Mesh adaptation is an iterative process which consists in changing locally the size and orientation of the mesh according the behavior of the studied physical solution. It generates the best mesh for a given problem and a fix number of degrees of freedom. Mesh adaptation methods have proven to be extremely effective in reducing significantly the mesh size for a given precision and reaching quickly an second-order asymptotic convergence for problems containing singularities when they are coupled to high order numerical methods. In metric-based mesh adaptation, two approaches have been proposed: Multi-scale methods based on a control of the interpolation error in Lp-norm and Goal oriented methods that control 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 Galerkin method, it becomes necessary to take into account the order of the numerical scheme in mesh adaptation process. Mesh adaptation is even more crucial for such schemes as they converge to first-order in flow singularities. Therefore, the mesh refinement at the singularities of the solution 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 the case of mesh adaptation for piecewise linear solutions to high order piecewise polynomial solutions. These solutions are represented using kth-order Lagrangian finite elements (k ≥ 2). This thesis will focus on modeling the local interpolation error of order k ≥ 3 on a continuous mesh. However, for metric-based mesh adaptation methods, the error model must be a quadratic form, which shows an intrinsic metric space. Therefore, to be able to produce such an area, it is necessary to decompose the homogeneous polynomial and to approximate it by a quadratic form taken at power k. This modeling allows us to define a metric field necessary to communicate with the mesh generator. The decomposition method will be an extension of the diagonalization method to high order homogeneous polynomials. Indeed, in 2D and 3D, symmetric tensor decomposition methods such as Sylvester decomposition and its 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 the overall 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 adaptation method for analytic functions and numerical simulations using kth-order solvers (k ≥ 3).
