When I connect the new production Database server with the front end application for the first time I faced the following error:
SQL Server does not exist or access denied
I turn off the Fire wall set up in my DB server and then the connection string started working well.
The following might also be the reasons if you face the same error:
1. Account does not have the access
2. Password is wrong
3. SQL Server Browser services should be started (for very few occasions)