Narrow Results

In a three-dimensional bounded possibly multiply connected domain, we give gradient and higher-order estimates of vector fields via div and curl in L^p-theory. Then, we prove the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, we consider the stationary Stokes equations with non-standard boundary conditions of the form u × n = g × n and π = π₀ on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are based on obtaining Inf–Sup conditions that play a fundamental role. We give a variant of the Stokes system with these boundary conditions, in the case where the compatibility condition is not verified. Finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as u⋅n and u × n on Γ.

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming $P1$ finite element. For the solutions which admit a regularity greater than $H_0^2(\mathrm{\Omega })$, in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in $C^4(\bar{\mathrm{\Omega }}),$ an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem.