论文标题
通过基于插值的离散扩展，强大的高阶不限制元素
Robust high-order unfitted finite elements by interpolation-based discrete extension
论文作者
论文摘要
在这项工作中，我们使用未固定网格上的有限元方法提出了一种新的公式，以解决部分微分方程的溶液。提出的公式依赖于在汇总有限元方法中提出的离散扩展运算符。相对于单元格中边界/界面的位置，此公式是可靠的。人们不仅在物理领域，而且在整个活动网格上都可以证明稳定性结果增强。但是，稳定性常数通过使用多项式顺序成倍增长，因为潜在的扩展运算符是通过外推定定义的。为了解决这个问题，我们引入了一个新的有限元元素的新变体，其中物理域中的扩展是高于两个高度的多项式的插值。结果，稳定性常数仅以近似值为多项式速率。我们证明，这种方法可以使用汇总有限元方法实现强大的高阶近似值。所提出的方法是一致的，最佳的收敛性，并且具有最佳尺度高阶近似值的条件号。
In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete extension operator proposed in the aggregated finite element method. This formulation is robust with respect to the location of the boundary/interface within the cell. One can prove enhanced stability results, not only on the physical domain, but on the whole active mesh. However, the stability constants grow exponentially with the polynomial order being used, since the underlying extension operators are defined via extrapolation. To address this issue, we introduce a new variant of aggregated finite elements, in which the extension in the physical domain is an interpolation for polynomials of order higher than two. As a result, the stability constants only grow at a polynomial rate with the order of approximation. We demonstrate that this approach enables robust high-order approximations with the aggregated finite element method. The proposed method is consistent, optimally convergent, and with a condition number that scales optimally for high order approximation.
