Q. Re- question 1437 and 1438 if there is a mitzva of simcha on Rosh Hashana. Is there a mitzva of simcha on Yom Kipur? Is the day called a chag or moed?

A. Talmud (Taanis 30b) teaches that there were no greater Yomim Tovim than Yom Kipur… since it was a day of forgiveness and redemption. Rabbenu Yonah (Shaarei Teshuva 4: 8) mentions that the seudos (feasts) one celebrates on Erev Yom Kippur, are related to seudos of Yom Tov, See Sichos Rav Sh. D. Pincus (p. 287). He also mentions (p. 339) that since it is also the day of the giving of the Torah, (2nd. Luchos) it became “Beyom Simchas Livo.” Yom Kipur is also included on the two Torah chapters of the moadim and is icluded in the hekesh comparison that unites them all.
The issue appears to hinge on a dispute among the tanna’im in the mishna in Moed Katan (19a) as to whether or not Rosh Hashanah and Yom Kipur cancel aveilus.  As we accept Rabban Gamliel’s ruling, that “Rosh Hashanah and Yom Kippur are like the Shalosh Regalim with respect to aveilus, we may deduce that according to Halacha, the simcha obligation indeed applies on Yom Kipur too.
If someone mistakenly recited the brocho of “Lehadlik ner shel YomTov” on Yom Kipur candle lightning instead of “Ner shel Yom Hakipurum,” Poskim maintain that after the fact he complies and he does not have to repeat the brocho. (Maharam Brisk 2: 44, Kovetz Sifsei Chachomim, Hisorerus Teshuva 122, Nitei Gavriel 25: 6. See also Kovetz Halochos 28: 2)
However, Maseches Sofrim (19: 6) maintains that we do not mention (on the tefilos) on Yom Kipur that it is a Yom Tov, Moed or simcha, since there is no simcha without eating.
Similarly,Mateh Efraim (582: 2) opines that when Yom Kipur coincides with Shabbos, we recite; “Bayi beshalom… betzahala,” and omit besimcha.
As we mentioned in regards to Rosh Hashana (Q. 1437 and 1438) the shalmei simcha offerings are not brought, and there is no requirement to partake of the sacrificial meat of shelamim; and on Yom Kippur we fast.
Horav Shlomo Miller’s Shlit’a opinion is that there is a mitzva of simcha on Yom Kippur, but different than on the other Yomim Tovim.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a