On Wavelet-Galerkin methods for Semilinear Parabolic Equations with Additive Noise
For researchers in numerical analysis of stochastic PDEs, this work offers a novel numerical scheme with error bounds, though it is incremental in nature.
The paper develops a wavelet-Galerkin method for semilinear stochastic heat equations with additive noise, combining non-adaptive and adaptive wavelet discretizations, and provides mean square error estimates.
We consider the semilinear stochastic heat equation perturbed by additive noise. After time-discretization by Euler's method the equation is split into a linear stochastic equation and a non-linear random evolution equation. The linear stochastic equation is discretized in space by a non-adaptive wavelet-Galerkin method. This equation is solved first and its solution is substituted into the nonlinear random evolution equation, which is solved by an adaptive wavelet method. We provide mean square estimates for the overall error.