An optimally blended finite-spectral element scheme with minimal dispersion for Maxwell equations Hafiz Abdul Wajid ⇑ , Sobia Ayub Department of Mathematics, Defence road, off raiwind road, M.A. Jinnah Campus, COMSATS Institute of Information Technology, Lahore, Pakistan article info Article history: Received 30 March 2011 Received in revised form 1 June 2012 Accepted 30 July 2012 Available online 10 August 2012 Keywords: Edge finite element Numerical dispersion Discrete dispersion relation abstract We study the dispersive properties of the time harmonic Maxwell equations for optimally blended finite-spectral element scheme using tensor product elements defined on rectan- gular grid in d-dimensions. We prove and give analytical expressions for the discrete dis- persion relations for this scheme. We find that for a rectangular grid (a) the analytical expressions for the discrete dispersion error in higher dimensions can be obtained using one dimensional discrete dispersion error expressions; (b) the optimum value of the blend- ing parameter is p=ðp þ 1Þ for all p 2 N and for any number of spatial dimensions; (c) ana- lytical expressions for the discrete dispersion relations for finite element and spectral element schemes can be obtained when the value of blending parameter is chosen to be 0 and 1 respectively; (d) the optimally blended scheme guarantees two additional orders of accuracy compared with standard finite element and spectral element schemes; and (e) the absolute accuracy of the optimally blended scheme is Oðp 2 Þ and Oðp 1 Þ times better than that of the pure finite element and spectral element schemes respectively. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Propagation of waves at a given frequency or wave number is highly important as it finds applications in several walks of human life covering domestic, commercial or even defence purpose needs. Complex nature of this phenomenon poses a chal- lenge to physists, engineers and mathematicians to find explicit solutions which is not realistically possible. Consequently, numerical methods are developed which fails to propagate waves at correct speed resulting in numerical dispersion and numerical dissipation [4,6,7,11]. A detailed understanding of the dispersive properties of a scheme is not only valuable the- oretically but serves as an a priori error estimate for practical problems. Several methods have been proposed by many [13,14] to reduce dispersion as it is well understood that dispersion cannot be completely avoided in higher dimensions [16]. Amongst these methods higher order finite element and spectral element methods are widely used for the approxima- tion of Acoustic [4,7,11], Maxwell [6,5] and Elastic [12] wave equations. Moreover, spectral element scheme has gain much popularity in seismologist community [8–10] because of the lumped mass matrix which is computationally inexpensive to invert and highly suitable for transient problems. Furthermore, numerical dispersion may result into leading phase or lag- ging phase and correspond to finite element and spectral element schemes respectively [1]. Relatively recently the detailed study of the dispersive properties of the pure finite element, spectral element and opti- mally blended schemes valid for elements of arbitrary order is conducted in [1,2,4]. Furthermore, extension of these schemes to any number of dimensions using tensor product meshes is also given there in. In [2] optimally blended finite-spectral ele- ment scheme was developed by taking a weighted averaging of the consistent and lumped mass matrices for a fixed order 0021-9991/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcp.2012.07.047 ⇑ Corresponding author. Tel.: +92 323 4860342. E-mail addresses: hawajid@ciitlahore.edu.pk (H.A. Wajid), sobia.ayub07@yahoo.com (S. Ayub). Journal of Computational Physics 231 (2012) 8176–8187 Contents lists available at SciVerse ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp