Abstract
The work of this thesis focuses on the development and analysis of Hybrid High-Order (HHO) discretization methods for complex problems in fluid mechanics. HHO methods are a new class of PDEs discretization methods, capable of handling general polytopic meshes. We are interested in problems involving non-Newtonian fluids, more precisely in a non-Hilbertian structure. The objective is to generalize discrete functional analysis theorems to the non-Hilbertian case in order to establish results of good position, convergence by compactness, and error estimates for HHO methods. Three main problems are studied for which we develop, analyze and numerically illustrate a HHO method. The first concerns the Stokes equations generalized to non-Newtonian fluids, which can consider fluids characterized by power-laws or Carreau-Yasuda laws. In this analysis, we introduce the notion of power-framed function, making it possible to handle the non-linearity of the problem. In addition, we generalize a discrete Korn inequality to the non-Hilbertian case in order to obtain the good position of the problem, as well as an error estimate. The second problem concerns the Leray-Lions problems, a classic example of which is that of the p-Laplacian. In the case of p<2, local degenerations can appear when the gradient of the solution vanishes or explodes. In this work, we establish new error estimates offering orders of convergence ranging from (k+1)(p-1) to k+1 depending on the degeneration of the problem, where k corresponds to the polynomial degree of the method. The third problem concerns the Navier-Stokes equations, generalized to incompressible non-Newtonian fluids, whose convection may follow a power-law. We introduce two Sobolev exponents characterizing the power-law behavior of the viscosity and convection laws of the fluid. In this work, an analysis of the continuous problem leads to relations between these Sobolev exponents which have repercussions at the discrete level. We thus establish convergence results under minimal regularity assumptions, as well as an error estimate for pseudoplastic fluids. Finally, we apply this method to the lid-driven cavity problem, illustrating the phenomena engendered by the introduction of the power-laws in the viscous and convective terms.