Adaptation de maillages pour des schémas numériques d'ordre très élevé. (Mesh adaptation for very high order numerical schemes)

Mesh adaptation is an iterative process which consists in changing locally thesize and orientation of the mesh according the behavior of the studied physical solution. Itgenerates the best mesh for a given problem and a fix number of degrees of freedom. Meshadaptation methods have proven to be extremely effective in reducing significantly the meshsize for a given precision and reaching quickly an second-order asymptotic convergence forproblems containing singularities when they are coupled to high order numerical methods.In metric-based mesh adaptation, two approaches have been proposed: Multi-scale methodsbased on a control of the interpolation error in Lp-norm and Goal oriented methods thatcontrol the approximation error of a functional through the use of the adjoint state. However,with the emergence of very high order numerical methods such as the discontinuous Galerkinmethod, it becomes necessary to take into account the order of the numerical scheme in meshadaptation process. Mesh adaptation is even more crucial for such schemes as they convergeto first-order in flow singularities. Therefore, the mesh refinement at the singularities of thesolution must be as important as the order of the method is high.This thesis deals with the extension of the theoretical and numerical results getting in thecase of mesh adaptation for piecewise linear solutions to high order piecewise polynomialsolutions. These solutions are represented using kth-order Lagrangian finite elements (k > 2).This thesis will focus on modeling the local interpolation error of order Pk+1 (k > 2) on acontinuous mesh. However, for metric-based mesh adaptation methods, the error model mustbe a quadratic form, which shows an intrinsic metric space. Therefore, to be able to producesuch an area, it is necessary to decompose the homogeneous polynomial and to approximateit by a quadratic form taken at powerk2 . This modeling allows us to define a metric fieldnecessary to communicate with the mesh generator. The decomposition method will be anextension of the diagonalization method to high order homogeneous polynomials. Indeed, in2D and 3D, symmetric tensor decomposition methods such as Sylvester decomposition andits extension to high dimensions will allow us to decompose locally the error function, then,to deduce the quadratic error model. Then, this local error model is used to control theoverall error in Lp-norm and the optimal mesh is obtained by minimizing this error.In this thesis, we seek to demonstrate the kth-order convergence of high order mesh adaptationmethod for analytic functions and numerical simulations using kth-order solvers (k > 3).