高阶多边形不连续Petrov-Galerkin（PolyDPG）方法使用超弱 formulations
High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations
AI总结：
本文首次利用超弱 formulations 实现高阶多边形有限元方法，通过不连续Petrov-Galerkin（DPG）方法构建了PolyDPG方法，无需额外稳定项，提供收敛性证明并验证了数值示例。
摘要：
本文首次尝试利用超弱 formulations 实现高阶多边形有限元方法，通过不连续Petrov-Galerkin（DPG）方法构建了PolyDPG方法。超弱变分 formulations 的特点是所有导数的权重都位于测试空间，而大部分试验空间可以是 $L^2$-离散化副本，无需在相邻元素间连续。此外，测试空间在网格界面处断裂。这允许通过限制边界三角形或盒子到多边形元素来构造符合的多边形有限元方法，即PolyDPG方法。唯一需要跨元素非平凡兼容性的变量是所谓的界面或骨架变量，这些变量可以直接定义在元素边界上。与其它高阶多边形方法不同，PolyDPG方法不需要额外的稳定项，因为DPG方法的构造稳定性。收敛性证明为 $h^p$ 形式，并通过多个示例验证。这些包括具有 $n$-边凸元素和高度扭曲凹元素的多边形网格，以及沿任意界面切割均匀网格的不连续材料属性建模。由于PolyDPG方法具有自然的后验误差估计器，因此开发了一种多边形自适应策略，并与基于约束悬挂节点的标准自适应方案进行了比较。本文还附带了一个开源的 $\texttt{PolyDPG}$ 软件，支持多边形和传统元素。
英文摘要：
This work represents the first endeavor in using ultraweak formulations to implement high-order polygonal finite element methods via the discontinuous Petrov-Galerkin (DPG) methodology. Ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space, while most of the trial space can be chosen as copies of $L^2$-discretizations that have no need to be continuous across adjacent elements. Additionally, the test spaces are broken along the mesh interfaces. This allows one to construct conforming polygonal finite element methods, termed here as PolyDPG methods, by defining most spaces by restriction of a bounding triangle or box to the polygonal element. The only variables that require nontrivial compatibility across elements are the so-called interface or skeleton variables, which can be defined directly on the element boundaries. Unlike other high-order polygonal methods, PolyDPG methods do not require ad hoc stabilization terms thanks to the crafted stability of the DPG methodology. A proof of convergence of the form $h^p$ is provided and corroborated through several illustrative numerical examples. These include polygonal meshes with $n$-sided convex elements and with highly distorted concave elements, as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid. Since PolyDPG methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes. This work is also accompanied by an open-source $\texttt{PolyDPG}$ software supporting polygonal and conventional elements.
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