How to speed up execution time in a SQL Server 2016 database which has restored from 2008 R2?
have restored the backup of the database from SQL V-2008R2 to SQL version 2016 and now facing the performance issue.


Tossing away considerations like hardware,

Are your configuration settings set? 

Is your max memory setting high enough? Is your Power plan set to high performance?

The database should be in the same state as it was on the SQL 2008 R2 Instance when the backup occured.
You could always update your statistics to get some better query plans.


  How can the issue be solved without changing database Compatibility


If you can set your compatibility level to 110 (2012) then you could enable traceflag 2312 to get the sql 2014 Cardinality estimator 
(this can improve but also degrade the performance of queries).
Traceflag 4199 might also help.

It is hard to help without more information.