76D05 Navier-Stokes equations [See also 35Q30]
We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.
In incompressible flows with vanishing
normal velocities at the boundary, irrotational forces in the momentum
equations should be balanced
completely by the pressure gradient.
Unfortunately, nearly all available discretization methods for incompressible flows violate this property.
The origin of the problem is that discrete velocity approximations
of incompressible flows are usually not
divergence-free. Hence, the use of divergence-free velocity reconstructions is
proposed wherever an $L^2$ scalar product appears in the discrete
variational formulation.
The approach is illustrated and applied to a nonconforming MAC-like discretization for unstructured Delaunay grids.
It is numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart-Thomas element
increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.
We apply the general framework developed by John et al. in [15] to
analyze the convergence of multi-level methods for mixed finite element
discretizations of the generalized Stokes problem using the
Scott-Vogelius element. Having in mind that semi-implicit operator
splitting schemes for the Navier-Stokes equations lead to this class of
problems, we take symmetric stabilization operators into account. The use
of the class of Scott-Vogelius elements seems to be promising since
discretely divergence-free functions are pointwise divergence-free.
However, to satisfy the Ladyzhenskaya-Babuska-Brezzi stability
condition, we have to deal in the multi-grid analysis with non-nested
families of meshes which are derived from nested macro element
triangulations.