## Grid adaptation via node movement.(English)Zbl 0889.65122

This paper gives a survey of methods for calculating grids adapted to the solution of the problem. Local mesh refinement is mentioned, but most of the discussion is devoted to grid selection by two methods: (1) by the equidistribution of some monitor function and (2) by the minimization of a norm of the error in the approximation. Both methods are illustrated by examples.

### MSC:

 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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### References:

 [1] Baines, M. J., Algorithms for optimal discontinuous piecewise linear and constant $$L_2$$ fits to continuous functions with adjustable nodes in one and two dimensions, Math. Comp., 62, 645-669 (1994) · Zbl 0801.41015 [2] Baines, M. J., Moving Finite Elements (1994), Oxford University Press: Oxford University Press Oxford · Zbl 0817.65082 [3] Baines, M. J., Approximate solutions of partial differential equations on optimal meshes using variational principles, Numerical Analysis Report 7/96 (1996), Department of Mathematics: Department of Mathematics University of Reading [4] Budd, C. J.; Huang, W.; Russell, R. D., Moving mesh methods for problems with blow-up, SIAM J. Sci. Comput., 17, 305-327 (1996) · Zbl 0860.35050 [5] Carey, G. F.; Dinh, H. T., Grading functions and mesh redistribution, SIAM J. Numer. Anal., 22, 1028-1040 (1985) · Zbl 0577.65076 [6] de Boor, C., Good Approximation by Splines with Variable Knots II (1973), Springer: Springer New York, Lecture Notes Series 363 · Zbl 0255.41007 [7] Huang, W.; Ren, Y.; Russell, R. D., Moving Mesh Partial Differential Equations (MMPDES) based on the equidistribution principle, SIAM J. Numer. Anal., 31, 709-730 (1994) · Zbl 0806.65092 [8] Huang, W.; Russell, R. D., A high-dimensional moving mesh strategy, Appl. Numer. Math., 26, 63-76 (1998), (this issue) · Zbl 0890.65101 [9] Hubbard, M. E.; Baines, M. J., Multidimensional upwinding and grid adaptation, (Morton, K. W.; Baines, M. J., Numerical Methods for Fluid Dynamics, V (1995), Oxford University Press: Oxford University Press Oxford), 431-438 · Zbl 0871.76056 [10] Jimack, P. K., A best approximation property of the moving finite element method, SIAM J. Numer. Anal., 33, 2286-2302 (1996) · Zbl 0863.65060 [11] Johnson, C.; Hansbo, P., Adaptive finite element methods in computational mechanics, Comput. Methods Appl. Mech. Engrg., 101, 143-181 (1992) · Zbl 0778.73071 [12] Süli, E.; Houston, P., Finite element methods for hyperbolic problems: stability, accuracy, adaptivity, Oxford University Computing Laboratory Report 96/09, Proceedings of the IMA Conference on the State of the Art in Numerical Analysis (1996), (OUP, to appear) [13] Tourigny, Y.; Baines, M. J., Analysis of an algorithm for generating locally optimal meshes for $$L_2$$ approximation by discontinuous piecewise polynomials, Math. Comp., 66, 623-650 (1997) · Zbl 0863.41013 [14] Wakelin, S. L., Variational principles and the finite element method for channel flows, Ph.D. Thesis (1993), Department of Mathematics: Department of Mathematics University of Reading [15] Wakelin, S. L.; Baines, M. J.; Porter, D., The use of variational principles in determining approximations to continuous and discontinuous shallow water flows, Numerical Analysis Report 13/95 (1995), Department of Mathematics: Department of Mathematics University of Reading [16] Zegeling, P. A., MFE revisited, Part 1: Adaptive grid-generation using the heat equation, Preprint No. 946 (February, 1996), Department of Mathematics: Department of Mathematics University of Utrecht
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