A high-order embedded-boundary method based on smooth extension and RBFs for solving elliptic equations in multiply connected domains

Abstract

This paper presents a new high-order embedded-/immersed-boundary method, based on point collocation, smooth extension of the solution and integrated radial basis functions (IRBFs), for solving the elliptic partial differential equation (PDE) defined in a domain with holes. The PDE is solved in the domain without holes, where the construction of the IRBF approximations is based on a fixed Cartesian grid and local five-point stencils, and the inner/immersed boundary conditions are included in the discretized equations. More importantly, nodal values of the second-, third- and fourth-order derivatives of the field variable are incorporated into the IRBF approximations, and the forcing term defined in the holes is constructed in a form that gives a globally smooth solution. These features enable the proposed scheme to achieve high level of sparseness of the system matrix, and high level of accuracy of the solution together. Numerical verification is carried out for problems with smooth and non-smooth inner boundaries. Highly accurate results are obtained using relatively coarse grids.