On Robust and Adaptive Multi-Grid Methods

In the present paper we discuss the development and practical application of robust multi-grid methods to solve partial differential equations on adaptively refined grids. We review several approaches to achieve robust multi-grid methods and describe two special new strategies for anisotropic and convection diffusion problems. The performance of these algorithms is investigated for three selected test problems.

[1]  K. Stüben.  Algebraic multigrid (AMG): experiences and comparisons , 1983 .

[2]  Wolfgang Hackbusch,et al.  The Frequency Decomposition Multi-Grid Algorithm , 1989 .

[3]  J. Douglas,et al.  A unified convergence theory for abstract multigrid or multilevel algorithms, serial and parallel , 1993 .

[4]  David E. Keyes,et al.  Flame sheet starting estimates for counterflow diffusion flame problems , 1987 .

[5]  D. Rose,et al.  Analysis of a multilevel iterative method for nonlinear finite element equations , 1982 .

[6]  Craig C. Douglas.  MGNet: a multigrid and domain decomposition network , 1992, SGNM.

[7]  A. Brandt.  Guide to multigrid development , 1982 .

[8]  Wolfgang Hackbusch,et al.  Theorie und Numerik elliptischer Differentialgleichungen , 1986, Teubner Studienbücher.

[9]  P. Wesseling.  Theoretical and Practical Aspects of a Multigrid Method , 1982 .

[10]  W. Mulder.  A new multigrid approach to convection problems , 1989 .

[11]  Alexandre Ern,et al.  Numerical solution of flame sheet problems with and without multigrid methods , 1993 .

[12]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[13]  Wolfgang Hackbusch.  The frequency decomposition multi-grid method , 1989 .

[14]  Peter Bastian.  Locally Refined Solution of Unsymmetric and Nonlinear Problems , 1993 .

[15]  W. Miranker,et al.  Acceleration by aggregation of successive approximation methods , 1982 .

[16]  Craig C. Douglas.  Implementing abstract multigrid or multilevel methods , 1993 .

[17]  S. McCormick,et al.  Multigrid methods for numerical simulation of laminar diffusion flames , 1993 .

[18]  Gabriel Wittum,et al.  Linear iterations as smoothers in multigrid methods: Theory with applications to incomplete decompositions , 1989, IMPACT Comput. Sci. Eng..

[19]  R. Bank,et al.  Sharp Estimates for Multigrid Rates of Convergence with General Smoothing and Acceleration , 1985 .

[20]  W. Hackbusch.  Multi-Grid Methods , 1994 .

[21]  Craig C. Douglas,et al.  A tupleware approach to domain decomposition methods , 1991 .

[22]  R. Bank,et al.  The hierarchical basis multigrid method , 1988 .

[23]  P. Wesseling.  An Introduction to Multigrid Methods , 1992 .

[24]  Anne Greenbaum.  A multigrid method for multiprocessors , 1986 .

[25]  W. Hackbusch,et al.  Analysis of a damped nonlinear multilevel method , 1989 .

[26]  P. Wesseling.  A robust and efficient multigrid method , 1982 .

[27]  María Cecilia Rivara,et al.  Design and data structure of fully adaptive, multigrid, finite-element software , 1984, ACM Trans. Math. Softw..

[28]  G. Wittum.  On the Robustness of ILU Smoothing , 1989 .

[29]  W. Miranker,et al.  Constructive interference in parallel algorithms , 1988 .

[30]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[31]  Rob Stevenson,et al.  New estimates of the contraction number of V-cycle multi-grid with applications to anisotropic equations , 1993 .

[32]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[33]  C. Douglas.  Multi-Grid Algorithms with Applications to Elliptic Boundary Value Problems , 1984 .

[34]  Cornelis W. Oosterlee,et al.  Multigrid Schemes for Time-Dependent Incompressible Navier-Stokes Equations , 1993, IMPACT Comput. Sci. Eng..

[35]  Jinchao Xu,et al.  Convergence estimates for multigrid algorithms without regularity assumptions , 1991 .