Journal of Scientific Computing

, Volume 74, Issue 2, pp 631–639

# Convergence Analysis in the Maximum Norm of the Numerical Gradient of the Shortley–Weller Method

• Jiwon Seo
• Seung-yeal Ha
• Chohong Min
Article

## Abstract

The Shortley–Weller method is a standard central finite-difference-method for solving the Poisson equation in irregular domains with Dirichlet boundary conditions. It is well known that the Shortley–Weller method produces second-order accurate solutions and it has been numerically observed that the solution gradients are also second-order accurate; a property known as super-convergence. The super-convergence was proved in the $$L^{2}$$ norm in Yoon and Min (J Sci Comput 67(2):602–617, 2016). In this article, we present a proof for the super-convergence in the $$L^{\infty }$$ norm.

## Keywords

Shortley–Weller Finite difference method Super-convergence Convergence analysis

## References

1. 1.
Caffarelli, L.A., Gilardi, G.: Monotonicity of the free boundary in the two-dimensional dam problem. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 7(3), 523–537 (1980)
2. 2.
Chorin, A.J.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 135(2), 118–125 (1997)
3. 3.
Ciarlet, P.G., Miara, B., Thomas, J.M.: Introduction to Numerical Linear Algebra and Optimisation. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1989)Google Scholar
4. 4.
Friedman, A.: Variational principles and free-boundary problems. Courier Corporation, North Chelmsford (2010)Google Scholar
5. 5.
Gibou, F., Min, C.: Efficient symmetric positive definite second-order accurate monolithic solver for fluid/solid interactions. J. Comput. Phys. 231, 3245–3263 (2012)
6. 6.
Gustafsson, I.: A class of first order factorization methods. BIT 18, 142–156 (1978)
7. 7.
Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8(3), 2182–2189 (1965)
8. 8.
Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1996)Google Scholar
9. 9.
Li, Z.-C., Hu, H.-Y., Wang, S., Fang, Q.: Superconvergence of solution derivatives of the shortley-weller difference approximation to poisson’s equation with singularities on polygonal domains. Appl. Numer. Math. 58(5), 689–704 (2008)
10. 10.
Ng, Y.T., Chen, H., Min, C., Gibou, F.: Guidelines for poisson solvers on irregular domains with dirichlet boundary conditions using the ghost fluid method. J. Sci. Comput. 41(2), 300–320 (2009)
11. 11.
Peskin, C.S.: Flow patterns around heart valves. In: Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, pp. 214–221. Springer (1973)Google Scholar
12. 12.
Shortley, G.H., Weller, R.: The numerical solution of Laplace’s equation. J. Appl. Phys. 9(5), 334–348 (1938)
13. 13.
Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. SIAM, New Delhi (2004)
14. 14.
Weynans, L.: A proof in the finite-difference spirit of the superconvergence of the gradient for the Shortley–Weller method. Ph.D. thesis, INRIA Bordeaux (2015)Google Scholar
15. 15.
Xiu, D., Karniadakis, G.E.: A semi-Lagrangian high-order method for Navier–Stokes equations. J. Comput. Phys. 172(2), 658–684 (2001)
16. 16.
Yoon, G., Min, C.: Analyses on the finite difference method by Gibou, et al. for poisson equation. J. Comput. Phys. 280, 184–194 (2015)
17. 17.
Yoon, G., Min, C.: Convergence analysis of the standard central finite difference method for Poisson equation. J. Sci. Comput. 67(2), 602–617 (2016)
18. 18.
Yoon, G., Min, C., Kim, S.: A stable and convergent method for hodge decomposition of fluid-solid interaction. (2016). arXiv:1610.03195