What should be the professional action? If someone faces problems to satisfy client and client is always in complaining mood without understanding the ground reality.

The professional action is acknowledge the complain, apologize, then thank the customer, smile in a most genuine way and leave it like that. If you've exhausted all the efforts and all the necessary actions to satisfy a client then there's nothing else left to be done but end it. No point to argue if the other end is not willing to understand from the very start. Be an example to your client.