to drain / let off / desist
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to drain
To let a liquid or gas escape from a container, a tire, or a system in a controlled way.
Der Mechaniker muss zuerst das alte Öl ablassen, bevor er neues einfüllt.
The mechanic first has to drain the old oil before putting in new oil.
'Ablassen' (to drain / let off / desist) is a separable verb. The prefix 'ab-' is placed at the end of the sentence in a main clause. In the infinitive with 'zu', 'zu' comes between the prefix and the verb.
Ich lasse das Badewasser ab.
I'm draining the bathwater.
The grammatical structure depends on the meaning. You can say 'etwas (Akk.) ablassen' (to drain something, e.g. water), 'Dampf (Akk.) ablassen' (to let off steam, fixed expression), or 'von etwas (Dat.) ablassen' (to give something up).
Der Bauer lässt das Wasser vom Feld ab.
The farmer drains the water from the field.
In the literal meaning, 'ablassen' is often used for liquids or gases. Typical combinations are 'Wasser ablassen' (to drain water, from the bathtub), 'Öl ablassen' (to drain oil, from the engine), or 'Luft ablassen' (to let air out, from a tire).
Vergiss nicht, die Luft aus der Matratze abzulassen, bevor du sie einpackst.
Don't forget to let the air out of the mattress before you pack it up.
'Dampf ablassen' (to let off steam) is a fixed idiom. It means releasing pent-up anger or frustration verbally or through an activity in order to feel relieved.
Er schrie laut, um endlich mal Dampf abzulassen.
He shouted loudly to finally let off some steam.
In the meaning 'aufgeben' (to give up), 'ablassen' is always used with the preposition 'von' and the dative. You desist from 'von einem Plan' (a plan), 'von einer Idee' (an idea), or even 'von einer Person' (a person).
Die Polizei ließ nicht von dem Verdächtigen ab.
The police did not let up on the suspect.
to give up / submit
'aufgeben' and 'ablassen von' can both mean that you do not continue with a plan or attempt. 'aufgeben' is more general and is used very often when someone no longer expects success or stops pursuing a hope. 'von etwas ablassen' sounds somewhat more elevated or formal and emphasizes that you are no longer pursuing an intention. It normally appears with 'von': you 'lässt von einem Plan oder einer Idee ab' (give up on a plan or idea). In everyday language, 'aufgeben' is usually more natural; 'ablassen von' fits better in more serious or written texts.
Nach der Warnung der Ärztin ließ Nico von seinem Plan ab, sofort wieder zu trainieren.
After the doctor's warning, Nico gave up on his plan to start training again immediately.
Zu Hause gebe ich die Suche nach dem alten Schlüssel heute auf.
At home, I am giving up the search for the old key today.