906 Error Analysis and Adaptivity for Advanced Galerkin Methods

Erwin Stein, Leibniz Universität Hannover
Jiun-Shyan Chen, University of California, San Diego
Marcus Rüter, University of California, Los Angeles
Since its birth in the 1940s and 1950s, the finite element method (FEM) - based on discrete (direct) variational approximations by W. Ritz, B. G. Galerkin, and R. Courant - has been the predominant Galerkin method for approximately solving (initial) boundary value problems that appear in Mathematics, Engineering, and Natural Sciences. However, certain disadvantages, e.g. shear and incompressibility locking, locking by inadequate approximations of the geometry, e.g. buckling of thin shells, or mesh dependencies, led to the development of advanced Galerkin methods. As examples, we mention the eXtended finite element method (XFEM) that allows to model arbitrary crack propagations, mixed finite element methods and exterior calculus, that bypass locking and other instability problems, or Galerkin meshfree methods, such as the reproducing kernel particle method (RKPM) or the element-free Galerkin method (EFG), that cope with problems arising from a mesh.

All these advanced Galerkin methods lead of course to numerical approximations and therefore require discretization error control to provide reliable numerical results that meet a prescribed user-defined accuracy at relatively low computational costs based on adaptive refinements. Such error estimation procedures have been mainly developed since the 1980s for mesh-based finite element methods. However, for some of the advanced Galerkin methods the development of error estimation procedures is still in an early stage.

The objective of this minisymposium therefore is to provide a platform for researchers to present and discuss their latest developments of fundamentals, error estimation procedures, and adaptivity that focus on advanced Galerkin methods. These include but are not limited to:

- generalized finite element method (GFEM)
- eXtended finite element method (XFEM)
- smoothed finite element method (SFEM)
- space-time finite element method
- mixed finite element method and exterior calculus
- hp-finite element method (hp-FEM)
- discrete element method (DEM)
- discontinuous Galerkin methods (DG methods)
- virtual element method (VEM)
- Galerkin meshfree methods (EFG, RKPM)
- peridynamics
- numerical methods for nonlocal mechanics