
vollkommen
completely
Means fully, entirely, or absolutely. It is often used as an adverb to intensify an adjective or an opinion.
Das ist eine vollkommen neue Situation für uns.
This is a completely new situation for us.
complete / perfect
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completely
Means fully, entirely, or absolutely. It is often used as an adverb to intensify an adjective or an opinion.
Das ist eine vollkommen neue Situation für uns.
This is a completely new situation for us.

perfect
Describes something that is perfect, flawless, or immaculate.
Auf der Suche nach der vollkommenen Harmonie reiste er um die Welt.
In search of perfect harmony, he traveled around the world.
'Vollkommen' is made up of 'voll' (full) and the participle of 'kommen' (to come). The literal idea is that something has 'come to fullness,' meaning it is complete or perfect.
Nach dem langen Spaziergang herrschte vollkommene Stille im Haus.
After the long walk, there was perfect silence in the house.
'Vollkommen' can be used as an adverb for emphasis (meaning 1) or as a normal adjective (meaning 2). As an adverb, it is not declined; as an adjective before a noun, it is.
Ich bin mit deiner Arbeit vollkommen zufrieden.
I am completely satisfied with your work.
In its role as an adverb, 'vollkommen' often intensifies adjectives or opinions. Typical combinations are 'vollkommen richtig' (completely right), 'vollkommen falsch' (completely wrong), 'vollkommen egal' (completely irrelevant), or 'vollkommen ausreichend' (completely sufficient).
Es ist vollkommen egal, ob wir heute oder morgen anfangen.
It is completely irrelevant whether we start today or tomorrow.
In the meaning of 'perfect,' 'vollkommen' is often combined with abstract nouns, such as 'vollkommene Harmonie' (perfect harmony), 'vollkommene Stille' (perfect silence), or 'vollkommene Dunkelheit' (complete darkness).
Für einen Moment herrschte vollkommene Stille im Raum.
For a moment, there was perfect silence in the room.
completely
Völlig and vollkommen can strongly intensify an adjective or a statement. Völlig neutrally emphasizes the full degree, while vollkommen can also sound more elevated and sometimes give a value judgment suggesting perfection. Use völlig in everyday situations and vollkommen when the stylistic or evaluative nuance fits.

Mit dieser eleganten Lösung war der Kunde vollkommen zufrieden.
The customer was completely satisfied with this elegant solution.
Nach der guten Nachricht war sie völlig zufrieden.
She was completely satisfied after hearing the good news.