Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise

Abstract

We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov assumptions, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the $L_{\omega }^p{L}_t^{\infty }{\dot{H}}^{1+\gamma }$-norm and a temporal Hölder regularity under the $L_{\omega }^p{L}_x^2$-norm for the solution of the proposed equation with an ${\dot{H}}^{1+\gamma }$-valued initial datum for $\gamma \in [0,1]$. Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $O(h^{1+\gamma }+{\tau }^{1/2})$ and $O(h^{1+\gamma }+{\tau }^{(1+\gamma )/2})$ for the Galerkin-based Euler and Milstein schemes, respectively.

References in corpus (7)

• Strong Approximation of Stochastic Allen-Cahn Equation with White Noise
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Cited by in corpus (4)

• Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise
• Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise
• On the Itô-Alekseev-Gröbner formula for stochastic differential equations
• Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise