課程題目： Construction of Immersed Finite Element Functions by the Differential Geometry of the Interface
授 課 人：林濤 教授 美國弗吉尼亞理工大學
課程時間：6月27日 星期四 09:30-10:30
6月28日 星期五 14:30-15:30
6月29日 星期六 14:30-15:30
課程地點： 正新樓209
校内聯系人：翟起龍 zhaiql@jlu.edu.cn
課程簡介：In this first class, we discuss a unified framework for the construction and analysis of immersed finite element (IFE) spaces designed to solve common two-dimensional elliptic interface problems with interface-independent meshes. This framework facilitates the construction of a range of IFE spaces depending on the choice of polynomials, such as P1 polynomials, Q1 polynomials, or rotated-Q1 polynomials. The shape functions within these IFE spaces are locally defined as piecewise polynomials based on the subelements formed by the interface itself, rather than its linear approximation. The unisolvence of these IFE spaces is established through the invertibility of the Sherman–Morrison matrix. Additionally, a set of universally valid estimates and identities are derived for both the interface geometry and IFE shape functions. Together with a multipoint Taylor expansion procedure, these tools enable us to demonstrate that these IFE spaces possess the expected optimal approximation capabilities.
The second class focus on the Immersed Finite Element (IFE) methods, which stand out for their ability to work on interface-independent meshes, by utilization of standard finite element functions on non-interface elements, together with IFE functions applied specifically on interface elements. However, IFE functions proposed in the literature, even though designed to adhere to interface jump conditions, diverge from the underlying Sobolev spaces. To mitigate this discrepancy, penalties are commonly employed in the related IFE methods, albeit at the cost of making these IFE methods less desirable. In this presentation, we introduce a new class of IFE functions, aiming to overcome the aforementioned shortcoming. Our approach leverages the Frenet-Serret apparatus from the differential geometry of the interface curve to establish an orthogonal curvilinear coordinate system in the vicinity of the interface. Each shape function in the local IFE space constructed in this method is a composition of a piecewise Q^m polynomial taking care of the interface jump conditions in Frenet cooridnates, and the Frenet transformation representing essential features of the differential geometry of the interface. Consequently, such IFE functions are locally conforming to the underlying Sobolev space of the interface problem, especially suitable for deployment in standard DG finite element schemes. Additionally, our method provides explicit formulas for the majority of functions in a basis of the local IFE space. This facilitates a more efficient construction of IFE shape functions within interface elements, particularly advantageous when using higher degree polynomials. We will showcase numerical results to illustrate features of IFE functions generated through this novel methodology.
The third class provides a detailed discussion on the computational issues and error analysis of a class of immersed finite element (IFE) functions constructed by the differential geometry of the interface curve. We hope it will aid in understanding and extending the related geometry-conforming IFE space.
This class 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 described in the previous lecture. 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 Frenet fictitious element of an interface element has a scale of O(h^2). We then consider the projections of a function that satisfies interface jump conditions onto the polynomial space Q^m on each of the two 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 then follows from a standard change of variables procedure.
授課人簡介：林濤，弗吉尼亞理工大學教授、博士生導師。1990年在懷俄明大學取得博士學位，1989年在弗吉尼亞理工大學數學系受聘為助理教授，2001年起擔任教授。他是計算數學和科學計算方面的專家，研究興趣涉及偏微分方程和積分微分方程的數值求解方法，尤其是在界面問題的浸入式有限元方法及其應用方面取得了開創性的研究成果。在SIAM Journal on Numerical Analysis、Numerische Mathematik和Journal of Computational Physics等國際頂級期刊發表研究性論文100餘篇，主持多項美國自然科學基金的科研項目。
