0213 Advances and Applications of Collocation Methods: Meshfree, IGA, Machine Learning for PDEs

  • Pratik Suchde, University of Luxembourg, Fraunhofer Institute for Industrial Mathematics ITWM
  • Isabel Michel, Fraunhofer Institute for Industrial Mathematics ITWM
  • Elena Atroshchenko, University of New South Wales
  • Stéphane P.A. Bordas, University of Luxembourg

Point collocation methods are the oldest numerical methods to solve partial differential equations (PDEs). They solve PDEs in their strong form over a set of computational points, rather than in an averaged sense. With the development of advanced discretization techniques in meshfree methods, isogeometric methods, and machine learning methods for PDEs, point collocation methods have been increasingly gaining traction today.

Meshfree methods have been an alternative to conventional mesh-based methods, as they avoid the issue of mesh generation and mesh dependency of the solution. Among these, meshfree collocation methods have the added advantage that they do not require a background mesh for numerical integration, as is often the case for many meshfree weak-form methods.

With the advent of isogeometric methods and machine learning based methods for PDEs, research in collocation methods has received a further boost. They are associated with higher continuity of the approximation functions which enable taking derivatives of higher order, and with lower computational costs.

Contributions are welcome on both method development aspects and applications of point collocation methods. Areas of interest for this mini-symposium include, but are not limited to, recent developments in

・Modelling with collocation methods
・Applications of collocation methods
・Collocation for multi-physics or multi-scale problems
・Coupling collocation methods with other numerical methods
・Development of efficient collocation schemes
・Collocation for discontinuities and singularities
・Mathematical analysis of stability and consistency of collocation methods

Contributions are welcome in all types of point collocation methods including, but not limited to,
・Generalized Finite Different Methods (GFDM)
・Radial Basis Functions- Finite Differences (RBF-FD)
・Moving Least Squares (MLS) collocation
・Generalized Moving Least Squares (GMLS)
・Spline Collocation
・Isogeometric collocation
・Maximum entropy collocation
・Reproducing Kernel Collocation Method (RKCM)
・Deep Learning based collocation methods
・Physical Informed Neural Networks (PINN)

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