Solution Of Time-harmonic Maxwell Equations Research Articles

Superconvergence Analysis for Linear Tetrahedral Edge Elements

Back in 1992, Monk (Numer Math 63: 243---261, 1992 ) found that the numerical solution of time-harmonic Maxwell equations shows a superconvergence result $$O(h^2)$$ O ( h 2 ) in the discrete maximu...

Fully automatic hp-adaptivity for Maxwell’s equations

I report on the development of a fully automatic hp-adaptive strategy for the solution of time-harmonic Maxwell equations. The strategy produces a sequence of grids that deliver exponential convergence for both regular and singular solutions. Given a (coarse) mesh, we refine it first globally in both h and p, and solve the problem on the resulting fine mesh. We consider then the projection-based interpolants of the fine mesh solution with respect to both current and next (to be determined) coarse grid, and introduce the interpolation error decrease rate equal to the difference of the old and new (coarse) mesh interpolation errors vs. number of degrees-of-freedom added. The optimal hp-refinements leading to the next coarse grid are then determined by maximizing the interpolation error decrease rate.

Modeling optical properties of liquid-crystal devices by numerical solution of time-harmonic Maxwell equations.

We consider numerical modeling of the optical properties of devices typical of beam-steering devices based on liquid-crystal materials: two-dimensional, anisotropic and inhomogeneous dielectric properties, periodic in one dimension. A mathematical formulation of the system of second-order partial differential equations for the components of the time-harmonic electric field is discretized by using a finite-element method based on curl-conforming edge elements. The discrete equations are also interpreted as equivalent finite-difference equations. It is shown how the resulting large sparse complex system of linear algebraic equations can be solved by an iterative method with convergence accelerated by a preconditioner based on fast Fourier transforms. Benchmarking results and the application to a realistic problem are reported. The practical limitations of the approach and its advantages and disadvantages compared with other approaches are discussed.