CONTINUOUS MESH FRAMEWORK PART II: VALIDATIONS AND APPLICATIONS

• Published in: SIAM journal on numerical analysis Vol. 49; no. 1/2; pp. 61 - 86
• Main Authors: LOSEILLE, ADRIEN, ALAUZET, FRÉDÉRIC
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Amplitude; Analysis of PDEs; Approximation; Approximations and expansions; Calculus of variations and optimal control; Convergence; Error analysis; Error rates; Errors; Euler equations; Exact sciences and technology; Interpolation; Linear interpolation; Mathematical analysis; Mathematical functions; Mathematical models; Mathematical sequences; Mathematics; Modeling; Norms; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Optimization; Partial differential equations; Sciences and techniques of general use; Strategy; Studies; Vertices; 65N15; Three-dimensional calculations; Metric space; Error estimation; anisotropic mesh adaptation; interpolation error; Duality; Adaptive control; Partial differential equation; Numerical approximation; 65N50; Variational calculus; 65L50; Analytical function; continuous mesh; Grid pattern; Riemannian metric space; Optimal convergence; optimal interpolation error bound; Interpolation; Numerical analysis; linear interpolate; 65D05; Optimal approximation; Two-dimensional calculations; unstructured mesh; optimal interpolation error bound AMS subject classifications. 65D05
• ISSN: 0036-1429; 1095-7170; 1095-7170
• DOI: 10.1137/10078654X

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 L p norm of the continuous interpolation error. Given the 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.