Robust multigrid technique for solving partial differential equations on structured grids


  • S.I. Martynenko


многосеточная технология, численные методы, итерационные алгоритмы, эллиптические дифференциальные уравнения, метод контрольного объема, краевые задачи


A new robust multigrid technique for solving elliptic partial differential equations is proposed. The technique is based on a united computational algorithm that consists of the following stages: 1) adaption of equations to numerical methods, 2) the control volume discretization, and 3) applying multigrid iterations. Special subgrids of the finest grid are generated to obtain the most powerful coarse grid correction strategy. Accuracy of the transfer operators is independent of the mesh size on coarse grids; therefore, a smoothing procedure and a multigrid cycle may be very simple. Expanded robustness of the multigrid technique is a result of adaption of equations, extremely accurate formulation of the discrete problems on the coarse grids, original coarsening, the most powerful coarse grid correction strategy, construction of problem-independent transfer operators, and absence of pre-smoothing and interpolation. The paper represents the algorithm, estimates of computational work, and results of numerical tests performed. Our numerical tests demonstrate robustness and efficiency of our multigrid technique.

Author Biography

S.I. Martynenko


  1. Hackbusch W. Robust multi-grid methods, the frequency decomposition multi-grid algorithm. Proc. 4th GAMM-seminar. Kiel, 1988.
  2. Wesseling P. An Introduction to Multigrid Methods. Chichester, 1991.
  3. Федоренко Р.П. Релаксационный метод решения разностных эллиптических уравнений // Журн. вычисл. матем. и матем. физ. 1961. T 1. 992-927.
  4. Patankar S. Numerical Heat Transfer and Fluid Flow. New York, 1980.
  5. Koren B. Defect correction and multigrid for an efficient and accurate computation of airfoil flows // J. Comput. Phys. 1988. T 77. 183-206.
  6. Hackbusch W. Multi-grid methods and applications. Berlin, 1985.
  7. Martynenko S.I. Template for the solution of partial differential equations: building blocks and diagnostic tools for the robust multigrid technique // Numerical Methods and Programming (to appear).
  8. Brandt A. Multi-level adaptive solutions to boundary value problems // Math. Comput. 1977. T 31. 333-390.
  9. Dendy Jr. J. E. Black box multigrid // J. Comput. Phys. 1982. T 48. 366-386.
  10. Wessiling P. Cell-centred multigrid for interface problem // J. Comput. Phys. 1988. T 79. 85-91.
  11. Launder B., Sharma B Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc // Letters in Heat and Mass Transfer. 1974. T 1. 131-138.



How to Cite

Мартыненко С. Robust Multigrid Technique for Solving Partial Differential Equations on Structured Grids // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2000. 1. 83-102



Section 1. Numerical methods and applications

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