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Hi T. M. Howe,
For accuracy point of view inside HFSS, convergence criterion defined based on S-parameter if ports are define and solution types are driven modal or terminal. You can change the converence criterion delta S, increases no of pasess for adaptive passes for mesh. You can specify multiple frequencies for adaptive meshing since sweep is a bradband 1 - 15 GHz. This will make sure that mesh is adapated as per frequency throgout the frequency range. In Analysis setup dialog, you can slect Solver option such as Direct and order of the basis fucntion as Second order.
With your current simulation you can plot the mesh at diferent frequency point and see how goo it is. Else you can apply meshin seeding on metametrial structure example length based meshing etc, If there is any curved surface or geometry is present accordingling you can enable curved mesh setting.
Apart from S-parameter based convergence you can define converence for adaptive meshing using Expression cache which is present in Setup dialog.
Most of this details you can look into HFSS help documentation.
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Thanks,