Adaptivity with moving grids

122 C. J. Budd, W. Huang and R. D. Russell
G. Beckett and J. A. Mackenzie (2001b), ‘Uniformly convergent high order finite
element solutions of a singularly perturbed reaction–diffusion equation using
mesh equidistribution’, Appl. Numer. Math. 39, 31–45.
G. Beckett, J. A. Mackenzie and M. L. Robertson (2001a), ‘A moving mesh finite
element method for the solution of two-dimensional Stefan problems’,
J. Comput. Phys. 186, 500–518.
G. Beckett, J. A. Mackenzie, A. Ramage and D. M. Sloan (2001b), ‘On the numerical
solution of one-dimensional PDEs using adaptive methods based on
equidistribution’, J. Comput. Phys. 167, 372–392.
J. D. Benamou and Y. Brenier (2000), ‘A computational fluid mechanics solution to
the Monge–Kantorovich mass transfer problem’, Numer. Math. 84, 375–393.
M. Berger and R. Kohn (1988), ‘A rescaling algorithm for the numerical calculation
of blowing-up solutions’, Comm. Pure. Appl. Math. 41, 841–863.
M. Berzins (1998), ‘A solution-based triangular and tetrahedral mesh quality indicator’,
SIAM J. Sci. Comput. 19, 2051–2060.
J. G. Blom and J. G. Verwer (1989), On the use of the arclength and curvature
monitor in a moving grid method which is based on the method of lines.
Technical Report NM-N8902, CWI, Amsterdam.
P. Bochev, G. Liao and G. d. Pena (1996), ‘Analysis and computation of adaptive
moving grids by deformation’, Numer. Methods PDEs 12, 489–506.
C. de Boor (1973), Good Approximations by Splines with Variable Knots II, Vol. 363
of Lecture Notes in Mathematics, Springer, Berlin.
J. U. Brackbill (1993), ‘An adaptive grid with directional control’, J. Comput.
Phys. 108, 38–50.
J. U. Brackbill and J. S. Saltzman (1982), ‘Adaptive zoning for singular problems
in two dimensions’, J. Comput. Phys. 46, 342–368.
L. Branets and G. F. Carey (2003), A local cell quality metric and variational
grid smoothing algorithm, in Proc. 12th International Meshing Roundtable,
Sandia National Laboratories, Albuquerque, NM.
Y. Brenier (1991), ‘Polar factorization and monotone rearrangement of vectorvalued
functions’, Comm. Pure Appl. Math. 44, 375–417.
C. J. Budd and V. A. Dorodnitsyn (2001), ‘Symmetry adapted moving mesh
schemes for the nonlinear Schrödinger equation’, J. Phys. A 34, 103887–
10400.
C. J. Budd and M. D. Piggott (2005), Geometric integration and its applications,
in Handbook of Numerical Analysis (F. Cucker, ed.), pp. 35–139.
C. J. Budd and J. F. Williams (2006), ‘Parabolic Monge–Ampère methods for
blow-up problems in several spatial dimensions’, J. Phys. A 39, 5425–5444.
C. J. Budd and J. F. Williams (2009), Mesh generation using the parabolic Monge–
Ampère method. Submitted.
C. J. Budd, W. Z. Huang, and R. D. Russell (1996), ‘Moving mesh methods for
problems with blow-up’, SIAM J. Sci. Comput. 17, 305–327.
C. J. Budd, S.-N. Chen and R. D. R. Russell (1999a), ‘New self-similar solutions
of the nonlinear Schrödinger equation, with moving mesh computations’,
J. Comput. Phys. 152, 756–789.