Metric-based mesh adaptation with curvilinear triangles

Curvilinear, also called high-order, meshes have been used for decades as a support for numerical simulations with finite element and finite volume methods. The added flexibility of high-order elements has historically been used to provide a more accurate representation of curved CAD boundaries than that offered by piecewise linear meshes. High-order meshes have gained interest in the last decade, as it has been shown that high-order numerical schemes, which enjoy accelerated convergence compared to classical second-order methods, may require high-order discretizations so as not to lose their approximation properties. Such body-fitted meshes are curved a posteriori by placing high-order vertices on the geometry. Curving boundary elements only may cause them to become invalid, i.e., tangled. Untangling methods cause edge or face curvature to be propagated inside the domain. Thus, the curvature of interior elements is never sought; it appears instead as a by-product of the untangling procedure, and the ideal element is always straight. Recently, the idea emerged of using the additional degrees of freedom provided by high-order meshes to further reduce the approximation error inside the computational domain. This is the topic of high-order anisotropic mesh adaptation, or simply curvilinear mesh adaptation. The classical theory of interpolation discourages arbitrarily curved elements, as they negatively impact the convergence rate of interpolation operators. It has been shown, however, that optimized curvilinear meshes may actually provide a substantial reduction in interpolation error, motivating their use in adaptivity. This thesis builds upon recent work in this direction and tackles the problem of metric-based curvilinear mesh adaptation in two dimensions. Specifically, we are concerned with the problem of generating, with the means of Riemannian metrics, meshes of quadratic triangles which minimize the interpolation error on a scalar field. This requires extending the classical questions of adaptivity, namely, quantifying the error committed on curved elements, deriving the Riemannian metric minimizing this error, describing the properties of ideal triangles for this metric, and finally, generating curved triangulations consisting of these ideal simplices. Each of these questions is addressed in more or less depth. We extend a recent linear error estimate to handle interpolation of arbitrary order on curved elements, then formulate the problem for the optimal metric, for which we provide a crude numerical scheme. The question of ideal simplices is treated by proposing a new definition of so-called unit elements based on Riemannian isometries, which encompasses the existing definitions. Finally, a meshing algorithm that outputs quasi-ideal triangulations for a given metric is presented.