在FEEC框架下求解泊松方程的自适应方法的收敛性与最优性
Convergence and Optimality of Adaptive Methods for Poisson's Equation in the FEEC Framework
AI总结：
本文研究了在一般拓扑域和空间维度上，利用FEEC框架构建求解泊松方程的自适应方法的收敛性和最优性，提出了可靠的误差估计器和更精确的准正交性结果，且在不考虑数据振荡标记的情况下，该方法在总误差中表现为收缩。
摘要：
有限元外算术（FEEC）是由Arnold、Falk、Winther等人在过去十年中开发的，目的是利用混合变分问题可以在希尔伯特复形上提出这一观察，并通过求解有限维子复形问题来获得伽辽金型混合方法。Chen、Holst和Xu（Math. Comp. 78（2009）35-53）建立了在二维可缩领域上使用Raviart-Thomas或Brezzi-Douglas-Marini元素的自适应混合有限元方法的收敛性和最优性，这可以视为de Rham复形上的边界问题。最近，Demlow和Hirani（Found. Math. Comput. 14（2014）1337-1371）开发了用于后验分析的de Rham复形的基本工具。在本文中，我们利用FEEC中的工具，在一般拓扑域和空间维度上构建收敛性和复杂性结果。特别地，我们使用一种新方法构建了可靠的误差估计器和更精确的准正交性结果。在不进行数据振荡标记的情况下，我们的自适应方法相对于包含误差估计器和数据振荡的总误差是收缩的。
英文摘要：
Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in two dimensions, which can be viewed as a boundary problem on the de Rham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371) developed fundamental tools for a posteriori analysis on the de Rham complex. In this paper, we use tools in FEEC to construct convergence and complexity results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.
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