Adaptation anisotrope pour maillages d'éléments d'ordre élevé

Anisotropic meshes have proved to be an efficient alternative to the classical and widely used isotropic meshes, as a support for finite element and finite volume computations. Such meshes allow for a significant reduction of the number of degrees of freedom, saving computational ressources that are time and memory. By allowing the elements to stretch along privileged directions in space, the adapted mesh follows the characteristics of both the geometry and the solution of the boundary value problem. While anisotropic adaptation on linear element meshes has become more and more popular, the lack of equivalence between the higher order error models and the metric tensor still hinders anisotropic adaptation when it comes to higher order element meshes, as those two quantities are represented by tensors of different orders. The present thesis introduces a methodology based on recent work to compute a metric tensor field, defined over the computational domain, in order to generate anisotropic meshes for elements of arbitrary order. The resulting mesh is constructed as the triangulation minimizing the interpolation error, evaluated in Lp norm. As most recent work on higher order meshes are focused on analytical problems, we also present here meshes adapted with respect to velocity fields obtained from fluid dynamics computations. The methodology is thus applied to simple flows, such as the lid-driven cavity and the backward facing step. The notion of continuous mesh links the classical discrete mesh, support for the finite element computations, to powerful continuous tools such as optmization and calculus of variations. This representation goes hand in hand with the metric tensor, which contains the anisotropic informations needed to perform the adaptation : indeed, the diagonalized form of its matrix representation shows the orientation and the length in each direction of an associated unit element. The classical hessian-based metric tensor, used for linear elements, can no longer be applied to higher order elements, whose interpolation error now features the higher order tensor of the derivatives of the solution. We shall thus look for a tensor of order two that is an upper bound for the higher order derivatives in the neighbourhood of a given mesh vertex. This tensor will then be scaled to obtain a metric tensor, allowing for the anisotropic adaptation. We then illustrate our methodology by computing metric tensors over fields featuring steep gradient regions : the resulting meshes show elements with a significant anisotropic ratio. Finally, a mesh convergence study is performed and a comparison between the obtained convergence rate and the one predicted by the continuous mesh theory is presented.