  Advanced transition elements are of utmost importance in many applications of
the finite element method (FEM) where a local mesh refinement is required.
Considering problems that exhibit singularities in the solution, an adaptive
hp-refinement procedure must be applied. Even today, this is a very demanding
task especially if only quadrilateral/hexahedral elements are deployed and
consequently the hanging nodes problem is encountered. These element types,
are, however, favored in computational mechanics due to the improved accuracy
compared to triangular/tetrahedral elements. Therefore, we propose a compatible
transition element - xNy-element - which provides the capability of coupling
different element types. The adjacent elements can exhibit different element
sizes, shape function types, and polynomial orders. Thus, it is possible to
combine independently refined h- and p-meshes. The approach is based on the
transfinite mapping concept and constitutes an extension/generalization of the
pNh-element concept. By means of several numerical examples, the convergence
behavior is investigated in detail, and the asymptotic rates of convergence are
determined numerically. Overall, it is found that the proposed approach
provides very promising results for local mesh refinement procedures.
