Implement smart search by text field in database

Suppose I have a table named movie with a field name, that is a VARCHAR field.

I want to implement a search bar in my website, so that when a user enters a string I can query movies with that name.

My first approach was quite naive:

select *
from movie
where name like '%user_string%';

The limitations are:

1. Special characters. Say the user string is "Lets go", I want it to return the movie with the name "Let's go" even if the apostrophe is missing.
2. Accents. Say the user string is "Pokemon", I want it to return the movie with the name "Pokémon" even if the accent is missing.

My idea was to create an additional normalized_name field, that is computed using the name field with all special characters and accents stripped. Then the query would become:

select *
from movie
where normalized_name like '%user_string%';

For instance: user searches for pokemon, the database query return a movie with the normalized_name = pokemon, which real name is Pokémon. Obviously the user string would be first normalized too - in order to allow searching by the movie real name too.

Now, is this a valid approach? What is the most widely used one - that may also make the search even better? Is there any literature on the matter?