Plural of 'performance'
MSWord grammar checker is correct as far as it goes. You can either use the plural "performances" with "are", or the singular (or uncountable) "performance" with "is".
The real question here is whether "performance" is countable or not.
According to Oxford Learners Dictionaries, this usage of "performance" (meaning 3) can be either countable or uncountable. So that doesn't really clear the matter up!
In your particular situation you could pluralise it, and use "are", but it doesn't quite feel right to me. Presumably it doesn't to you either, or you wouldn't be asking the question! If you say:
The performances of the new and existing methods are compared
then it sounds to my ear as if both the new method and the existing method have multiple performances, which are being compared (presumably Performance n new method being compared with Performance n existing method).
The problem, though, is that you can't compare one "uncountable" item to another, because you'd have to have two of them to compare one against the other, and you can't have two of something that is uncountable. So uncountable "performance" with "is" isn't an option.
Your only other option is to reword the sentence to avoid the situation.
I would suggest:
The performance of the new method is compared to that of the existing method
or possibly
The relative performance of the new and existing method is evaluated.
More context is needed. If you are referring to two different performances (the new one and the existing one), it should be plural, and the verb of which 'performances' is the subject should also be plural. This, in that case, is the correct sentence:
The performances of the new and existing methods are compared.
On the other hand, there might be one performance that features both methods. In that case, use this:
The performance of the new and existing methods is compared.
In either case, this is incorrect:
The performance of the new and existing methods are compared.
because the verb and subject do not agree.