Solving a Singularly Perturbed Elliptic Problem by a Cascadic Multigrid Algorithm with Richardson Extrapolation
A two-dimensional linear elliptic equation with regular boundary layers is considered in the unit square. It is solved by using an upwind difference scheme on the Shishkin mesh which converges uniformly with respect to a small parameter \(\varepsilon \). It is known that the application of multigrid methods leads to essential reduction of arithmetical operations amount. Earlier we investigated the cascadic two-grid method with the application of Richardson extrapolation to increase the \(\varepsilon \)-uniform accuracy of the difference scheme. In this paper multigrid algorithm of the same structure is studied. We construct an extrapolation of initial guess using numerical solutions on two coarse meshes to reduce the arithmetical operations amount. The application of the Richardson extrapolation method based on numerical solutions on the last three meshes leads to increase the \(\varepsilon \)-uniform accuracy of the difference scheme by two orders. The different components of a cascadic multigrid method are studied. The results of some numerical experiments are discussed.
KeywordsSingularly perturbed elliptic problem Regular boundary layers Difference scheme Shishkin mesh \(\varepsilon \)-uniform accuracy Cascadic multigrid method Richardson extrapolation
Research has been supported by the program of fundamental scientific researches of the SB RAS No I.1.3., project No 0314-2016-0009.
- 6.Fedorenko, R.P.: The speed of convergence of one iterative process. Zh. Vychisl. Mat. Mat. Fiz. 4(3), 559–564 (1964). (in Russian)Google Scholar
- 12.Angelova, I.T., Vulkov, L.G.: Comparison of the two-grid method on different meshes for singularly perturbed semilinear problems. In: Applications of Mathematics in Engineering and Economics, pp. 305–312. American Institute of Physics (2008). https://doi.org/10.1063/1.3030800
- 13.Vulkov, L.G., Zadorin, A.I.: Two-grid algorithms for the solution of 2D semilinear singularly perturbed convection-diffusion equations using an exponential finite difference scheme. In: Application of Mathematics in Technical and Natural Sciences, pp. 371–379. AIP Conference Proceedings (2009). https://doi.org/10.1063/1.3265351
- 25.Wessiling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)Google Scholar
- 26.Han, H., Il’in, V.P., Kellogg, R.B.: Flow directed iterations for convection dominated flow. In: Proceeding of the Fifth International Conference on Boundary and Interior Layers, pp. 7–17 (1988)Google Scholar
- 27.Ilin, V.P.: Finite Difference and Finite Volume Methods for Elliptic Equations. ICMMG Publishers, Novosibirsk (2001). (in Russian)Google Scholar