Geometry-Conforming Immersed Finite Element Functions

报告专家:林涛(Virginia Tech)



报告摘要:In solving interface problems of partial differential equations, Immersed Finite Element (IFE) methods stand out for their ability to work on interface-independent meshes, by utilization of standard finite element functions on non-interface elements, juxtaposed with IFE functions applied specifically on interface elements. These IFE functions, even though designed to adhere to interface jump conditions, often diverge from the underlying Sobolev spaces. To mitigate this discrepancy, penalties are commonly employed in the related IFE methods, albeit at the cost of introducing less desirable algebraic systems. In this presentation, we introduce a novel method for constructing two-dimensional IFE functions, aiming to overcome the aforementioned limitation. 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. This enables us to transform the interface curve into a simple line segment along the vertical axis within this local Frenet coordinate system. Consequently, IFE functions can be made in this local coordinate system in terms of Q^m polynomials to precisely meet the stipulated jump conditions. This construction method yields conforming IFE functions akin to isoparametric finite elements by a change of variables from the local Frenet coordinates to the x-y coordinates used by the partial differentia equation. Additionally, unlike construction methods in the literature, 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 dealing with higher degree polynomials. We will showcase numerical results to illustrate features of IFE functions generated through this novel methodology.