The Shortley–Weller embedded finite-difference method for the 3D Poisson equation with mixed boundary conditions

• Published in: Journal of computational physics Vol. 229; no. 10; pp. 3675 - 3690
• Main Authors: Jomaa, Z., Macaskill, C.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.05.2010; Elsevier
• Subjects: Boundary conditions; Computational techniques; Dirichlet problem; Embedding; Errors; Exact sciences and technology; Finite difference method; Finite differences; Irregular boundaries; Mathematical analysis; Mathematical methods in physics; Mathematical models; Mixed boundary conditions; Physics; Poisson equation; Three dimensional; Poisson equation; Embedding; Mixed boundary conditions; Finite differences; Irregular boundaries; Boundary conditions; Calculation; Calculation methods; Finite difference method
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.01.021

This paper describes a method for the solution of the 3D Poisson equation, subject to mixed boundary conditions, on an irregularly shaped domain. A finite difference method is used, with the domain embedded in a rectangular grid. Quadratic treatment of the boundary conditions is shown to be necessary to obtain uniform error of O ( Δ 2 ) . This contrasts with the Dirichlet case where both quadratic and linear treatments give O ( Δ 2 ) error, although the coefficient of error may be much larger for the linear case. Explicit error estimates demonstrating this behaviour are found for the 1D case with similar behaviour found in 2D and 3D numerical examples. Finally, the extension of this approach to the N-dimensional case is given, where N > 3 .