Algebraic and discrete Velte decompositions (Q5952342)

This paper is devoted to numerical methods for solving the first boundary value problem associated with systems of Stokes type. The authors discuss the basic decomposition of the type \[ V\equiv[H_0^1(\Omega)]^3=V_0\oplus V_1\oplus V_2 , \] where \(V_0\) and \(V_1\) correspond to kernels of divergence and rotation operators (an analog of the Helmholtz decomposition for \([L_2(\Omega)]^3\)). They pay special attention to dicrete variants of this splitting for difference and projective (finite element) methods since they believe that they are essential for getting accurate estimates for the constant in the inf-sup (LBB) condition (this condition is connected with the normal invertibility of the divergence operator and the estimate for \(\|[\div]^{(-1)}\|\)).