As far as I know that Eid , fasting/Ramadan can only be observed with reference to the moon-sighting only and not pre-calculated scientific knowledge of moon’s appearance. I personally do not know if that method is accurate or not but for so many of my friends how do I convince them that method is not as per the Islamic laws? But they insist that the scientific method is correct and accurate. Please advise me as how do I convince them as which method is correct based on the teachings of Quran and Sunnat e Rasool ( May Allah bless him and his progeny)

There are different schools in the pre-calculated method regarding the possibility of moon sighting. Some of them are very simple, while others are very tough in their methods.
Hence, if the toughest method said that the crescent is there and could be seen, and you became confident that it is the case, then there is no need for observation.