Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem

• Published in: Computer methods in applied mechanics and engineering Vol. 344; pp. 538 - 568
• Main Authors: Zhang, Qinghui, Banerjee, Uday, Babuška, Ivo
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.02.2019; Elsevier BV
• Subjects: Conditioning; Convergence; Enrichment; Finite element analysis; Finite element method; GFEM/XFEM; Interface; Mathematical analysis; Nodes; Nonlinear programming; Robustness (mathematics); Scaled condition number; Singularity; SSGFEM; GFEM/XFEM; Singularity; SSGFEM; Scaled condition number; Interface; Convergence
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2018.10.018

In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of “subtracting the interpolant.” A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface. •GFEM for 2D non-smooth interface problem.•Singular enrichment functions in addition to distance based enrichment functions.•Proof of optimal convergence of the GFEM.•Experimental study of conditioning and robustness of GFEM.•The notion of Strongly Stable GFEM (SSGFEM).