This article develops a mesh space mapping (MSM) technique between low-fidelity (coarse) and high-fidelity (fine) models to accelerate the EM simulation of microwave structures such as antennas, filters and substrate ICs at the circuit level and enhance the optimization efficiency. The radial basis function technique is incorporated into a mesh morphing algorithm applied to the coarse mesh structure to overcome discontinuities that arise from mesh reconstruction. Approaches to sharpen structural processing and simplify structures for the coarse space are introduced. The Matrix Padé via Lanczos (MPVL) method is incorporated into the finite element calculations to significantly reduce the computation workload. Several design examples illustrate the effectiveness of the technique.