Analysis of adaptive nonconforming Galerkin finite element methods

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In this project, we want to analyse adaptive nonconforming Galerkin finite element methods for stationary and evolutionary problems. The relaxation of the conforming `rigidity' often provides better approximation properties, enhanced stability properties and simplifies accommodating structural properties of the problem. Beyond that, weakening of global couplings facilitates the parallelisation of solvers for the resulting discrete systems. Adaptivity has been a fundamental tool in the efficient numerical solution of problems in engineering and scientific computing for more than three decades. Based on a posteriori error indicators, adaptive finite element methods with heuristic marking strategies typically show an optimal cost-accuracy balance in practice. The mathematical understanding of this outstanding performance has gained substantial maturity for conforming discretisations, however, only very few results exist for nonconforming methods. In the first application period, we have concentrated on basic convergence properties of adaptive `discontinuous Galerkin' methods. Beside this qualitative characteristics, for the practical usability of a method, often its pre-asymptotic features like stability and robustnessare decisive. Moreover, our recently developed nonconforming methods show structural similarities with discontinuous Galerkin schemes, but also with `hybrid-high-order'-, `recoverd finite element'- and `virtual finite element' methods. We will therefore extend the pool ofconsidered methods and focus on quantitative properties like optimal convergence rates and pre-asymptotic aspects; specifically in situations when nonconforming methods unveilsuperior characteristics compared to conforming approaches. Examples are the preservation of structural properties like the conservation of mass in Stokes and Navier-Stokes problems or locking phenomena in linear elasticity and singularly perturbed problems. Though our focus is on a mathematical strict analysis, we will give particular attention to test newly developed methods not only within academic case-scenariors, but also to apply them tosuitable benchmark problems. In this way, the proposed research will significantly improve thetheoretical understanding of nonconforming finite element Galerkin methods and lay the foundation for its application in real-life problems on high performance computers.