To convert your example string (which I assume is not actually a date value):
to_date('12.August 2020', 'd.MMMM yyyy')
This will give a datetime value.
To convert it directly to the new format:
format_date(to_date('12.August 2020', 'd.MMMM yyyy'), 'yyyy-MM-dd')
Producing 2020-08-12
You will need to substitute your field name for '12.August 2020' for this to work, i.e.:
format_date(to_date("myDate", 'd.MMMM yyyy'), 'yyyy-MM-dd')
See here for information on date formats and string conversion, or look at the help in the field calculator for format_date()
.