Spurious solutions in mixed finite element method for Maxwell’s equations: Dispersion analysis and new basis functions Luis Tobón a,b , Jiefu Chen a , Qing Huo Liu a,⇑ a Department of Electrical and Computer Engineering at Duke University, 130 Hudson Hall, Box 90291, Durham, NC 27708, United States b Departamento de Ciencias e Ingeniería de la Computación, Pontificia Universidad Javeriana, Cali, Colombia article info Article history: Received 29 November 2010 Received in revised form 18 May 2011 Accepted 28 May 2011 Available online 16 June 2011 Keywords: Mixed finite element method Spurious solutions Dispersion analysis abstract The finite element method is a well known computational technique used to obtain numer- ical solutions to boundary-value problems including Maxwell’s equations. This paper first presents a brief description of the mathematical structure, based on the De Rham diagram, to discretize Maxwell’s equations. Then it uses a numerical dispersion analysis of the mixed finite element method with both electric and magnetic fields as unknowns to eval- uate the presence of spurious solutions for different basis functions. These unwanted spu- rious solutions appear when the same order of element is used for electric and magnetic fields, while the system is free of spurious modes when different orders of elements are employed for electric and magnetic fields. In this work, finite elements in both frequency and time domain are studied, and the effects of these spurious solutions in both domains are analyzed in one- and three-dimensional cases. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction One of the most active areas of research in the finite element method is the appropriate choices of basis functions that can solve correctly the topological and constitutive laws in electromagnetics by Maxwell’s equations [1–5]. A problematic situ- ation is the presence of extraneous or spurious solutions when inappropriate basis functions are used [5–13]. These spurious solutions are non-physical. We can differentiate two categories of spurious solutions [6]: the first set has modes with very low (theoretically zero) frequencies, and are several orders of magnitude away from the rest in the spectrum. The origin of this kind of solutions is that the divergence condition for electric and magnetic fields is not satisfied. A typical situation for such modes is when the curl–curl operator is used, without taking into account of the divergence operator [5,9,8], therefore solutions in the form of gradient of a potential are valid mathematical solutions but are not physical. A correct formulation of the problem based on the four Maxwell’s equations, and the inclusion of compatibility relations applied to boundaries be- tween elements [9,14] solve this problem. The second category of spurious solutions intermingle with the correct ones [15–17,7]. These non-physical solutions appear when the mixed finite element method is used to solve the first order Maxwell’s equations with the same order ele- ments for electric and magnetic fields. Previously the cause of this subset of spurious solutions has not been completely clear, and it is supposed to be related to a compatibility problem between the electric and magnetic field formulation [2,3,18,13]. Some characteristics of these solutions are their rapid spatial variations (high numerical wavenumbers), and high and negative group velocities [19]. Under these considerations a dispersion analysis [10] is presented in this work as a simple and practical procedure to detect the second kind of spurious solutions in finite element formulations. The mathematical 0021-9991/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2011.05.035 ⇑ Corresponding author. E-mail addresses: luis.tobonllano@duke.edu (L. Tobón), jiefu.chen@duke.edu (J. Chen), qhliu@ee.duke.edu (Q.H. Liu). Journal of Computational Physics 230 (2011) 7300–7310 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp