Can I stay in Schengen area after my visa expires if I enter while the visa is valid?

You can enter a Schengen country on the last day of validity of your visa but, unlike US visas, you must also legally leave the Schengen area before your visa expires (or obtain some other visa or legal means to stay). It's also perfectly possible to enter on a visa and stay and leave on another one.

If you are entering on the last day of validity and you don't have any other (Schengen) visa, I would expect the border guards to be somewhat suspicious so it would be best to have a good explanation or a plan to leave the area in time. As you might know, your passport and your visa should also be checked upon leaving the Schengen area so if you have not been caught before, you are likely to be found out when trying to return to your country of residence.

If you are found to have overstayed, the exact penalty will depend on the country (those rules have not been harmonized in the whole Schengen area), on whether you were caught at the exit port or in another context and, to some extent, on the border guard or police officer (if you are already leaving and are only a couple of days late, you might get lucky but if you are caught within the country after a few months, it's something else).

Generally speaking, a fine is likely and an entry ban is possible. Such a ban would be registered in a database called the SIS and would make it impossible for you to enter the whole Schengen area (and probably also more difficult to obtain a visa in the future after that). As an example, here are the rules for the Netherlands.

Schengen countries check and stamp your passport when you leave so even if they'd let you in the country if you left on an expired visa that could make future attempts to get a visa more complicated.

I would strongly advise against it.

I can categorically say most of the post above are too judgemental. I have been late on my visa more than twice. It really all depends from where you exit the Schengen zone. Amsterdam is very accommodating, no fine and just a warning that you might be on the SIS list but more than likely not. The key for me was just honesty. I overstayed because I love my stay so much.

one year multiple-entry Schengen visa

Read it (and the conditions) VERY carefully. "Visa" (the word) is frequently mis-used. Strictly speaking, a visa gets you across the border, at which time you are given permission to remain for a period of time.

So, if your visa says something like "remain for 90 days within a 180-day period" you can use up your remaining days beginning on the last day of your visa.

If there is another term like "within a 180-day period from the date of first entry" then you better check your calendar.

Where I live, visas are usually valid for a year. This means you can enter the country any time within one year from the issue date on the visa. At the airport they stamp "USED" on the visa and put another permit beside it. Officially that's called "Status of Residence" but everyone calls it a visa.

Overstaying is usually a very bad idea. Depending on the duration, the country involved, and the mood of the border guard that day, penalties range from "Don't do it again" to significant jail time.