120 C. J. Budd, W. Huang and R. D. Russell V θ 9000 9000 8000 8000 7000 7000 z 6000 5000 z 6000 5000 4000 4000 3000 3000 2000 2000 1000 1000 −1 −0.5 0 0.5 1 x x 10 6 −5 0 5 x x 10 5 Figure 5.11. The (x, z) contours of the longitudinal velocity and potential temperature of the Eady problem close to the formation of a tropical storm. 10000 Arclength V monitor function 10000 Max eigenvalue monitor function 9000 9000 8000 8000 7000 7000 z 6000 5000 4000 3000 2000 1000 0 −1 −0.5 0 x 0.5 1 x 10 6 z 6000 5000 4000 3000 2000 1000 0 −1 −0.5 0 0.5 1 x x 10 6 Figure 5.12. Two meshes adapted to the solution of the Eady problem presented above. In the first figure we use the arclength monitor function M 1 and in the second the potential vorticity monitor function M 2 .
Acknowledgements <strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 121 It is a pleasure to thank Emily Walsh for Figures 5.7, 5.8, 5.9, 5.11 and 5.12 and J. F. Williams for Figures 3.7, 3.8, 3.9, 3.11, 5.2 and 5.4. Both also helped by reading this text and through many useful discussions. This work was supported in part by the EPSRC Critical Mass Grant GR/586525/01. REFERENCES S. Adjerid and J. E. Flaherty (1986), ‘A <strong>moving</strong> finite element method <strong>with</strong> error estimation and refinement for one-dimensional time dependent partial differential equations’, SIAM J. Numer. Anal. 23, 778–795. M. Ainsworth and J. T. Oden (2000), A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics, Wiley-Interscience. V. F. Almeida (1999), ‘Domain deformation mapping: Application to variational mesh generation’, SIAM J. Sci Comput. 20, 1252–1275. D. A. Anderson and M. M. Rai (1983), The use of solution adaptive <strong>grids</strong> in solving partial differential equations, in Numerical Grid Generation (J. H. Thompson, ed.), pp. 317–338. V. B. Andreev and N. B. Kopteva (1998), ‘On the convergence, uniform <strong>with</strong> respect to a small parameter, of monotone three-point difference approximations’, Diff. Urav. 34, 921–929. U. Ascher, J. Christiansen and R. D. Russell (1981), ‘Collocation software for boundary value ODEs’, ACM Trans. Math. Software 7, 209–222. I. Babuška and W. C. Rheinboldt (1979), ‘Analysis of optimal finite element meshes in R 1 ’, Math. Comput. 33, 435–463. M. J. Baines (1994), Moving Finite Elements, Clarendon Press, Oxford. M. J. Baines and S. L. Wakelin (1991), Equidistribution and the Legendre transformation. Numerical Analysis report 4/91, University of Reading. M. J. Baines, M. E. Hubbard, and P. K. Jimack (2005), ‘A <strong>moving</strong> mesh finite strategy for the adaptive solution of time-dependent partial differential equations <strong>with</strong> <strong>moving</strong> boundaries’, Appl. Numer. Math. 54, 450–469. M. J. Baines, M. E. Hubbard, P. K. Jimack, and A. C. Jones (2006), ‘Scaleinvariant <strong>moving</strong> finite elements for nonlinear partial differential equations in two dimensions’, Appl. Numer. Math. 56, 230–252. M. L. Balinski (1986), ‘A competitive (dual) simplex method for the assignment problem’, Math. Program. 34, 125–141. R. E. Bank and R. K. Smith (1997), ‘Mesh smoothing using a posteriori error estimates’, SIAM J. Numer. Anal. 34, 979–997. G. I. Barenblatt (1996), Scaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics, Cambridge Texts in Applied Mathematics, Cambridge University Press. G. Beckett and J. A. Mackenzie (2000), ‘Convergence analysis of finite-difference approximations on equidistributed <strong>grids</strong> to a singularly perturbed boundary value problem’, Appl. Numer. Math. 35, 87–109. G. Beckett and J. A. Mackenzie (2001a), ‘On a uniformly accurate finite difference approximation of a singularly perturbed reaction–diffusion problem using grid equidistribution’, J. Comput. Appl. Math. 131, 381–405.