После возвращения домой я почувствовал облегчение.

Because после requires the genitive case.

The base noun is возвращение (return / returning).
In the genitive singular, it becomes возвращения.

So:

• после = after
• возвращение = return / returning
• после возвращения = after returning / after the return

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после приезда = after arriving

Russian often uses a verbal noun where English might use an -ing form.

So English after returning home becomes literally something like:

после возвращения домой = after the return home / after returning home

Here, возвращение is a noun made from the verb возвращаться / вернуться (to return).

This structure is very natural in Russian. Another common way to say the same idea would be:

После того, как я вернулся домой, я почувствовал облегчение.
= After I returned home, I felt relief.

That version uses a full clause instead of a noun.

Домой means homeward / to home / home in the sense of movement toward home.

Compare:

• дома = at home (location)
• домой = home (direction, motion toward home)

So:

• Я дома. = I am at home.
• Я иду домой. = I am going home.

In your sentence, the idea is returning home, so direction is involved. That is why домой is used.

В дом would usually mean into the house, focusing on entering a building physically, not the idiomatic English idea of going home.

Почувствовал is the past tense form of почувствовать (to feel, to sense in a completed sense).

In Russian past tense, verbs show gender in the singular:

• я почувствовал = I felt — if the speaker is male
• я почувствовала = I felt — if the speaker is female
• оно почувствовало = it felt
• мы почувствовали = we felt

So -л is part of the masculine singular past tense form.

This is about aspect.

• чувствовать = imperfective
• почувствовать = perfective

Почувствовать means to feel / to come to feel / to experience at a certain moment.
It often emphasizes the onset of a feeling.

So:

• я почувствовал облегчение = I felt relief / I experienced relief
• я чувствовал облегчение = I was feeling relief / I felt relief over a period of time

In this sentence, the speaker is describing the moment or result that came after returning home, so the perfective почувствовал fits very well.

Yes. Облегчение is the direct object of почувствовал, so it is in the accusative case.

The noun is:

• nominative: облегчение
• accusative: облегчение

These look the same because облегчение is a neuter inanimate noun, and for many such nouns, nominative and accusative singular are identical.

So:

• Это облегчение. = This is relief.
• Я почувствовал облегчение. = I felt relief.

Same form, different grammatical role.

Yes. Russian word order is fairly flexible.

The sentence

После возвращения домой я почувствовал облегчение.

is neutral and natural. It puts the time/background part first: after returning home.

You could also say:

Я почувствовал облегчение после возвращения домой.

This is also correct. It sounds slightly more focused first on what I felt, and then adds when/after what.

Russian word order often changes for focus, emphasis, or style, not because the grammar changes.

Yes, absolutely.

Compare:

• После возвращения домой я почувствовал облегчение.
• После того, как я вернулся домой, я почувствовал облегчение.

Both are natural.

The difference is mainly structural:

1. После возвращения домой...

   • more compact
   • uses a noun (возвращение)

2. После того, как я вернулся домой...

   • more explicit
   • uses a full clause (I returned home)

English learners often find the second one easier at first because it maps more directly onto English grammar.

Russian often leaves out possessive words like my when they are obvious from context.

In English, you might think of:

• After my return home
• After returning to my home
• After I returned home

But in Russian, после возвращения домой already strongly suggests that it is the speaker’s return home, especially because the main clause has я почувствовал.

Russian does this a lot when the meaning is clear:

• Я поднял руку. = I raised my hand.
• Он закрыл глаза. = He closed his eyes.

Adding a possessive is possible when needed, but often unnecessary.

Yes, облегчение is the standard and very natural noun for relief in this context.

The phrase

почувствовать облегчение

is a very common collocation and means:

• to feel relief
• to experience relief

It can be used for emotional, mental, or even physical relief.

Examples:

• Я почувствовал облегчение после экзамена. = I felt relief after the exam.
• Она почувствовала облегчение, когда всё закончилось. = She felt relief when everything ended.

So the phrase in your sentence is very idiomatic Russian.