Vortragsankündigung
Im Rahmen des Kolloquiums spricht
Herr Prof. Dr. Andreas Rademacher (Universität Bremen)
Der Vortrag findet statt am Mittwoch, den 25.06.2025 um 17.15 Uhr im Raum W01 0-006
Titel: The Finite Cell Method: Basic Idea, Adaptivity, and Mixed Problems
Abstract: The meshing of complex or time-dependent domains is one of the fundamental challenges in the finite element method (FEM). Fictitious domains are one approach to meeting this challenge. The actual domain is embedded in a larger and simply structured domain. The approximation of the actual domain is then carried out by introducing a regularized truncation function into the partial differential equation (PDE) to be solved. The application of the FEM to the resulting problem leads to the finite cell method (FCM). Here, the truncation function must now be approximated with sufficient accuracy within the framework of the numerical calculation of integrals, for which suitable summed quadrature formulas must be constructed. In this talk, we discuss the basic idea of FCM and especially the construction of the summed quadrature formulas. Furthermore, we discuss the existing a priori error estimates.
In practical applications, the question arises whether the quadrature formulas are sufficiently accurate or not. In this talk, we present an approach based on a posteriori error estimators, which constructs a suitable quadrature based on adaptive algorithms. The other errors (error from the domain approximation, discretization errors of the FEM, numerical error) are also included and equilibrated. The derivation of the error estimators uses the dual weighted residual (DWR) method. The fundamental challenge lies in the separation of the individual error components and the appropriate approximation of the analytical error identity.
A second fundamental problem with the FCM is the realization of (inhomogeneous) Dirichlet boundary conditions on the real boundary, if it does not coincide with the fictitious boundary, since the techniques commonly used in the FEM cannot be applied. One approach that is commonly used in the literature is the integration by means of an additional penalty term with all the associated advantages and especially disadvantages, such as the difficult choice of the penalty parameter. Therefore, mixed approaches for the realization of Dirichlet boundary conditions are presented and analyzed. However, ensuring inf-sup stability contradicts numerical performance. This is why we provide an outlook on techniques that ensure both based on stabilized mixed methods.