You might want to try this...

When the form is submitted, send a cookie to the client's browser. So when the page is refreshed, have your code check for the presence of the cookie. If the cookie is present, it means the entry has already been added so have the script print a message such as "Your submisstion has already been processed".

What if the user has disabled cookies? In this case, when the form is accessed, set a cookie to the client's browser. When the form is submitted, check for the presence of this form-accessed cookie. If no cookie is present, it means cookies have been disabled and so the submission will not be processed. In other words, have the script check for the presence of cookies before allowing further processing.