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To present numerical approaches to the solution of physically coupled non‐linear problems, which frequently happen to be characterized by their multi‐domain character.

By adopting coupled solution strategies a considerable attention is devoted, in order to obtain a computationally efficient numerical algorithm, to the selection of appropriate space and time discretization, as well as to a proper discrete approximation method used.

Coupling of two methods, the finite element method and the boundary element method, respectively, has proved to be computationally exceedingly advantageous, particularly in case of moving domains.

As specific case studies computer simulation of an induction heating problem and a mushy‐state forming problem are considered. A thorough discussion on the coupling effects, characterizing the evolutions of respective physical quantities' fields, is given, and their impact on those evolutions is identified.

This paper presents efficient numerical strategies for the solution of a certain class of multi‐physics and multi‐domain problems.