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The phrase "element by" is grammatically correct and can be used in written English.
It is typically used to describe a step-by-step process or to indicate a specific aspect or factor of something. Example: "The recipe requires you to add the ingredients element by element, stirring continuously." In this sentence, "element by element" means that each ingredient should be added separately, one after the other. Another example: "The company's success can be attributed to their attention to detail, element by element." Here, "element by element" emphasizes that the company's success is a result of paying attention to each individual aspect or factor.
Exact(48)
The Modern Element, by Adam Kirsch (Norton; $25.95).
Maybe the Giants will create that element by committee.
The first creatures originated from the moist element by evaporation.
They synthsised this element by bombarding plutonium-242 with neon-22.
Element by element, "Tom Sawyer" turns into a near-perfect production.
Element by Westin Times Square West; 311 West 39th Street, New York; (212) 643-0770; elementtimessquare.com.com
Similar(12)
where denotes the element-by-element multiplication.
⊙ denotes the Hadamard product, i.e., element-by-element product.
where ⊙ denotes the Hadamard (element-by-element) multiplication.
and the operator denotes an element-by-element matrix division.
The operation.∗ represents the element-by-element multiplication of two matrixes.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com