Convergence rates and classification for one-dimensional finite-element meshes

摘要

Recent advances in adaptive gridding strategies have highlighted the need to consider the finite-element method on irregular meshes. In this paper, we consider irregular grids in the context of refinement paths, i.e. sequences of grids such that h0. For a two-point boundary value problem with piecewise linear elements for which O(h2) convergence is expected in the L2 sense, we demonstrate a family of grids that give||u - uh||L2 Chp where p can take any value in the interval [2,2.5]. Here, as usual, h is the mesh size and u and uh are the actual and approximate solutions. Despite this apparent superconvergence, these grids actually define a suboptimal refinement path. This is due to the high degree of irregularity of the meshes. In this situation, it is shown that the classical h-convergence rate is not the appropriate rate to analyze. We also propose a simple classification that discriminates when this behaviour can and cannot occur.