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Least squares and approximate equidistribution in multidimensions
Author(s) -
Baines M. J.
Publication year - 1999
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199909)15:5<605::aid-num7>3.0.co;2-9
Subject(s) - mathematics , residual , conservation law , piecewise , partition (number theory) , minification , mathematical analysis , norm (philosophy) , vector field , combinatorics , geometry , law , mathematical optimization , algorithm , political science
In this article it is shown that, under a natural condition, least squares minimization of the residual of the divergence of a vector field is equivalent to that of a least squares measure of equidistribution of the residual. More specifically, consider the conservation law div f = 0, when the vector field f is approximated by a conforming piecewise differentiable function F on a partition of a polygonal region Ω into triangles. Then, we show that, if F has a prescribed flux across the outer boundary ∂Ω of Ω, minimization of the l 2 norm of the average residual of div F over all internal parameters of the partition (including nodal positions as well as solution amplitudes) is equivalent to minimization of the l 2 norm of the differences in the average residuals of F , taken over all pairs of triangles of the partition. The result is of importance in the approximate solution of conservation laws, where alignment of the mesh is often of considerable benefit in deriving extra accuracy. The property is readily extended to systems of conservation laws. Moreover it holds for the average vorticity residual of F over a triangle as well as for l 2 ‐type norms combining both the divergence and the vorticity (as in the case of the Cauchy‐Riemann equations). © 1999 Wiley & Sons. Inc. Numer Methods Partial Differential Eq 15:605–615, 1999