Liu, Yang; Mori, Yoichiro Properties of discrete delta functions and local convergence of the immersed boundary method. (English) Zbl 1268.65143 SIAM J. Numer. Anal. 50, No. 6, 2986-3015 (2012). The paper deals with the regularization of the delta function in order to provide numerical approximations for constant coefficient elliptic partial differential equations with singular source terms on a manifold (immersed structure). The authors focus on the pointwise convergence properties and show how the local convergence behaviour is influenced by the order of the differential operator, the order of the finite difference discretization and properties of the discrete delta function. The main technical contribution of the paper is the estimation of the immersed boundary error. The paper highlights the role played by the smoothing order in determining the rate of convergence of immersed boundary-type methods. The grid line effect is diminished if the discrete delta function has high smoothing order. Reviewer: Adrian Carabineanu (Bucureşti) Cited in 1 ReviewCited in 24 Documents MSC: 65N06 Finite difference methods for boundary value problems involving PDEs 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 58J05 Elliptic equations on manifolds, general theory Keywords:immersed boundary method; discrete delta function; smoothing order; moment order; regularization; constant coefficient elliptic partial differential equations; manifold; pointwise convergence; finite difference discretization PDFBibTeX XMLCite \textit{Y. Liu} and \textit{Y. Mori}, SIAM J. Numer. Anal. 50, No. 6, 2986--3015 (2012; Zbl 1268.65143) Full Text: DOI