Convergence of a spectral method for the stochastic incompressible Euler equations

Abstract

We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by a multiplicative noise. We show that the SVM solution converges to a dissipative measure-valued martingale solution of the underlying problem. These solutions are weak in the probabilistic sense i.e. the probability space and the driving Wiener process are an integral part of the solution. We also exhibit a weak (measure-valued)-strong uniqueness principle. Moreover, we establish strong convergence of approximate solutions to the regular solution of the limit system at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system.