Analysis of adaptive nonconforming Galerkin finite element methods

在这个项目中，我们想分析平稳和进化问题的自适应非协调伽辽金有限元方法。一致性“刚性”的松弛通常提供了更好的近似性质，增强了稳定性，简化了问题的可容纳结构性质。除此之外，弱化全局耦合有利于求解得到的离散系统的并行化。三十多年来，自适应一直是工程和科学计算中有效数值求解问题的基本工具。基于后验误差指标，启发式标记策略的自适应有限元方法在实践中通常表现出最优的成本-精度平衡。对这一杰出性能的数学理解对于一致性离散已经获得了实质性的成熟，然而，对于非一致性方法只有很少的结果存在。在第一个应用阶段，我们重点研究了自适应“不连续伽辽金”方法的基本收敛性质。除了这些定性特征之外，对于一种方法的实际可用性，通常它的前渐近特征，如稳定性和鲁棒性是决定性的。此外，我们最近开发的非一致性方法显示出与不连续Galerkin格式的结构相似性，但也与“混合高阶”-，“恢复有限元”-和“虚拟有限元”方法相似。因此，我们将扩展所考虑的方法池，并专注于最优收敛率和前渐近方面等定量性质；特别是当不符合标准的方法比符合标准的方法揭示出更优越的特性时。例如结构性质的保存，比如Stokes和Navier-Stokes问题中的质量守恒，或者线性弹性和奇摄动问题中的锁定现象。虽然我们的重点是数学上的严格分析，但我们将特别注意测试新开发的方法，不仅在学术案例场景中，而且将它们应用于合适的基准问题。因此，本文的研究将大大提高非协调有限元伽辽金方法的理论认识，并为其在高性能计算机上的实际应用奠定基础。

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.