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It has been noted that the traditional scaling argument cannot be directlyapplied to the error analysis of immersed finite elements (IFE) because, ingeneral, the spaces on the reference element associated with the IFE spaces ondifferent interface elements via the standard affine mapping are not the same.By analyzing a mapping from the involved Sobolev space to the IFE space, thisarticle is able to extend the scaling argument framework to the errorestimation for the approximation capability of a class of IFE spaces in onespatial dimension. As demonstrations of the versatility of this unified erroranalysis framework, the manuscript applies the proposed scaling argument toobtain optimal IFE error estimates for a typical first-order linear hyperbolicinterface problem, a second-order elliptic interface problem, and thefourth-order Euler-Bernoulli beam interface problem, respectively.

A unified immersed finite element error analysis for one-dimensional interface problems