This thesis is concerned with new numerical and algorithmic tools for flows with pressure and shear
dependent viscosity together with the necessary background of the generalized Navier-Stokes equations.
In general, the viscosity can depend on the density, for instance in the case of compressible flow, and if the relation
between the pressure and the density is reversible, the viscosity may depend on the pressure. This, however, is
not the case for incompressible powder flow where the viscous stresses may change with varying pressure whereas
density changes remain negligible. Meanwhile, the viscosity of the flow may change as the shear rate is varied,
for instance in Bingham flow. Both dependencies can occur together, something that holds for the so-called nonflowing
or slow-flowing materials, namely slow-flowing smaller-sized bulk powder. The Navier-Stokes equations
in primitive variables (velocity-pressure) are regarded as the privilege answer to incorporate these phenomena.
The modification of the viscous stresses leads to generalized Navier-Stokes equations extending the range of their
validity to such flow.
The resulting equations are mathematically more complex than the Navier-Stokes equations. From the numerical
point of view several problems arise. Firstly, the difficulty of approximating incompressible velocity fields. Secondly,
poor conditioning and possible lack of differentiability of the involved nonlinear functions due to the material laws
and finally, the eventual domination of convective effects.
The difficulty related to the approximation of incompressible velocity fields was treated by applying the nonconforming
Rannacher-Turek Stokes element. However, with this approach another problem arises related to the
low order nonconforming approximation for problems involving the symmetric part of the gradient: the classical
discrete Korn’s inequality is not satisfied. To handle this, the interior penalty stabilized finite element method
in the frame of Nitsche’s method was used. It requires a modification of the discrete bilinear form by adding an
interface term, penalizing the jump of the velocity over edges. This was achieved via a modified procedure in the
derivation of a discontinuous Galerkin formulation. This general approach is referred to as edge-oriented finite
element method.
The lack of differentiability was treated by regularization. Then, the continuous Newton method as linearization
technique was applied. The method consists of working directly on the variational integrals. Then, the
corresponding continuous Jacobian operators were derived. Consequently, a convergence rate of the nonlinear
iterations independent of the mesh refinement was achieved. This continuous approach is advantageous: Firstly,
the explicit accessibility of the Jacobian allows adaptive treatment which leads to a robust method with respect
to the starting guess. Secondly, it avoids the delicate task of choosing the step-length which is required for divided
differences approaches.
Another criterion for the success of any numerical method is to overcome the numerical instabilities at high
Reynolds number such that the Galerkin formulation fails and causes spurious oscillations. Although the edgeoriented
method was introduced in the frame of hybrid finite element methods, it has been shown that the
appropriate choice of edge-oriented stabilization is able to provide simultaneously excellent results regarding
robustness and accuracy for both apparent problems of the numerical solution: the lack of coercivity and a
convection dominated problem.
A fundamental issue for edge-oriented stabilization is the growing stencil for the stiffness matrix. The jump terms
involve more than the adjacent elements; the problem of storing the new stabilization matrix arises. To overcome
the storage problem, the stiffnes matrix would contain a part of stabilization matrix which fits into the same
standard FEM data structure, while the complementary part is assembled via elementwise operations. Moreover,
the construction of preconditioners for the corresponding linear systems would only include parts of the matrix.
In parallel, a nonstandard edge-oriented data structure has been developed to support the additional contributed
elements. So, the local element and edge matrices are easily deduced from the global one. Accordingly, efficient
Vanka smoothers were introduced, namely a full cell-oriented Vanka smoother and edge-oriented Vanka smoother.
In doing so, it became possible to privilege edge-oriented stabilization for CFD simulations. Moreover, the velocitypressure
coupling was treated in linearized saddle point problems via local Multilevel Pressure Schur Complement
methods.
The principle of empiric verification stands behind the wide range of well known benchmark calculations introduced
to fulfill the objective need of performance.
Key words: Edge-oriented Finite Element Method, Multilevel Pressure Schur Complement, Newton’s
Method, Nonlinear Fluids, Granular Material.