an:05303107
Zbl 1155.65016
Towers, John D.
A convergence rate theorem for finite difference approximations to delta functions
EN
J. Comput. Phys. 227, No. 13, 6591-6597 (2008).
00220417
2008
j
65D15
delta function; integral; level set method; finite difference; approximation; regular grid; convergence rate
A new rate of convergence for approximations to certain integrals over codimension one manifolds in \(\mathbb R^n\), is proved. The type of manifold is defined by the zero level set of a smooth mapping \(u:\mathbb R^n\to\mathbb R\). The approximation method used, is based on two finite difference methods for the discretizing of the delta function, originally presented by the author in a previous research work [J. Comput. Phys. 220, No. 2, 915--931 (2007; Zbl 1115.65028)], were empirical convergence rates had indicated the first and second order accuracy. In this work these convergence rates are proved for the two proposed algorithms, under fairly general hypotheses.
Vasilis Dimitriou (Chania)
Zbl 1115.65028