Moving finite element, least squares, and finite volume approximations of steady and time-dependent PDEs in multidimensions

We review recent advances in Galerkin and least squares methods for approximating the solutions of first- and second-order PDEs with moving nodes in multidimensions. These methods use unstructured meshes and minimise the norm of the residual of the PDE over both solutions and nodal positions in a unified manner. Both finite element and finite volume schemes are considered, as are transient and steady problems. For first-order scalar time-dependent PDEs in any number of dimensions, residual minimisation always results in the methods moving the nodes with the (often inconvenient) approximate characteristic speeds. For second-order equations, however, the moving finite element (MFE) method moves the nodes usefully towards high-curvature regions. In the steady limit, for PDEs derived from a variational principle, the MFE method generates a locally optimal mesh and solution: this also applies to least squares minimisation. The corresponding moving finite volume (MFV) method, based on the l2 norm, does not have this property however, although there does exist a finite volume method which gives an optimal mesh, both for variational principles and least squares.