Numerical Approximation of Optimal Convergence for Fractional Elliptic Equations with Additive Fractional Gaussian Noise

Abstract

We study numerical approximation for one-dimensional stochastic elliptic equations with integral fractional Laplacian and the additive Gaussian noise of power-law: $1/f^\beta$ noise and fractional Brownian noise. We present an optimal convergence of our method using spectral expansions of noises. We first establish the well-posedness of a corresponding deterministic problem and show the stability of solutions for the rough data via negative norms in weighted Sobolev spaces. We also analyze the regularity of the noise and approximation properties of their finite truncations. Next, we show the optimal error estimates of our method for a wide range of parameters in the order of fractional operator and the fractional Gaussian noise. Finally, we present several numerical examples to illustrate the mean-square convergence orders and verify our optimal convergence rates.