Two-Level Brezzi-Pitkäranta Discretization Method Based on Newton Iteration for Navier-Stokes Equations with Friction Boundary Conditions

We present a new stabilized finite element method for Navier-Stokes equations with friction slip boundary conditions based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of numerical solutions in some norms are derived for standard one-level method. Combining the techniques of two-level discretization method, we propose two-level Newton iteration method and show the stability and error estimate. Finally, the numerical experiments are given to support the theoretical results and to check the efficiency of this two-level iteration method

Two-Level Brezzi-Pitkäranta Stabilized Finite Element Methods for the Incompressible Flows

We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods

Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. Here we derive such schemes in 2D as the follow up of the companion paper treating the 1D case. Numerical tests include in particular a generalized 2D benchmark for Bingham codes (the Bingham–Couette flow with two non-zero boundary conditions on the velocity) and a simulation of the avalanche path of Taconnaz in Chamonix—Mont-Blanc to show the usability of these schemes on real topographies from digital elevation models (DEM)

Analysis of the Brezzi-pitkäranta stabilized Galerkin Scheme for creeping flows of Bingham fluids

In this paper we propose and analyze a finite element scheme for a class of variational nonlinear and nondifferentiable mixed inequalities including balance equations governing incompressible creeping flows of Bingham fluids. For numerical efficiency reasons, equal-order piecewise linear approximations are used for both velocity and pressure, and the numerical scheme is stabilized by a Brezzi-Pitkäranta perturbation term. We obtain error estimates of the same order as for stable discretizations, namely h1/2 for velocity and pressure solutions in [H2(Ω)]d and H 1(Ω), respectively. A decomposition-coordination algorithm to solve the discrete nonlinear algebraic system is presented, together with its convergence properties. Finally, numerical tests are performed. The solution of the problem under consideration presents particular regularity properties that are shown to permit convergence order improvement to h|log(h)|1/2. This estimate is confirmed by numerical results. © 2004 Society for Industrial and Applied Mathematics