Discrete Helmholtz Decomposition for Electric Current Volume Integral Equation Formulation

Abstract

A volume integral equation formulation for the equivalent current is investigated by decomposing the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -conforming unknown current into orthogonal functions. The decomposition shows that the solenoidal, irrotational and harmonic subspaces scale differently with respect to the material parameter. This has a negative effect on the conditioning of the system, and thus, the convergence of the iterative solution slows down with increasing permittivity. We construct discrete decomposition operators, and use them as a preconditioner for the electric current volume integral equation. The eigenvalues of the resulting system are almost independent on the permittivity. Numerical examples show that the proposed preconditioner improves the condition number and decreases the number of iterations required to solve the system. However, efficient evaluations of the projection operators require additional regularization techniques such as algebraic multigrid preconditioners.