Nonlinear reduction using the extended group finite element method

• Published in: Computational & applied mathematics 2024-04, Vol.43 (3), Article 125
• Main Authors: Tolle, Kevin, Marheineke, Nicole
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Finite element analysis; Finite element method; Interpolation; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Model reduction; Nonlinearity; Proper Orthogonal Decomposition
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-024-02637-2

In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the extended group finite element formulation achieves a noticeable reduction in the computational overhead associated with nonlinear finite element problems. However, the problem’s size still leads to long solution times in most applications. Aiming to make real-time and/or many-query applications viable, we consider model order reduction and complexity reduction techniques to reduce the problem size and efficiently handle the reduced nonlinear terms, respectively. For proof of concept, we focus on the proper orthogonal decomposition and discrete empirical interpolation methods. While similar approaches based on the group finite element method only focus on semilinear problems, our proposed framework is also compatible with quasilinear problems. Compared to existing methods, our reduced models prove to be superior in many different aspects as demonstrated in three numerical benchmark problems.