22 research outputs found
Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise
The stochastic Allen-Cahn equation with multiplicative noise involves the
nonlinear drift operator . We use the fact that
satisfies a weak monotonicity property to deduce uniform bounds in strong norms
for solutions of the temporal, as well as of the spatio-temporal discretization
of the problem. This weak monotonicity property then allows for the estimate for all
small , where is the strong variational solution of the
stochastic Allen-Cahn equation, while solves a
structure preserving finite element based space-time discretization of the
problem on a temporal mesh of size which
covers
A splitting semi-implicit method for stochastic incompressible Euler equations on
The main difficulty in studying numerical method for stochastic evolution
equations (SEEs) lies in the treatment of the time discretization (J. Printems.
[ESAIM Math. Model. Numer. Anal. (2001)]). Although fruitful results on
numerical approximations for SEEs have been developed, as far as we know, none
of them include that of stochastic incompressible Euler equations. To bridge
this gap, this paper proposes and analyses a splitting semi-implicit method in
temporal direction for stochastic incompressible Euler equations on torus
driven by an additive noise. By a Galerkin approximation and the
fixed point technique, we establish the unique solvability of the proposed
method. Based on the regularity estimates of both exact and numerical
solutions, we measure the error in and show that the
pathwise convergence order is nearly and the convergence order in
probability is almost