A scaling argument for analyzing immersed finite element spaces


报告时间:2023年10月18日:10:30 — 11:20



报告摘要:The scaling argument is a commonly-used technique in error analysis for finite elementmethods. A fundamental ingredient in this potent technique is to map finite element functions from any element within a mesh—irrespective of its position, geometry, or size—to THE SAME polynomial space on the reference element. However, applying the traditional scaling argument directly to error analysis for immersed finite elements (IFE) presents challenges. This is primarily because, in general, the spaces on the reference element linked to the IFE spaces on various interface elements via thestandard affine mapping are NOT THE SAME. As a result, the error analysis of IFE approximations in the literature istypically conducted on a per-case basis. In this presentation, we will discuss our efforts to broaden the scope of the scaling argument framework for assessing the approximation performance of a specific group of IFE spaces. Our approach involves a detailed examination of the mapping from the relevant Sobolev space to the IFE space. To illustrate the versatility of this newly established error analysis framework, we employ it to derive optimal error estimates for IFE spaces design to solve a range of interface problems. These encompass, but are not limited to, a first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and an interface problem for the fourth-order Euler Bernoulli beam equation.