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 $\mathbf{L}^p$ norm of the continuous interpolation error. Given the $\mathbf{L}^p$ 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.