduced in [18] to yield a finite element scheme for the Maxwell equations. In this WG method, two types of weak functions are used: u h = {u 0 , u b } ∈ [P s (T )] 3 × [P t (e)] 3 and p h = {p 0 , p b } ∈ P (T ) × P ι (e), with u h = u 0 and p h = p 0 inside of each element and u h = u b and p h = p b on the boundary of the element. Error estimates of optimal order are established for the WG approximations in appropriate norms for the case of s = t = k and = k − 1, ι = k with k ≥ 1.…”