Computational Issues and Analysis for Geometry-Conforming Immersed Finite Element Functions

报告专家:林涛(Virginia Tech)



报告摘要:This talk provides a detailed discussion on the computational issues and error analysis of the newly introduced geometry-conforming immersed finite element (IFE) space. We hope it will aid in understanding and extending this new geometry-conforming IFE space. The presentation consists of two parts. The first part addresses computational issues in constructing the geometry-conforming IFE functions. We begin with the construction of the first 𝑚+1 basis functions in the local IFE space on the Frenet fictitious element, demonstrating that these IFE functions follow the specific format presented previously. We will then discuss the evaluation of eta partial derivatives of the Laplacian of a function along the interface line segment in the Frenet fictitious element. Regarding the condition of the matrix used to compute the coefficients of these m+1 basis functions, we will show that it remains stable even when one of the sub-elements becomes smaller. The second part focuses on the approximation capability of the geometry-conforming IFE space. In contrast to the traditional scaling argument in finite element error analysis, our error estimations are primarily conducted on the Frenet fictitious element of an interface element. First, we show that the height of the Frenet fictitious element of an interface element is approximately O(h). We then consider the projections of a function that satisfies interface jump conditions onto the polynomial space Q^m on each of the sub-elements formed by the interface in the Frenet fictitious element. We demonstrate that these L^2 projections are close to each other along the interface. Next, we extend one of these two projections to the other sub-element to form an IFE function, showing that it can optimally approximate the given function. The optimal error bound for the L^2 projection of a function on the interface element follows from a standard change of variables procedure.