A group of immersed finite-element spaces for elliptic interface problems. (14th December 2017)

• Record Type: Journal Article
• Authors: Guo, Ruchi; Lin, Tao
• Is Part Of: IMA journal of numerical analysis. Volume 39:Number 1(2019)
• Page Start: 482
• Page End: 511
• Publication Date: 2017-12-14
• Subjects: interface problems -- discontinuous coefficients -- finite-element spaces -- Cartesian mesh; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drx074
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: We present a unified framework for developing and analyzing immersed finite-element (IFE) spaces for solving typical elliptic interface problems with interface-independent meshes. This framework allows us to construct a group of new IFE spaces with either linear, or bilinear, or the rotated-$Q_1$ polynomials. Functions in these IFE spaces are locally piecewise polynomials defined according to the subelements formed by the interface itself instead of its line approximation. We show that the unisolvence for these IFE spaces follows from the invertibility of the Sherman–Morrison matrix. A group of estimates and identities are established for the interface geometry and shape functions that are applicable to all of these IFE spaces. These fundamental preparations enable us to develop a unified multipoint Taylor expansion procedure for proving that these IFE spaces have the expected optimal approximation capability according to the involved polynomials.