André Harnist

Hybrid High-Order methods are a recent and highly innovative class of new generation numerical methods for PDEs that aim at overcoming the limitations of traditional discretization methods such as Finite Element or Finite Volumes. Their most prominent features include:

• support of general polytopal meshes in arbitrary space dimension,
• arbitrary approximation order,
• compliance with the physics,
• reduced computational cost thanks to hybridization, static condensation, and compact stencil.

The objective of this Ph.D. thesis is to develop, analyze, and implement novel HHO discretizations of complex problems in fluid mechanics. We specifically aim at treating generalized Newtonian fluids where the shear stress function exhibits a nonlinear dependency on the shear rate, and possibly include the challenging variable density case. The convergence analysis will rely on both standard error estimates and compactness arguments. This will require to develop discrete functional analysis lemmas whose interest will go beyond applications to computational fluid mechanics. The implementation will rely on the spafedte library, and will benefit from the latest advances in C++ programming.