用于DPG平流离散化的Fortin算子
Fortin operators for DPG advection discretizations
- Portland State University（波特兰州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对任意空间维度和多项式次数的平流方程DPG离散化构造了Fortin算子，证明其相关性质并应用于全离散残差最小化方法，得到拟最优性等结论。
AI中文摘要：
我们针对空间维度为N的任意单纯形网格、试验空间多项式次数k≥1的情况，构造了散度为零的分段常数平流向量β的平流方程的不连续Petrov-Galerkin（DPG）离散化的Fortin算子。在每个单元上，从面和内部泡函数构建最小测试空间，随后对其进行扩充。该Fortin算子在自然的β加权破碎测试图范数下是有界的，对于每一个1<q<∞（其中q是试验指数p的共轭指数），该范数均在形状规则网格族上一致有界。当k≥2时，需满足非特征面条件，而最低阶情况无需此类条件，且允许特征面。作为应用，我们证明了实用的全离散残差最小化方法在每一个1<p<∞时，在DPG能量范数下是拟最优的，其拟最优常数仅由Fortin算子决定；该方法的可计算残差是全局可靠且高效的后验误差估计器；在最低阶情况下，扩充测试空间并改变测试范数可改善结果。
英文摘要：
We construct Fortin operators for discontinuous Petrov-Galerkin (DPG) discretizations of the advection equation with a piecewise constant divergence-free advection vector $β$, on simplicial meshes of any spatial dimension $N$ and for any polynomial degree $k \ge 1$ of the trial space. A minimal test space is built on each element from facet and interior bubbles and later augmented. The Fortin operator is shown to be bounded, uniformly over shape-regular mesh families, in the natural $β$-weighted broken test graph norm built on $L_q$ for every $1 < q < \infty$, where $q$ is the exponent conjugate to the trial exponent $p$. A non-characteristic facet condition is assumed when $k \ge 2$, while the lowest-order case requires no such condition and admits characteristic facets. As applications we prove that the practical fully discrete residual minimization method is quasioptimal in the DPG energy norm for every $1 < p < \infty$, with a quasioptimality constant governed solely by the Fortin operator, that its computable residual is a globally reliable and efficient a posteriori error estimator, and that augmenting the test space and changing the test norm improves the results in the lowest-order case.
