On the property of the curl-curl matrix in finite element analysis with edge elements

Abstract

Abstract-This paper discusses properties of the curl-curl matrix in the finite element formulation with edge elements. Moreover the observed deceleration in convergence of the CG and ICCG methods applied to magnetostatic problems through the tree-cotree gauging is explained on the basis of the eigenvalue separation property. From the eigenvalue separation property it follows that neither minimum nonzero eigenvalue of the curl-curl matrix nor maximum one increase through the tree-cotree gauging. Hence it is concluded that the condition number of the curl-curl matrix tends to grow by its definition. Moreover the maximum eigenvalue tends to keep constant whereas the minimum nonzero eigenvalue reduces. This property also makes the condition number worse.

On convergence of ICCG applied to finite-element equation for quasi-static fields

“…Only 1/8 of the model is considered due to the symmetry. The tree-cotree gauging has not been applied because it is known that the gauging makes the matrix condition worse [7]. The number of iterations until the convergence of the CG process is evaluated.…”

“…Here, they are not exactly zero due to the numerical errors in the eigenvalue computation. Moreover, these zero singular values do not give any influence on the convergence [7].…”

“…The singular value of a complex matrix satisfies , , where denotes Hermitian conjugation and is the eigenvalue of . It is known that convergence of the CG method becomes better (worse) when the condition number, which would be defined by (7) becomes smaller (larger) [7], [8], where and are the maximum singular value and minimum nonzero one of the matrix, respectively. Fig.…”

Effect of Preconditioning in Edge-Based Finite-Element Method

“…Note here that (2) is dependent on (1) because divergence of both sides of (1) yields (2 The weak form of (1) and (2) can be written in the form (3) (4) where and represent edge-based and scalar basis functions for approximation of and , respectively. The FE discretization of (3) and (4) with edge elements provides In (6) and (7), matrices and , which are and matrices with entries 1 and 0, represent the discrete counterparts of curl and grad, where , , and denote the number of nodes, edges, and faces, respectively [3].…”

“…The matrices and are positive definite and positive semi-definite symmetric matrices, respectively. The matrix , whose rank is proved to be [4], corresponds to the FE matrix for static magnetic fields.…”