ON GENERALIZED FINITE ELEMENT METHOD’S NUMERICAL STABILITY PRESERVATION

Resumo

The Generalized Finite Element Method (GFEM) proposes the expansion of FEM’s approximation

space by combining Partition of Unity (PU) functions with enrichment functions. Such combination provides lo-
cally better capacity to approximate the desired solution. PU functions combine local approximations, forming

the corresponding global approximation. The Galerkin method is employed, leading to a system of equations to
be solved for global nodal parameters. However, it is important that the extended approximation space is linearly
independent, as to not jeopardize GFEM’s system’s conditioning which could lead to inaccurate results. Recent

developments propose alternatives for preserving the solving system’s linear independence, improving its condi-
tioning. This work offers new contributions in two manners. On one hand, a new set of PU functions to compose

the enriched space is explored. On the other hand, the Singular Value Decomposition (SVD) methodology is ex-
plored to obtain adequate solutions even in the presence of linear dependencies. Examples that normally lead to

dependencies are explored, emphasizing the advantages of the strategies. It is shown that GFEM’s system’s condi-
tioning is close to FEM’s while employing the new PU. Also, SVD’s efficiency for solving the system disregarding

its dependencies is demonstrated, adding robustness to the most conventional versions of the method.