Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions

Abstract

We describe a 2D finite difference algorithm for inverting the Poisson equation on an irregularly shaped domain, with mixed boundary conditions, with the domain embedded in a rectangular Cartesian grid. We give both linear and quadratic boundary treatments and derive 1D error expressions for both cases. The linear approach uses a five point formulation and is first order accurate while the quadratic treatment uses a nine point stencil and is second order accurate. The key aspect of the quadratic treatment is the use of a suitably chosen directional derivative to find the second order accurate approximation to the normal derivative at the boundary. References F. Bouchon and G. H. Peichl, {A second-order immersed interface technique for an elliptic Neumann problem}, Numer. Methods Partial Differential Equations. 23 (2007), no. 2, 400--420. doi:10.1002/num.20187 J. H. Bramble and B. E. Hubbard, {Approximation of solutions of mixed boundary value problems for Poisson's equation by finite differences}, J. Assoc. Comput. Mach. 12 (1965) 114--123. doi:10.1145/321250.321260 P.G. Ciarlet, {Introduction to numerical linear algebra and optimisation}, 1st English edition, Cambridge University Press, 1989. doi:10.2277/0521339847 L. Collatz, {Bemerkungen zur fehlerabschaetzung fuer das differnzenverfahren bei partiellen differentialgleichungen}, Z. Angew. Math. Mech. 13 (1933) 56--57. doi:10.1002/zamm.19330130110 D. Greenspan, {On the numerical solution of problems allowing mixed boundary conditions}, J. Franklin Inst. 277 (1964) 11--30. doi:10.1016/0016-0032(64)90035-3 Z. Jomaa and C. Macaskill, {The Shortley--Weller embedded finite-difference method for the 3D Poisson equation with mixed boundary conditions}, {submitted}. Z. Jomaa and C. Macaskill, {The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions}, J. Comp. Phys. 202 (2005) 488--506. doi:10.1016/j.jcp.2004.07.011 X. Liu, R. Fedkiw and M. Kang, {A boundary condition capturing method for Poisson's equation on irregular domains}, J. Comp. Phys. {{160}} (2000) 151--178. doi:10.1006/jcph.2000.6444 G. H. Shortley and R. Weller, {The numerical solution of Laplace's equation}, J. Appl. Phys. 9 (1938) 334--348. doi:10.1063/1.1710426