title: Some new analysis results for a class of interface problems.
abstract: Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Due to these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergency of the Immersed Boundary method is presented. The IB method is shown to be first order convergent inL∞norm.
authors.affiliation: Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA, and School of Mathematical Sciences, Nanjing Normal University, Nanjing, China.
authors.forename: Zhilin
authors.initials: Z
authors.lastname: Li
authors.affiliation: School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, China.
authors.forename: Li
authors.initials: L
authors.lastname: Wang
authors.affiliation: Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA.
authors.forename: Eric
authors.initials: E
authors.lastname: Aspinwall
authors.affiliation: Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, VA 23284, USA.
authors.forename: Racheal
authors.initials: R
authors.lastname: Cooper
authors.affiliation: Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA.
authors.forename: Paul
authors.initials: P
authors.lastname: Kuberry
authors.affiliation: Department of Mathematics, Jackson State University, Jackson, MS 39217, USA.
authors.forename: Ashley
authors.initials: A
authors.lastname: Sanders
authors.affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang Province, 310058, P. R. China.
authors.forename: Ke
authors.initials: K
authors.lastname: Zeng
keywords: Dirac delta function; Immersed Boundary (IB) method; Immersed Interface Method (IIM); boundary singularity; convergence of IB method; discontinuous coefficient; equivalent boundary conditions; jump conditions; weak solution
references.citation: Symp Soc Exp Biol. 1995;49:265-76
references.pmid: 8571229