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2020-06-16

Unconditionally Stable Time Stepping Method for Mixed Finite Element Maxwell Solvers

By Zane Crawford, Jie Li, Andrew Christlieb, and Balasubramaniam Shanker
Progress In Electromagnetics Research C, Vol. 103, 17-30, 2020
doi:10.2528/PIERC20021001

Abstract

Time domain finite element methods (TD-FEM) for computing electromagnetic fields are well studied. TD-FEM solution is typically effected using Newmark-Beta methods. One of the challenges of TD-FEM is the presence of a DC null-space that grows with time. This can be overcome by solving Maxwell equations directly. One approach, called time domain mixed finite element method (TD-MFEM), discretizes Maxwell's equations using appropriate spatial basis sets and leapfrog time stepping. Typically, the basis functions used to discretize field quantities have been low order. It is conditionally stable, and there is a strong link between time step size and mesh dependent eigenvalues, much like the Courant-Friedrichs-Lewy (CFL) condition. This implies that the time step sizes can be very small. To overcome this challenge, we use the Newmark-Beta approach. The principal contribution of this work is the development of, and rigorous proof of, unconditional stability for higher order TD-MFEM for different boundary conditions. Further, we analyze nullspaces of the resulting system, and demonstrate stability and convergence. All results are compared against the conditionally stable leapfrog approach.

Citation


Zane Crawford, Jie Li, Andrew Christlieb, and Balasubramaniam Shanker, "Unconditionally Stable Time Stepping Method for Mixed Finite Element Maxwell Solvers," Progress In Electromagnetics Research C, Vol. 103, 17-30, 2020.
doi:10.2528/PIERC20021001
http://www.jpier.org/PIERC/pier.php?paper=20021001

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