I have a character who is a refugee from another country. She comes to the protagonist, and they have a couple of scenes together, viewed from the protagonist.
I read about how important it is to make dialogue convey information about the character, to prevent adverb overdose (e.g. "Blabla, he said wisely."). Intelligence I would indicate with a broader vocabulary. But how do I do that when someone is intelligent but should not have that vocabulary? If I write everything using described gestures, broken grammar and disconnected words - as it would realistically be - that person will quickly sound like an idiot.
Insight. Or, if you're so smart --- Prove It!
I think you misunderstand intelligent people, and I wouldn't rely on vocabulary to indicate it in the first place.
I am a professor in a university, intelligent by conventional standards, but in my work and in my speech I do not use a complex vocabulary, because I consider it far more important to be understood by the people I am speaking to than to impress them.
If I am speaking to students, they are basically high school graduates, so my vocabulary (without slang of my generation or theirs) is set for their level. If I am talking to colleagues I will use all the specialized terminology and references of our fields, but my vocabulary is tuned to what I expect that person to understand, for efficiency's sake. Note this is not a high bar: Many professors struggle with common English outside their fields. They represent the real life example of what you are talking about; highly intelligent with difficulty communicating.
The solution to your problem is to show the reader the consequences of intelligence. Don't try to tell them. Have your foreign speaker use their intelligence to get their point across. Draw a diagram. Solve an equation: here is an example from math.
Frustrated, she rises and walks to the whiteboard. she points at an integral amongst his dense equations, and held up a marker with eyebrows raised. He'd been staring at this problem all week. He picked up his phone, zoomed and snapped the board, then gestured at her to proceed.
She rubbed out a foot of space after the integral and wrote a transformation of it, then looked at him.
Stokes? That would help but you can't do it without proving compact support on a smooth manifold! He had neither! He tried to think of a sign for compact, and held two cupped hands together to make a ball.
She nodded once, and pointed higher on the board to a previous equation, pointed at her head, then cupped her hands as he did, then rubbed one flat hand against the other.
Compact and smooth? He stared at it; and shook his head. How do you sign corners? He touched fingertips with fingers at right angles, then did it again moving his hands around, many corners. She nodded, erased more of the board and began writing ... dividing his domain. He watched for sixty seconds and his throat caught. She was right, the corners were countable, the support would be compact and by Whitney's generalization ... Stokes would hold!
Why hadn't he seen that? He felt a strange combination of thrill with a flush of embarrassment: His problem was solved, and he was an idiot. If it was even still his problem. When she finished, he nodded, lips tight. She turned to erase the rest of his equations after Stokes, and began writing, quickly and without pause, to finish the rest of his manifold transformation in two long lines. She stood back to examine it, then capped the marker and put it back in the tray, turning back to him.
He stood, and quietly applauded, then pointed at her.
"Sue Jen?"
She pointed at herself. "Suzhien. Sue. Zhee. En. Suzhien."
He held out a hand. "Suzhien. Richard."
"Reeshard?" she said, pointing at him. He nodded. Close enough. He lowered his hand.
She pointed at the board. "Suzhien." She made a lifting motion with her hand, "Reeshard."
Then she pointed at Richard. "Reeshard." Again she made a lifting motion; "Suzhien."
With hesitation, she put her hands together flat, in prayer, and pushed them toward him.
Ah, not lift. Help. Please help.
"Yes," he said, and nodded. "Help. Richard will help Suzhien."
Of course, in whatever way your character is intelligent, you must invent a scene where that intelligence can shine without language; not always easy. I chose math because it is an easy choice, an arena in which the "language" of symbols and notation is truly universal throughout the modern world.
One might say the same thing about chemistry, physics, mechanical engineering, and perhaps other scientific fields like biology. It would be less true in fields dependent on language or culture that your highly intelligent person has not learned. She can't be an expert on Shakespeare if she has never read him and can't understand what the actors are saying.
The hallmark of high intelligence is NOT, as is commonly portrayed in nerd fiction, an inability to express yourself, or an inability to understand what the less intelligent and less articulate are saying. Nor is it opaque language others have difficulty understanding! I have spent near a lifetime in colleges, and in my experience "elevated" language is pretty much always a shield for banal ideas. When restated simply such ideas are cliché, pedestrian, or just plain laughable.
The hallmark of high intelligence is, like Sherlock, seeing clues others overlook, and thinking quickly to solve problems using those clues. Not just math problems or physics problems or computer problems. All problems. Including how to deal with and work around a handicap like not knowing the language.
Truly intelligent people are smart enough to make their complex ideas understandable in terms that others can understand. If your character is a genius with the vocabulary and language comprehension of a two-year old, find ways for her to show her genius by taking action to solve problems in creative ways that nobody around her can. The benefit of her skill will make others do the work necessary to understand her.
