Boundary Value Problems Numerical Solution on Multiblock Grids

Authors: Martynenko S.I., Varaksin A.Yu. Published: 15.02.2021
Published in issue: #1(94)/2021  
DOI: 10.18698/1812-3368-2021-1-18-33

Category: Mathematics and Mechanics | Chapter: Computational Mathematics  
Keywords: boundary value problems, multigrid methods, multi-block grids

Results of theoretical analysis of the geometric multigrid algorithms convergence are presented for solving the linear boundary value problems on a two-block grid. In this case, initial domain could be represented as a union of intersecting subdomains, in each of them a structured grid could be constructed generating a hierarchy of coarse grids. Multigrid iteration matrix is obtained using the damped nonsymmetric iterative method as a smoother. The multigrid algorithm contains a new problem-dependent component --- correction interpolation between grid blocks. Smoothing property for the damped nonsymmetric iterative method and convergence of the robust multigrid technique are proved. Estimation of the multigrid iteration matrix norm is obtained (sufficient convergence condition). It is shown that the number of multigrid iterations does not depend on either the step or the number of grid blocks, if interpolation of the correction between grid blocks is sufficiently accurate. Results of computational experiments are presented on solving the three-dimensional Dirichlet boundary value problem for the Poisson equation illustrating the theoretical analysis. Results obtained could be easily generalized to multiblock grids. The work is of interest for developers of highly efficient algorithms for solving the (initial-) boundary value problems describing physical and chemical processes in complex geometry domains


[1] Il’in V.P. Matematicheskoe modelirovanie. Ch. 1. Nepreryvnye i diskretnye modeli [Mathematical simulation. P. 1. Continuous and discrete models]. Novosibirsk, SB RAS Publ., 2017.

[2] Fedorenko R.P. A relaxation method for solving elliptic difference equations. USSR Comput. Math. Math. Phys., 1962, vol. 1, iss. 4, pp. 1092--1096. DOI: https://doi.org/10.1016/0041-5553(62)90031-9

[3] Fedorenko R.P. The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys., 1964, vol. 4, iss. 3, pp. 227--235. DOI: https://doi.org/10.1016/0041-5553(64)90253-8

[4] Bakhvalov N.S. On the convergence of a relaxation method with natural constraints on the elliptic operator. USSR Comput. Math. Math. Phys, 1966, vol. 6, iss. 5, pp. 101--135. DOI: https://doi.org/10.1016/0041-5553(66)90118-2

[5] Astrakhantsev G.P. An iterative method of solving elliptic net problems. USSR Comput. Math. Math. Phys, 1971, vol. 11, iss. 2, pp. 171--182. DOI: https://doi.org/10.1016/0041-5553(71)90170-4

[6] Hackbusch W. Robust multi-grid methods, the frequency decomposition multi-grid algorithm. In: Robust Multi-Grid Methods. Viewig, Braunschweig, 1989, pp. 96--104.

[7] Trottenberg U., Oosterlee C.W., Schuller A. Multigrid. Academic Press, 2000.

[8] Brandt A., McCormick S.F., Ruge J. Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Institute for Computational Studies, 1982.

[9] Wesseling P. An introduction to multigrid methods. Wiley, 1992.

[10] Samarskiy A.A., Gulin A.V. Chislennye metody matematicheskoy fiziki [Numerical methods of mathematical physics]. Moscow, Nauchnyy mir Publ., 2003.

[11] Martynenko S.I. Mnogosetochnaya tekhnologiya: teoriya i prilozheniya [Multi-grid technology: theory and applications]. Moscow, FIZMATLIT Publ., 2015.

[12] Martynenko S.I. The robust multigrid technique. De Gruyter, 2017.

[13] Vanka S.P. Block-implicit multigrid solution of Navier --- Stokes equations in primitive variables. J. Comput. Phys., 1986, vol. 65, iss. 1, pp. 138--158. DOI: https://doi.org/10.1016/0021-9991(86)90008-2

[14] Xu J. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured meshes. Computing, 1996, vol. 56, no. 3, pp. 215--235. DOI: https://doi.org/10.1007/BF02238513

[15] Martynenko S.I. Potentialities of the robust multigrid technique. Comp. Meth. Appl. Math., 2010, vol. 10, iss. 1, pp. 87--94. DOI: https://doi.org/10.2478/cmam-2010-0004

[16] Hackbusch W. Multi-grid methods and applications. Springer Series in Computational Mathematics. Berlin, Heidelberg, Springer, 1985. DOI: https://doi.org/10.1007/978-3-662-02427-0

[17] Ol’shanskiy M.A. Lektsi i iuprazhneniya po mnogosetochnym metodam [Lectures and exercises on multigrid methods]. Moscow, FIZMATLIT Publ., 2005.