Analysis of an algorithm for generating locally optimal meshes for 𝐿₂ approximation by discontinuous piecewise polynomials

Abstract

Abstract. This paper discusses the problem of constructing a locally optimal mesh for the best L 2 approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines' algorithm [M. J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

“…The L 2 error can further be minimized over node positions as well as nodal values to give an optimal initial mesh [5,138].…”

A Local Convergence Proof for the Minvar Algorithm for Computing Continuous Piecewise Linear Approximations

“…Algorithms similar in flavor to the scalar specialization of minvar were introduced in [2,25,26]. A treatment of discontinuous piecewise polynomial approximations on two dimensional triangulations is provided in [40]. Also, [41] provides an algorithm for a moving mesh finite element solution to variational problems.…”

“…A specialization of this moving mesh algorithm is finding the best L p , p finite and even, continuous piecewise polynomial approximation to a function. Both of these algorithms [40,41], as well as the piecewise polynomial literature in general [2,25,26], assume that the function to be approximated is available directly, and the algorithms entail steps, such as root finding, that incorporate the function intrinsically. In contrast, the minvar algorithm is defined for arbitrary (finite) dimension and can either use a finite set of data or directly use of the function to be approximated.…”

Invertible piecewise linear approximations for color reproduction

“…Tourigny and Baines [7] present an algorithm for the two-dimensional function approximation problem, which can be generalized to higher dimensions. There are no corresponding analytical results.…”

“…Analytical results are available for the one dimensional case for certain functioq families [5]. Algorithms exist for one dimension [6] as well as for higher dimensions [7] (i.e. monotonic).…”