论文标题
超定位的正交分解，用于对流为主导的扩散问题
Super-localized orthogonal decomposition for convection-dominated diffusion problems
论文作者
论文摘要
本文提出了一种多尺度的方法，用于在大péclet数字方面进行对流为主的扩散问题。解决方案操作员在某些任意的粗网格上的固定右侧的固定右侧定义了有限维的粗ansatz空间，具有有利的近似属性。对于某些相关的错误度量，包括$ l^2 $ -Norm，Galerkin对此通用有限元空间的投影甚至会产生$ \ varepsilon $非依赖性错误界，$ \ varepsilon $是单数扰动参数。通过构建近似局部基础，该方法以超定位正交分解（Slod）精神的精神成为一种新颖的多规模方法。可以以A-tosterii的方式估算由基础定位引起的误差。与现有的多尺度方法相反，数值实验表明$ \ varepsilon $非依赖性收敛，即使在大型网格péclet数量的不足的状态下，也没有预隔效应。
This paper presents a multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The application of the solution operator to piecewise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the $L^2$-norm, the Galerkin projection onto this generalized finite element space even yields $\varepsilon$-independent error bounds, $\varepsilon$ being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a-posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate $\varepsilon$-independent convergence without preasymptotic effects even in the under-resolved regime of large mesh Péclet numbers.
