Numerical Enclosures for Solutions of the Navier-stokes Equation for Small Reynolds Numbers

We describe a method to compute veriied enclosures for solutions of the stationary Navier-Stokes equation in two-dimensional bounded domains. In order to obtain error bounds for numerical approximations , we use the theorem of Newton{Kantorovich. Therefore, we compute approximations for the stream function of the ow and upper bounds of the defect in H ?2 (); we determine an upper bound for the norm of the inverse of the linearization by a perturbation argument; upper bounds for some embedding constants yield a Lipschitz constant. For small Reynolds numbers we apply our method to the driven-cavity problem. We consider solutions (u; p) 2 V L 2 () of the Navier-Stokes equation ?u + Re(u r)u + rp = f; where V = fv 2 H 1 0 (; R 2) j r v = 0g, Re denotes the Reynolds number and R 2 is a bounded Lipschitz domain. We explain a method to determine error bounds kr(u ? ~ u)k L 2 () r for some numerical approximation ~ u 2 V : We have to compute upper bounds for the defect. Therefore, we use the stream function { vorticity formulation of the Navier-Stokes equation (cf. 1]). This guarantees that the approximation is in V. Furthermore, the defect can be calculated without determining the pressure (cf. sec. 2). We need an estimate for the norm of the inverse of the linearization at ~ u. Up to now, we can only handle the case where the Reynolds number is suuciently small. Then we consider the linearization as a perturbation of the Stokes operator.