Summary
We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.
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References
Axelsson, O., Barker, V.A.: Finite Element Solution of Boundary Value Problems: Theory and Computation. New York: Academic Press 1984
Bank, R.E.: PLTMG User's Guide, Edition 4.0; Technical Report, Department of Mathematics. University of California at San Diego 1985
Bank, R.E., Dupont, T.: Analysis of a Two-Level Scheme for Solving Finite Element Equations. Report CNA-159; Center for Numerical Analysis, University of Texas at Austin 1980
Bank, R.E., Dupont, T.: An Optimal Order Process for Solving Finite Element Equations. Math. Comput.36, 35–51 (1981)
Bank, R.E., Sherman, A., Weiser, A.: Refinement Algorithms and Data Structures for Local Mesh Refinement. In: Scientific Computing (R. Stepleman et al. eds.), Amsterdam: IMACS/North Holland 1983
Braess, D.: The Contraction Number of a Multigrid Method for Solving the Poisson Equation. Numer. Math.37, 387–404 (1981)
Bramble, J.H., Pasciak, J.E., Schatz, A.H.: An Iterative, Method for Elliptic Problems and Regions Partioned into Substructures. Math. Comput.46, 361–369 (1986)
Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The Construction of Preconditioners for Elliptic Problems by Substructuring. I. Math. Comput.47, 103–134 (1986)
Hackbusch, W.: Multigrid Methods and Applications. Berlin Heidelberg New York: Springer 1985
Hageman, L.A., Young, D.M.: Applied Iterative Methods. New York: Academic Press 1981
Maitre, J.F., Musy, F.: The Contraction Number of a Class of Two-Level Methods; an Exact Evaluation for some Finite Element Subspaces and Model Problems. In: Multigrid Methods (W. Hackbusch, U. Trottenberg eds.), Lect. Notes Math. 960, Berlin Heidelberg New York: Springer 1982
Rivara, M.C.: Algorithms for Refining Triangular Grids Suitable for Adaptive and Multigrid Techniques. Int J. Numer. Methods Eng.20, 745–756 (1984)
Young, D.M.: Convergence Properties of the Symmetric and Unsymmetric Successive Overrelaxation Method and Related Methods. Math. Comput.24, 793–807 (1970)
Yserentant, H.: On the Multi-Level Splitting of Finite Element Spaces. Numer. Math.49, 379–412 (1986)
Yserentant, H.: Hierarchical Bases Give Conjugate Gradient Type Methods a Multigrid Speed of Convergence. Appl. Math. Comput.19, 347–358 (1986)
Yserentant, H.: On the Multi-Level Splitting of Finite Element Spaces for Indefinite Elliptic Boundary Value Problems. Siam J. Numer. Anal.23, 581–595 (1986)
Yserentant, H.: Hierarchical Bases of Finite Element Spaces in the Discretization of Nonsymmetric Elliptic Boundary Value Problems. Computing35, 39–49 (1985)
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Bank, R.E., Dupont, T.F. & Yserentant, H. The hierarchical basis multigrid method. Numer. Math. 52, 427–458 (1988). https://doi.org/10.1007/BF01462238
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DOI: https://doi.org/10.1007/BF01462238