DRAFT 1 Parallel and Explicit Finite-Element Time-Domain Method for Maxwell Equations Joonshik Kim and Fernando L. Teixeira, Senior Member, IEEE Abstract— We construct a parallel and explicit finite-element time-domain (FETD) algorithm for Maxwell equations in simpli- cial meshes based on a mixed E-B discretization and a sparse approximation for the inverse mass matrix. The sparsity pattern of the approximate inverse is obtained from edge adjacency information, which is naturally encoded by the sparsity pattern of successive powers of the mass matrix. Each column of the approximate inverse is computed independently, allowing for different processors to be used with no communication costs and hence linear (ideal) speedup in parallel processors. The convergence of the approximate inverse matrix to the actual inverse (full) matrix is investigated numerically and shown to exhibit exponential convergence versus the density of the approximate inverse matrix. The resulting FETD time-stepping is explicit is the sense that it does not require a linear solve at every time step, akin to the finite-difference time-domain (FDTD) method. Index Terms— Finite elements, FETD, differential forms, Maxwell equations, parallel computing. I. I NTRODUCTION T HE finite-element time-domain (FETD) method has been extensively used to simulate Maxwell equations in com- plex geometries [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. FETD in simplicial meshes typically requires a linear solve at every time-step. This is a basic drawback of the method, and is in contrast to, for example, the finite-difference time-domain (FDTD) method in rectangular meshes, which is a “matrix-free” algorithm [12]. In [7], the construction of an “explicit” FETD time-stepping (“marching-on-time”) algorithm, i.e. with no need for a linear solve at every time step, was proposed in irregular meshes. This explicit FETD based on a FE discretization with mixed basis functions [3], [13] (denoted as “mixed E-B FETD” [9], [10]) and a SPAI (SParse Approximated Inverse) algorithm [14] to obtain a sparse approximate inverse mass (Hodge) matrix. In this work, we construct a parallel version of the SPAI- based, explicit FETD scheme in simplicial meshes and ana- lyze its performance in three-dimensional (3-D) simulations of Maxwell equations. A key step in the proposed SPAI approximation is the a priori setup of a sparsity pattern for the approximate inverse. Here, we employ sparsity patterns given by successive powers of the mass matrix, which describe successive high-level adjacencies between the edges of the Manuuscript received MM,DD,YYYY. The authors are with the ElectroScience Laboratory and the Department of Electrical and Computer Engineering, 1320 Kinnear Road, The Ohio State University, Columbus, Ohio, 43212, USA. This work was supported in part by NSF grants ECCS-0347502 and ECCS-0925272, and AFOSR grant FA 9550-04-1-0359. Supercomputing resources were provided by the Ohio Supercomputing Center under grants PAS-0061 and PAS-0110. mesh. We examine the convergence of the approximate system matrix inverse (in general, a combination of mass and stiffness matrices) and of the approximate mass matrix inverse to their respective exact inverses (which are full), for different sparsity levels and time step increments. We show that exponential con- vergence is obtained for time scales below the Courant limit, where the system matrix converges to the mass matrix. We discuss how this exponential convergence is associated with properties of the Hodge star operator, which is the continuum equivalent to the mass matrix. Because no communication cost between processors is incurred in the computation of sparse approximate inverses, linear (ideal) speed up can be achieved in parallel processor systems. II. DISCRETIZATION A. Mixed E-B FETD We write Maxwell equations as [15], [16], [18] dE = − ∂ ∂t B (1) d⋆ µ -1 B = ∂ ∂t ⋆ ǫ E (2) where E is a 1-form, B is a 2-form, and d is the exterior derivative operator. The Hodge star (or simply Hodge) op- erators ⋆ ǫ and ⋆ µ -1 generalize the constitutive relations to incorporate all metric information as well [19], [20], [21], [22], [23], [24], [25]. The discretization of the above is done by expanding E and B in terms on Whitney forms [16], [17]. By denoting the Whitney p-form associated with the i- th p-cell (viz., nodes, edges, faces, volumes for p =0, 1, 2, 3 respectively) as W p i , the expansion writes [7], [9] E = Ne  i=1 e i W 1 i (3) B = N f  i=1 b i W 2 i (4) where N e , N f are the total number of edges and faces of the 3-D FE mesh, respectively. The semi-discrete counterpart to (1) and (2) is [7], [9], [10], [13] Ce = − d dt b (5) C T [⋆ -1 µ ]b = d dt [⋆ ǫ ]e (6) where e =[e 1 , ··· ,e Ne ] T (7) b =  b 1 , ··· ,b N f  T (8) (9)