Equidistributing Meshes with Constraints
01 Dec 1980-Siam Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics)-Vol. 1, Iss: 4, pp 499-511
TL;DR: In this article, a technique for equidistributing a function subject to constraints on the ratios of adjacent steps in the mesh is presented and a theoretical analysis of the procedure is presented, and numerical algorithms for implementing the method are given.
Abstract: Adaptive methods which “equidistribute” a given positive weight function are now used fairly widely for selecting discrete meshes. The disadvantage of such schemes is that the resulting mesh may not be smoothly varying. In this paper a technique is developed for equidistributing a function subject to constraints on the ratios of adjacent steps in the mesh. Given a weight function $f \geqq 0$ on an interval $[a,b]$ and constants c and K, the method produces a mesh with points $x_0 = a,x_{j + 1} = x_j + h_j ,j = 0,1, \cdots ,n - 1$ and $x_n = b$ such that \[ \int_{xj}^{x_{j + 1} } {f \leqq c\quad {\text{and}}\quad \frac{1} {K}} \leqq \frac{{h_{j + 1} }} {{h_j }} \leqq K\quad {\text{for}}\, j = 0,1, \cdots ,n - 1 . \] A theoretical analysis of the procedure is presented, and numerical algorithms for implementing the method are given. Examples show that the procedure is effective in practice. Other types of constraints on equidistributing meshes are also discussed.The principal application of the procedure is to the solution of boundary value problems, where the weight function is generally some error indicator, and accuracy and convergence properties may depend on the smoothness of the mesh. Other practical applications include the regrading of statistical data.
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277 citations
Cites background from "Equidistributing Meshes with Constr..."
...(2.9) Such grids are termed quasi-uniform (Li et al. 1998, Zegeling 2007, Kautsky and Nichols 1980, Kautsky and Nichols 1982), and normally lead to truncation (and approximation) errors of the same order as uniform meshes (Veldman and Rinzema 1992)....
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01 Jan 1988
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References
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TL;DR: In this article, various adaptive mesh selection strategies for solving two-point boundary value problems are brought together and a limited comparison is made, and the mesh strategies are applied using collocation met...
Abstract: Various adaptive mesh selection strategies for solving two-point boundary value problems are brought together and a limited comparison is made. The mesh strategies are applied using collocation met...
204 citations
TL;DR: The combination of automatic variable order (via deferred corrections) and automatic (adaptive) mesh selection produces, as in the case of initial value problem solvers, a versatile, robust, and efficient algorithm.
Abstract: A variable order variable step finite difference algorithm for approximately solving m-dimensional systems of the form y'' = f(t,y), t $\in$ [a,b] subject to the nonlinear boundary conditions g(y(a),y(b)) = 0 is presented. A program, PASVAR, implementing these ideas has been written and the results on several test runs are presented together with comparisons with other methods. The main features of the new procedure are: a) Its ability to produce very precise global error estimates, which in turn allow a very fine control between desired tolerance and actual output precision. b) Non-uniform meshes allow an economical and accurate treatment of boundary layers and other sharp changes in the solutions. c) The combination of automatic variable order (via deferred corrections) and automatic (adaptive) mesh selection produces, as in the case of initial value problem solvers, a versatile, robust, and efficient algorithm.
198 citations
149 citations
TL;DR: For a two-point boundary-value problem the existence of a unique optimal mesh distribution is proved and its properties are analyzed, allowing for rather straightforward extensions to more general problems in one dimension as well as to higher-order elements.
Abstract: A theory of a posteriori estimates for the finite element method has been developed. On the basis of this theory, for a two-point boundary-value problem the existence of a unique optimal mesh distribution is proved and its properties are analyzed. This mesh is characterized in terms of certain, easily computable local error indicators which in turn allow for a simple adaptive construction of the mesh and also permit the computation of a very effective a posteriori error bound. While the error estimates are asymptotic in nature, numerical experiments show the results to be excellent already for 10% accuracy. The approaches are not restricted to the model problem considered here only for clarity; in fact, they allow for rather straightforward extensions to more general problems in one dimension as well as to higher-order elements. 11 tables.
137 citations
TL;DR: This paper extends techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes to meshes on which the local truncation error of the method is approximately constant in some norm.
Abstract: In order to use finite difference approximations with non-uniform meshes in boundary value problems, it is necessary to develop procedures for mesh selection. In this paper we extend techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes. By this term we mean meshes on which the local truncation error of the method is approximately constant in some norm. Improved error estimates for methods which use equidistributing meshes are obtained.
136 citations