Continuous Mesh Framework Part II: Validations and Applications

This paper gives a numerical validation of the continuous mesh framework introduced in Part I [A. Loseille and F. Alauzet, SIAM J. Numer. Anal., 49 (2011), pp. 38–60]. We numerically show that the interpolation error can be evaluated analytically once analytical expressions of a mesh and a function are given. In particular, the strong duality between discrete and continuous views for the interpolation error is emphasized on two-dimensional and three-dimensional examples. In addition, we show the ability of this framework to predict the order of convergence, given a specific adaptive strategy defined by a sequence of continuous meshes. The continuous mesh concept is then used to devise an adaptive strategy to control the Lᵖ norm of the continuous interpolation error. Given the Lᵖ norm of the continuous interpolation error, we derive the optimal continuous mesh minimizing this error. This exemplifies the potential of this framework, as we use a calculus of variations that is not defined on the space of discrete meshes. Anisotropic adaptations on analytical functions correlate the optimal predicted theoretical order of convergence. The extension to a solution of nonlinear PDEs is also given. Comparisons with experiments show the efficiency and the accuracy of this approach.