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We consider the reliable implementation of high-order unfitted finite elementmethods on Cartesian meshes with hanging nodes for elliptic interface problems.We construct a reliable algorithm to merge small interface elements with theirsurrounding elements to automatically generate the finite element mesh whoseelements are large with respect to both domains. We propose new basis functionsfor the interface elements to control the growth of the condition number of thestiffness matrix in terms of the finite element approximation order, the numberof elements of the mesh, and the interface deviation which quantifies the meshresolution of the geometry of the interface. Numerical examples are presentedto illustrate the competitive performance of the method.

An Arbitrarily High Order Unfitted Finite Element Method for Elliptic Interface Problems with Automatic Mesh Generation