Main navigation | Main content

HOME » PROGRAMS/ACTIVITIES » Annual Thematic Program

PROGRAMS/ACTIVITIES

Annual Thematic Program »Postdoctoral Fellowships »Hot Topics and Special »Public Lectures »New Directions »PI Programs »Industrial Programs »Seminars »Be an Organizer »Annual »Hot Topics »PI Summer »PI Conference »Applying to Participate »

Talk Abstract

Multigrid Method for Maxwell's Equations

Multigrid Method for Maxwell's Equations

The problem under consideration is the wave equation for the electric field in a 3D cavity with perfectly conducting walls. When treated in the time domain an implicit timestepping is highly desirable due to its unconditional stability. In a finite-element setting each timestep involves the solution of a discrete variational problem for the bilinear form $(\cdot,\cdot)_0+(${\bf curl}$\cdot,$ {\bf curl}$\cdot)_0$ posed over {\bf H(curl; $\Omega$)}. I relied on Nedelec's {\bf H(curl; $\Omega$)}-conforming finite elements (edge elements), which yield a viable discretization for Maxwell's equations.

A multigrid method is employed as a fast iterative solution method. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the {\bf curl}--operator, discrete Helmholtz--decompositions of the finite element spaces are crucial for the design and analysis of the multigrid scheme.

Under certain assumptions on the computational domain and material functions, a rigorous proof of asymptotic optimality of the multigrid method can be given; it shows that convergence does not deteriorate on very fine grids. The results of numerical experiments confirm the practical efficiency of the method.