Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier–Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a penalty parameter \(\varepsilon \) $\epsilon$ . Space and time are discretized through a finite element approximation and an Euler method. The convergence analysis of the suggested numerical scheme is investigated throughout this paper. It is based on a local monotonicity property permitting the convergence toward the unique strong solution of the stochastic Navier–Stokes equations to occur within the originally introduced probability space.