The Orthonormalized Generalized Finite Element Method–OGFEM: Efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions

• The shortcomings of generalized/extended finite element methods are illustrated.
• A procedure to construct orthonormal enriched basis functions is proposed (OGFEM).
• The condition number of the unconstrained system matrix is equal to one in the OGFEM.
• We elaborate on the applicability of the OGFEM in finite precision arithmetic.
• The OGFEM improves the rate at which the approximation solution converges to the exact solution substantially for the 1D modified Helmholtz and Poisson equations.

An extension of the Generalized Finite Element Method (GFEM) is proposed with which we efficiently reduce approximation errors. The new method constructs a stiffness matrix with a conditioning that is significantly better than the Stable Generalized Finite Element Method (SGFEM) and the Finite Element Method (FEM). Accordingly, the risk of a severe loss of accuracy in the computed solution, which burdens the GFEM, is prevented. Furthermore, the computational cost of the inversion of the associated stiffness matrix is significantly reduced. The GFEM employs a set of enriched basis functions which is chosen to improve the rate at which the approximation converges to the exact solution. The stiffness matrix constructed from these basis functions is often ill-conditioned and the accuracy of the solution cannot be guaranteed. We prevent this by orthonormalizing the basis functions and refer to the method as the Orthonormalized Generalized Finite Element Method (OGFEM). Because the OGFEM has the flexibility to orthonormalize either a part or all of the basis functions, the method can be considered as a generalization of the GFEM. The method is applicable with single or multiple global and/or local enrichment functions. Problems in blending elements are avoided by a modification of the enrichment functions. The method is demonstrated for the one-dimensional modified Helmholtz and Poisson equations and compared with the FEM, GFEM and SGFEM.