A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas
A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas
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DOI：
10.1137/s0036142901388731
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发表时间：
2002-03
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影响因子：
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通讯作者：
S. Christiansen;J. Nédélec
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文献类型：
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作者：
S. Christiansen;J. Nédélec
我们描述了电场积分方程（EFIE）的伽辽金近似的预处理技术，该技术出现在谐波电磁波的散射理论中。它基于卡尔德隆公式和亥姆霍兹分解的离散化。我们证明了该方法的几个属性，特别是它在伽辽金空间的子空间上产生了变分解，对于该子空间，我们有 LBB inf-sup 条件。当与连续算子相关的 Krylov 空间是非简并的时，我们证明离散 Krylov 空间随着网格细化趋于零而收敛；此外，当 EFIE 在连续 Krylov 空间上非简并时，离散 Krylov 迭代会向连续迭代收敛。我们还认为，对于某些 C > 0 和 $\alpha>0$，人们可能期望连续 Krylov 迭代表现出 $n \mapsto C^n(n!)^{-\alpha}$ 形式的超线性收敛。最后，我们用数值实验来说明该理论。
We describe a preconditioning technique for the Galerkin approximation of the electric field integral equation (EFIE), which arises in the scattering theory for harmonic electromagnetic waves. It is based on a discretization of the Calderon formulas and the Helmholtz decomposition. We prove several properties of the method, in particular that it produces a variational solution on a subspace of the Galerkin space for which we have an LBB inf-sup condition. When the Krylov spaces associated with the continuous operators are nondegenerate we prove that the discrete Krylov spaces converge as the mesh refinement goes to zero; when, moreover, the EFIE is nondegenerate on the continuous Krylov spaces, the discrete Krylov iterates converge towards the continuous ones. We also argue that one might expect the continuous Krylov iterates to exhibit superlinear convergence of the form $n \mapsto C^n(n!)^{-\alpha}$ for some C > 0 and $\alpha>0$. Finally, we illustrate the theory with numerical experiments.