Extended group finite element method

• Published in: Applied numerical mathematics 2021-04, Vol.162, p.1-19
• Main Authors: Tolle, Kevin, Marheineke, Nicole
• Format: Article
• Language: English
• Subjects: Galerkin approximation; Group finite element method; Multi-linear forms; Nonlinear partial differential equations; Tensor actions
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2020.12.008

•Development of an efficient method for nonlinear finite element problems.•Extension of the group finite element formulation results in a more powerful method.•Reduction of the computational costs through interpolation onto finite element spaces.•Exploitation of tensor structure for nonlinear forms in quasi-linear problems. Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation interpolates nonlinear terms onto the finite element approximation space, we propose the use of a separate approximation space that is tailored to the nonlinearity. In many cases, this allows for the exact reformulation of the discrete nonlinear problem into a quadratic problem with algebraic constraints. Furthermore, the substitution of the nonlinear terms often shifts general nonlinear forms into trilinear forms, which can easily be described by third-order tensors. The numerical benefits as well as the advantages in comparison to the original group finite element method are studied using a wide variety of academic benchmark problems.