Commun. Comput. Phys., 10 (2011), pp. 509-576.

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

M. J. Baines 1*, M. E. Hubbard 2, P. K. Jimack 2

1 Department of Mathematics, The University of Reading, RG6 6AX, UK.
2 School of Computing, University of Leeds, LS2 9JT, UK.

Received 20 October 2010; Accepted (in revised version) 4 May 2011
Available online 1 June 2011


This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

AMS subject classifications: 35R35, 65M60, 76M10

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Key words: Time-dependent nonlinear diffusion, moving boundaries, finite element method, Lagrangian meshes, conservation of mass.

*Corresponding author.
Email: (M. Baines), (M. E. Hubbard), (P. K. Jimack)

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