Equidistributing Meshes with Constraints

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 0 on an interval $[a,b]$ and constants c and K, the method produces a mesh with points x₀ = a,xj + 1 = xⱼ + hⱼ ,j = 0,1, ⋯ ,n - 1 and xₙ = b such that \[ ∫ₓⱼ^{xj + 1 } {f c {and} 1/K} {{hj + 1 }} {{h_j }} K {for}\, j = 0,1, ⋯ ,n - 1 . \] 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.