to accept / compensate
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to come to terms with
To reluctantly accept an unpleasant or unchangeable situation and stop resisting it.
Er musste sich damit abfinden, dass er den Job nicht bekommen hatte.
He had to come to terms with the fact that he had not gotten the job.
'Abfinden' is a separable verb. The prefix 'ab' stands at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er findet sich mit der Situation ab.
He comes to terms with the situation.
In the meaning of 'akzeptieren' (to accept), the verb is always reflexive and is used with the fixed preposition 'mit' and the dative. The structure is: sich mit etwas (Dativ) abfinden.
Sie musste sich mit dem Gedanken abfinden, umziehen zu müssen.
She had to come to terms with the idea of having to move.
In the meaning of 'entschädigen' (to compensate), the verb is transitive. You compensate someone (accusative), often with a sum of money. The structure is: jemanden (Akkusativ) mit etwas (Dativ) abfinden.
Nach der Kündigung fand die Firma ihn mit einer hohen Summe ab.
After the dismissal, the company compensated him with a large sum.
You can easily distinguish the two main meanings by the sentence structure. If the verb appears with the reflexive pronoun 'sich', it is about acceptance. If it appears with a person in the accusative as the object, it is about financial compensation.
Ich finde mich damit ab. (Akzeptanz)
I come to terms with it. (acceptance)
to accept
'akzeptieren' and 'sich abfinden mit' can both mean that you accept a situation. 'Akzeptieren' is more general and can sound neutral or even positive. You acknowledge that something is the way it is, or you accept it. 'Sich abfinden mit' is almost always used for something unpleasant that you cannot change. It often sounds reluctant: you are not happy with it, but you stop fighting against it. A memory aid: you can 'akzeptieren' something and agree with it; you often only have to 'sich mit etwas abfinden' because there is no better option.
Lisa muss sich mit dem kleineren Büro abfinden, obwohl sie unzufrieden ist.
Lisa has to come to terms with the smaller office, even though she is unhappy.
Nach dem Gespräch akzeptiert du die neue Regel im Verein.
After the conversation, you accept the new rule in the club.