keep plugging away
idiom

- to keep working in a determined way

(Cambridge Dictionary Of American Idioms)

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WORD ORIGIN

The verb "to plug" has been used in the sense of working hard at something since around the turn of the 20th century. In this context, it is frequently combined with "along" or "away" to form the idioms "plug along" and "plug away." Plug can be used by itself in the same context, such as "You're doing a fine job—just keep plugging."

In addition, the verb plug also means to publicise something in a favourable manner, like in a radio or television broadcast: "He plugged his new book on all the major talk shows this week."

Plug is also part of two common phrasal verbs:

plug in - to connect something to an electrical outlet

plug into - to connect or be connected to in the manner of an electrical device (The local system is plugged into the national telephone network. This computer plugs into a server.)

plug into (slang) - to be closely familiar with or connected to something (The CEO is plugged into high levels of government. She's plugged into the local music scene.)

Finally, plug is a noun that among other things refers to an object used to fill a hole or an opening, a dense mass of material that obstructs a passage, and a cylindrical piece cut from something larger to be used as a sample.

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SYNONYMS

apply, attend to, buckle down, concentrate, devote, dig, direct, engage in, focus on, give, go at, go for, grind, hammer away, have at, knuckle down, peg away, persevere, pitch into, plod, struggle, sweat, take up, throw, try hard, undertake

(Roget's New Millennium™ Thesaurus)

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ANTONYMS

cut back, give up, let up, mellow out, slack off, slow down, tail off

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SMUGGLE OWAD INTO TODAY'S CONVERSATION:
say something like:

"Don’t worry! We won’t give up, we’ll keep plugging away until we solve the problem."