Analysis of adaptive nonconforming Galerkin finite element methods
Final Report Abstract
The aim of the project was to analyse and construct non-conforming Galerkin methods for solving partial differential equations. The focus was on the development of a qualitative convergence analysis for adaptive methods, as well as on quantitative aspects, such as a posteriori error estimators and structure-preserving properties based on the freedom offered by non-conforming methods with respect to e.g. pressure robustness or parameter robustness. The former was essentially dealt with during the first project phase, so we focussed on aspects such as stability and robustness in this project phase. For example, we have developed a general framework for the construction of strictly equivalent a posteriori error bounds and demonstrated its practicability by means of important modelling problems. Furthermore, we have demonstrated the advantages of pressure robust methods in the approximation of nonlinear problems from fluid dynamics. To this end, we have designed pressure-resistant, non-conforming methods lowest order, which in contrast to comparable conformal methods compliant methods have optimal order of convergence of the velocity error. As further central results, we have developed a new mathematical theory for models for porous media flows and used it to derive a design concept for numerical methods. In particular, we expect these ideas to have a major impact on the construction and analysis of both new discretisations and competitive numerical solvers.
Publications
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Inf-sup stable discretization of the quasi-static Biot’s equations in poroelasticity
C. Kreuzer & P. Zanotti
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Inf-sup theory for the quasi-static Biot’s equations in poroelasticity
C. Kreuzer & P. Zanotti
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Pressure robust finite element discretization of nonlinear Stokes equations
L. Diening, A. Hirn, C. Kreuzer & P. Zanotti
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Strictly equivalent a posteriori estimators for quasi-optimal nonconforming methods
C. Kreuzer, M. Rott, A. Veeser & P. Zanotti
