On a High-Order Generalized Finite Element Method

Resumo

In recent years, technological development has grown exponentially. In this context, numerical methods
consist of an attractive tool that guarantees flexibility and ease of access in modeling Science and Engineering
problems. In particular, the High-Order version of the Finite Element Method (FEM), based upon orthogonal
polynomials as a means for constructing hierarchic approximation spaces, is of special interest, due to its high
convergence rate and adequate matrix conditioning. However, albeit FEM achieves good results in a large class

of problems, it is not as adequate when non-smooth solutions are expected. Aiming to circumvent such a limita-
tion, the Generalized Finite Element Method (GFEM) introduces enrichment functions, selected on the basis of a

previous knowledge about the solution of the problem, in order to enlarge FEM’s approximation space. Despite
providing scope and generality expansion to the FEM, such technique may lead, nevertheless, to ill-conditioned
systems of equations, therefore penalizing numerical precision. Taking this into account, this paper proposes a
methodology for integrating positive features of the two aforementioned versions of the FEM, resulting in a stable,
precise and high performing numerical tool. The methodology herein presented allows for the possibility of being
easily implemented in previously existing codes, already designed to handle GFEM. Planar elasticity applications
are considered – including Linear Elastic Fracture Mechanics problems, for which GFEM is more suitable – in

order to demonstrate the previously mentioned convergence and conditioning properties of the proposed formula-
tion.