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Summary form only given. Numerical methods, particularly their applications in solving Electromagnetic problems, have been very active at ElectroScience Lab., Electrical Engineering Dept., the Ohio State University. Part I of this talk will focus on recent developments in the frequency domain approaches. Specifically, we have been making significant advancements in the integral equation methods as well as finite element methods in the frequency domain. Topics that will be addressed in this talk will include our collective efforts on 1. Automatic mesh generation. 2. Multilevel FMM methods. 3. Matrix compression methods based on low-rank approximation. 4. Error estimation and adaptive finite element methods. 5. Preconditioning techniques in both finite element as well as integral equation methods.