FE/BE coupling for
time-dependent interface problems in electromagnetics
We present an h-version of the FE/BE coupling method to
the eddy current problem for time dependent Maxwell's equations. For
the time discretization we use the discontinuous Galerkin method with
piecewise linear test and trial functions; for the space discretization
we take -conforming vector-valued polynomials to
approximate the electric field in the conductor and surface curls of continuous
piecewise polynomials on the boundary of
to approximate the twisted tangential trace of the magnetic field on . In matrix form the fully discrete scheme of
the discontinuos Galerkin method is the following linear system.
and are the singular
layer, double layer and hypersingular boundary integral operators,
The linear system is symmetric and indefinite, and
iteratively solved by Hybrid Modified Conjugate Residual Method (HMCR).
Also, we derive a priori and a posteriori error estimates
implement the adaptive refinement procedures with the software package Maiprogs
The a priori error estimate shows an convergence rate
chosen proportional to the mesh size . Figure 1 shows the convergence of the error
in the energy norm and the behavior of the error estimators (of
residual type). denotes
the degrees of freedom and the time step
We use as
preconditioner the inverse of the diagonal blocks. A comparison of the
condition numbers is shown in Table 1.
The Figure (2) - (4) show the adaptive meshes with hanging nodes.
Figure 2 Start mesh
Figure 3 after 3 refinement
|Figure 4 after 5 refinement
This project includes
- The ongoing PhD-Thesis (Ricardo Prato): "FE/BE
for time-dependent interface
problems in electromagnetics"
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