Special finite elements for use with high-order boundary conditions

Abstract

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 ( C N ) regularity along B and C 0 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 C 1 regularity on B and C 0 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 ‘ C 0 theory’ and a domain governed by a ‘ C N theory.’