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Moving finite element, least squares, and finite volume approximations of steady and time-dependent PDEs in multidimensions by M.J. Baines is a Mathematics article available to read on EtoBox.

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We review recent advances in Galerkin and least squares methods for approximating the solutions of ÿrst-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 uniÿed manner. Both ÿnite element and ÿnite volume schemes are considered, as are transient and steady problems. For ÿrst-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 ÿnite 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 ÿnite volume (MFV) method, based on the l2 norm, does not have this property however, although there does exist a ÿnite volume method which gives an optimal mesh, both for variational principles and least squares.

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Author

M.J. Baines

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0377-0427)

Published

2001

Language

EN

Field

Mathematics (Physical Sciences)