FE/BE coupling for time-dependent interface problems in electromagnetics

We present an h-version of the FE/BE coupling method to solve 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 $\mathbf{H}(\mathbf{curl},\Omega)$ -conforming vector-valued polynomials to approximate the electric field in the conductor $\Omega_{}$ and surface curls of continuous piecewise polynomials on the boundary $\Gamma$ of $\Omega_{}$ to approximate the twisted tangential trace of the magnetic field on $\Gamma$ . In matrix form the fully discrete scheme of the discontinuos Galerkin method is the following linear system.

$\displaystyle \left( \begin{array}{cc\vert cc} \scriptstyle \rule{0mm}{5mm} \fr... ...idetilde{F}_2 \rule{0mm}{5.5mm} 0 \rule{0mm}{5.5mm} 0 \end{array} \right)$

with

$\begin{displaymath} \begin{array}{ccc} {\cal M}^{*} = [(\rho \Phi_i,\Phi_j)]^{i=... ..._i,\psi_j \rangle]^{i=1,\ldots,m}_{j=1,\ldots,m} && \end{array}\end{displaymath}$

where $\mathcal{V}$ , $\mathcal{K}$ and $\mathcal{W}$ are the singular layer, double layer and hypersingular boundary integral operators, respectively.

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 and implement the adaptive refinement procedures with the software package Maiprogs.

The a priori error estimate shows an $O(h)=O(N^{-\frac{1}{3}})$ convergence rate where $\Delta t$ is 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

**Figure 1:** error estimators
$\begin{figure}\begin{center} \epsfig{file=b1.eps,scale=0.4}\end{center}\end{figure}$

We use as preconditioner the inverse of the diagonal blocks. A comparison of the condition numbers $\kappa({\mathcal{ A} })$ is shown in Table 1.

$\scalebox{0.8}{ \begin{tabular}[htb]{*{3}{\vert c}\vert} \hline &\multicolumn{1}... ...8\ \hline \multicolumn{3}{c}{}\\ \multicolumn{3}{c}{Table 1}\\ \end{tabular}}$

The Figure (2) - (4) show the adaptive meshes with hanging nodes.

$\epsfig{file=adap11.eps,scale=0.2}$

$\epsfig{file=adapD14.eps,scale=0.2}$

Figure 2 Start mesh

Figure 3 after 3 refinement


$\epsfig{file=adapD16.eps,scale=0.2}$
Figure 4 after 5 refinement

This project includes

The ongoing PhD-Thesis (Ricardo Prato): "FE/BE coupling for time-dependent interface problems in electromagnetics"

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FE/BE coupling for time-dependent interface problems in electromagnetics

This project includes

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