Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations

We consider the numerical discretization of the time-domain Maxwell's equations with an energy-conserving discontinuous Galerkin finite element formulation. This particular formulation allows for higher order approximations of the electric and magnetic field. Special emphasis is placed on an efficient implementation which is achieved by taking advantage of recurrence properties and the tensor-product structure of the chosen shape functions. These recurrences have been derived symbolically with computer algebra methods reminiscent of the holonomic systems approach.Comment: 16 pages, 1 figure, 1 table; Springer Wien, ISBN 978-3-7091-0793-

Energy Conserving Higher Order Mixed Finite Element Discretizations of Maxwell's Equations

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 mixed finite element spatial discretization. We demonstrate stability and energy conservation for the variational formulation of this Maxwell's system. We then discuss two implicit, energy conserving schemes for its temporal discretization: the classical Crank-Nicholson scheme and an implicit leapfrog scheme. We next 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-Nicholson 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. Finally, we show computational results to validate our theoretical claims using linear and quadratic Whitney forms for the finite element discretization for some model problems in two and three spatial dimensions

High-accuracy finite-element methods for positive symmetric systems

AbstractA nonstandard-type “least'squares” finite-element method is proposed for the solution of first-order positive symmetric systems. This method gives optimal accuracy in a norm similar to the H1 norm. When a regularity condition holds it is optimal in L2 as well. Otherwise, it gives errors suboptimal by only h12 (where h is the mesh diameter). Thus, it has greater accuracy than usual finite-element, finite-difference or least-squares methods for such problems. In addition, the spectral condition number of the associated linear system is only O(h−1) vs. O(h−2) for the usual least-squares methods.Thus, the method promises to be an efficient, high-accuracy method for hyperbolic systems such as Maxwell's equations. It is also equally promising for mixed-type equations that have a formulation as a positive symmetric system

Parallel Sparse Matrix Solver on the GPU Applied to Simulation of Electrical Machines

Nowadays, several industrial applications are being ported to parallel architectures. In fact, these platforms allow acquire more performance for system modelling and simulation. In the electric machines area, there are many problems which need speed-up on their solution. This paper examines the parallelism of sparse matrix solver on the graphics processors. More specifically, we implement the conjugate gradient technique with input matrix stored in CSR, and Symmetric CSR and CSC formats. This method is one of the most efficient iterative methods available for solving the finite-element basis functions of Maxwell's equations. The GPU (Graphics Processing Unit), which is used for its implementation, provides mechanisms to parallel the algorithm. Thus, it increases significantly the computation speed in relation to serial code on CPU based systems

Finite elements with divergence-free shape function and the application to inhomogeneously-loaded waveguide analysis

A simple mixed triangular edge element is proposed for the finite elements, with which inhomogeneously-loaded and arbitrarily shaped waveguides are analyzed. The shape functions used for approximating the fields are found analytically to be divergence-free. The formulation has been found to encounter spurious-free solutions. As evidence, the non-physical solutions that appeared in the longitudinal component finite element formulation are shown to be absent in the present formulation. A comparison with another mixed element is furnished here in order to demonstrate the advantages provided by the present element </p