Continuously deforming finite elements

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Abstract
A general class of time‐dependent co‐ordinate transformations is introduced in a variational formulation for evolution problems. The variational problem is posed with respect to both solution and transformation field variables. An approximate analysis using finite elements is developed from the continuous variational form. Modified forms of the variational functional are considered to ensure the deforming mesh is not top irregular. ODE system integrators are utilized to integrate the resulting semidiscrete systems. In Part I we consider the formulation for problems in one spatial dimension and time, including, in particular, convection‐dominated flows described by the convection‐diffusion, Burgers' and Buckley–Leverett equations. In Part II the extension of the method to two dimensions and supporting numerical experiments are presented.

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