Analysis of adaptive nonconforming Galerkin finite element methods

在这个项目中，我们想要分析定常和演化问题的自适应非协调Galerkin有限元方法。协调“刚性”的松弛通常提供更好的逼近性质、增强的稳定性性质和简化适应问题的结构性质。除此之外，全局耦合的减弱促进了最终离散系统的解算器的并行化。三十多年来，适应性一直是工程和科学计算问题的有效数值解的基本工具。基于后验误差指标，具有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佳的成本-精度平衡。对这一突出性能的数学理解对于协调离散化已经获得了实质性的成熟，然而，对于非协调方法只有很少的结果。在第一个应用阶段，我们集中讨论了自适应“间断Galerkin”方法的基本收敛性质。除了这种定性特征外，对于一种方法的实际可用性来说，它的稳定性和健壮性等预渐近特征往往是决定性的。此外，我们最近发展的非协调方法不仅在结构上与间断Galerkin格式相似，而且在结构上也与“混杂-高阶”、“恢复有限元”和“虚拟有限元”方法相似。因此，我们将扩展考虑的方法库，并将重点放在定量性质上，如最优收敛速度和预渐近方面；特别是在非协调方法揭示出优于一致方法的特征的情况下。例如，在Stokes和Navier-Stokes问题中的质量守恒，或者在线弹性和奇异摄动问题中的锁定现象等结构性质的保持。虽然我们的重点是严格的数学分析，但我们将特别注意测试新开发的方法，不仅在学术案例编制者中，而且还将它们应用于合适的基准问题。这样，本文的研究将大大提高对非协调有限元Galerkin方法的理论认识，并为其在高性能计算机上应用于实际问题奠定基础。

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.