Scale-inariant moving finite elements for time-dependent nonlinear partial differential equations

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

may be carried out using a scale-invariant time-stepping. The accuracy and

reliability of the algorithm is tested against exact self-similar

solutions, where available, and a state-of-the-art $h$-refinement scheme

for a range of second and fourth order problems with moving boundaries.

The monitor functions used are the dependent variable and a monitor

related to the surface area of the solution manifold.