'However', 'nevertheless', and 'nonetheless' connect two sentences and show an unexpected result to what has been stated in the preceding sentence. 'However' is the most common of these expressions, whereas 'nevertheless' and 'nonetheless' are more formal are generally used in written English.
"However, she decided to continue with an optical storage system."
"However, I optimized my system independently."
"Nevertheless, an Ethernet connection is not always the most reliable choice. "
"Nevertheless, we increased our rate of output."
"Nonetheless, we are going to analyze every packet."
"Nonetheless, we are going to continue to implement SEO tactics and increase our page ranking."