Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions
Baines, M. J., Hubbard, M. E., Jimack, P. K. and Jones, A. C. (2006) Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions. Applied Numerical Mathematics, 56 (2). pp. 230-252. ISSN 0168-9274
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A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.