Special finite elements for use with high-order boundary conditions Original Research Article

A hierarchy of special two-dimensional finite elements is constructed for use in a layer near a boundary or an interface B. The construction of the elements guarantees high-order (CN) regularity along B and C0 regularity elsewhere. All the elements in the hierarchy are proved to be fully conforming. The most important finite element in this class is the first-order one, which provides C1 regularity on B and C0 regularity elsewhere. It is used here to improve the numerical solution of Dirichlet problems, by prescribing the tangential derivative of the solution on B in addition to the solution itself. This element and higher-order ones are also useful near artificial boundaries where high-order local non-reflecting boundary conditions are used, and as transition elements near the interface between a domain governed by a ‘C0 theory’ and a domain governed by a ‘CN theory.’