In this paper we introduce general transfer operators between high-order and low-order refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving and high-order accurate. While the proofs apply to affine geometries, numerical experiments indicate that the results hold for more general curved and mixed meshes. These operators also have applications in the context of coarsening solution fields defined on meshes with nonconforming refinement. The transfer operators for $H^1$ finite element spaces require a globally coupled solve, for which robust and efficient preconditioners are developed. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multi-discretization coupling.
翻译:在本文中,我们引入了高顺序和低顺序精细的有限元素空间之间的一般转移操作员,这些空间可用于对高顺序和低顺序模拟。在对低顺序精密空间的自然限制下,我们证明高到低顺序线性绘图和低到高顺序线性绘图都是保守的、不断保存的和高顺序准确的。虽然这些证据适用于近距离地理比例,但数字实验表明,结果支持了更一般曲线和混合的网目。这些操作员还应用了在模贝上定义的粗化溶域中和不达标的改进。$H$1$的有限元素空间的转移操作员需要一种全球结合的解决方案,为此,我们提出了一些数字结果,证实了我们的分析,并展示了在适应性网目精细和保守的多分解组合中新绘图的效用。