Integral equation formulations and discretizations
Convergence history of an ill-conditined and a well-conditioned formulation.
The surface integral equation method (method of moments) is a popular numerical method for solution of many open-region scattering and radiation problems. This technique essentially reduces the dimensionality of the problem by one and considers the unknowns on the surfaces and interfaces of homogeneous regions only.
One of the major shortcomings of the surface integral equation method is that the resulting matrix can be very poorly conditioned. The condition number of the matrix typically grows rapidly as the discretization density is increased or the frequency is decreased. This problem is called dense mesh or low frequency breakdown and leads to poorly converging iterative solutions and prevents efficient use of the fast solvers. Our group is investigating various strategies to obtain well-conditioned and stable formulations and to improve the condition of the matrix.
- Well-conditioned and stable formulations
- Mixed discretization schemes and dual basis functions
- Calderon preconditioning
References
- P. Ylä-Oijala, S. P. Kiminki and S. Järvenpää: Calderon preconditioned surface integral equation method for composite structures with junctions, IEEE Transactions on Antennas and Propagation, Vol. 59, No. 2, pp. 546-554, Feb. 2011.
- P. Ylä-Oijala, S. P. Kiminki and S. Järvenpää: Solving IBC-CFIE with dual basis functions, IEEE Transactions on Antennas and Propagation, Vol. 58, No. 12, pp. 3997-4004, Dec. 2010.
- P. Ylä-Oijala, M. Taskinen and S. Järvenpää: Analysis of surface integral equations in electromagnetic scattering and radiation problems, Engineering Analysis with Boundary Elements, 32, 196-209, 2008.
- P. Ylä-Oijala, M. Taskinen and S. Järvenpää: Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods, Radio Science, 40(6), RS6002, 2005.