Higher order space and time discretizations of Maxwell’s equations

Abstract

Computational electromagnetics involves the numerical solution of Maxwell’s equations and has been one of the fundamental pillars of modern electrical engineering. Given the ubiquity of electrical and electronic devices at all power levels, from micro- to giga-watts in our everyday world, the importance of a provably correct and accurate finite element discretization of Maxwell’s equations is both theoretically useful and practically relevant. In our work, we demonstrate such a structure-preserving higher-order numerical discretization for Maxwell’s equations. Our work shows how modern finite elements, incorporating ideas from differential geometry and algebraic topology, can be used in conjunction with higher-order time discretization schemes and provide for highly accurate energy-conserving schemes. In our work, we study a system of Maxwell’s equations that describes the time evolution of electromagnetic fields with an additional electric scalar variable to make the system amenable to a compatible, mixed finite element spatial discretization. We demonstrate stability and energy conservation for the variational formulation of this Maxwell’s system. We discuss two implicit, energy conserving schemes for its temporal discretization: the classical Crank-Nicolson scheme and an implicit leapfrog scheme. We show discrete stability and discrete energy conservation for the semi-discretization using these two time integration methods. We complete our discussion by showing that the error for the full discretization of the Maxwell’s system with each of the two implicit time discretization schemes and with spatial discretization through a conforming sequence of de Rham finite element spaces converges quadratically in the step size of the time discretization and as an appropriate polynomial power of the mesh parameter in accordance with the choice of approximating polynomial spaces. Our results for the Crank-Nicolson method are generally well known but have not been demonstrated for this Maxwell’s system. Our implicit leapfrog scheme is a new method to the best of our knowledge, and we provide a complete error analysis for it. Moreover, we show computational results to validate our theoretical claims using linear and quadratic Whitney forms for the finite element discretization for a model problem each in two and three spatial dimensions. We then propose two energy conserving fourth-order time discretizations of the same three-field formulation of Maxwell’s equations in conjunction with a spatial discretization using higher-order and compatible de Rham finite element spaces. Toward this end, we delineate two broad classes of strategies for general higher-order time discretizations which we term spatial and temporal strategies. We provide a description of these two strategies and develop fourth-order time accurate schemes in the context of our Maxwell’s system. Moreover, our description can be used to prescribe similar fourth- or even higher-order time-integration methods for any linear (or quasi-linear) system of time-dependent partial differential equations. Our organizing principle in our proposed two strategies is to Taylor expand the unknown solution in time by assuming sufficient regularity. Then, in the spatial strategy, we use Maxwell’s equations themselves to replace the fourth-order time derivatives in an appropriately truncated Taylor expansion with corresponding higher-order spatial derivatives. On the other hand, in the temporal strategy, we simply use higher-order finite difference schemes for the various higher-order time derivative terms in the truncated Taylor approximation. In both cases, we then defer to a standard finite element exterior calculus manner of compatible discretization for the spatial component of the Maxwell’s solution. Finally, we generalize our proposed time discretization scheme using spatial strategy to an arbitrarily higher (even) order implicit leapfrog scheme for time discretization of our Maxwell’s system. We use this in conjunction with an arbitrarily higher-order and compatible discretization using finite element spaces that form a de Rham complex. We prove stability, demonstrate energy conservation, and characterize the asymptotic convergence of the error for the time semidiscretization as well as for the full spatial and temporal discretization of this Maxwell’s system. Code Availability: Python code for all examples in this thesis to generate the various pieces output including tables and figures are available at the following GitHub archive: