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Properties of discrete delta functions and local convergence of the immersed boundary method. (English) Zbl 1268.65143
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.

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
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