Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

Abstract

In this paper, we present a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for solving a linear stochastic parabolic partial differential equation driven by additive noises. When the diffusion coefficient in the stochastic parabolic partial differential equation varies in multiple scales, it is very challenging to resolve all scales for the model using traditional finite element methods. To overcome the difficulty, we use the CEM-GMsFEM to solve the stochastic multiscale parabolic differential equation and construct a coarse computational model. For numerical computation, the infinite dimensional additive noise is approximated by a finite dimensional noise. Convergence analysis is carried out for semi-discretization and full discretization. The convergence rate is characterized by the coarse grid size, eigenvalue decay of local spectral problems and the stochastic noise approximation. A few numerical results for the stochastic parabolic equation driven by different noises are presented to confirm the theoretic analysis and show the computational performance of the approach.

Introduction

Stochastic partial differential equations (SPDEs) have recently gained much attention for their ability of fully describing intrinsic uncertainty in complex phenomena from sciences and engineering. SPDEs have been used for various applications, such as the approximative pricing of interest-rate based financial derivatives [1], nonlinear filtering [2], population biology [3], turbulence [4], and climate prediction [5]. It is rarely obtained the analytic solutions to SPDEs, so numerical methods have been developed to solve SPDEs [6], [7], [8], [9], [10], [11], [12].

Many media and materials in science and engineering have a multiple scale nature. At the same time, they are heterogeneous at a certain scale. Typical examples are human bones, composite materials, porous media, and other engineering materials. It is a great challenge to resolve all scales in the models using traditional numerical methods in a fine scale. There have been many existing approaches in the literatures to handle multiscale problems. One of these approaches is based on averaging idea and computing effective properties, such as homogenization approaches [13], numerical upscaling methods [14], [15]. Another approach is to construct multiscale basis functions on a coarse grid, such as multiscale finite element method (MsFEM) [16], [17], [18], generalized multiscale finite element method (GMsFEM) [19], [20], and constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) [21], [22] and so on. The multiscale basis functions solve some local problems, and the multiscale information is embedded in the multiscale basis functions. Thus a coarse computational model is built through a variation form and the multiscale basis functions. This can significantly reduce the computation complexity of the original fine scale model. In this paper, we will use the CEM-GMsFEM to construct multiscale basis functions. The CEM-GMsFEM shares the framework of the GMsFEM. To construct multiscale basis functions, a constraint energy minimization is imposed on auxiliary basis functions via local spectral problems. Because the basis functions have a fast decay outside the target coarse block, we can compute the basis functions in a local region.

Stochastic partial differential equations (SPDEs) with multiscale structures can be used to model accurately multiscale dynamical systems affected by noises. The purpose of this paper is to present a numerical multiscale method and solve SPDEs efficiently. We consider linear stochastic parabolic partial differential equations driven by additive noises with multiscale diffusion coefficients, and carry out the convergence analysis for the CEM-GMsFEM. The main idea of the CEM-GMsFEM is to use the multiscale basis functions which can be accurately computed to capture complex multiscale features. The challenge in the convergence analysis of multiscale finite element approximation for the SPDEs is the lack of regularity of white noises. For instance, as shown in [23], the required regularity conditions of the SPDEs solution are not satisfied for the standard error estimates of finite element methods. To overcome this difficulty, Allen, Novosel, and Zhang [23] have given a piecewise constant approximation of the white noises, but the regularity of white noises are not improved much due to the nature of standard Wiener process. In order to obtain “a more regular” noises and improve the regularity of the solution, Du and Zhang [24] represent noises term as a linear combination of Fourier modes. Following this idea to approximate white noises, the approximation solutions of SPDEs converge to the exact solutions as the white noise approximation becomes finer. We have proved that the CEM-GMsFEM achieves the first order convergence with respect to coarse grid size in $L^2$-norm under some mild assumption on the SPDEs. The stability analysis and convergence analysis are also presented for fully discretizing spatial space, temporal space and white noises.

This paper is organized as follows: In Section 2, we present the white noise approximation for a stochastic parabolic equation. The error estimate between the exact solution and approximate solution is established. In Section 3, the CEM-GMsFEM is briefly introduced. In Section 4, we present a convergence analysis for the SPDEs using the CEM-GMsFEM and discretized white noises. In Section 5, the full discrete scheme is presented. In Section 6, a few numerical results are presented to confirm the error analysis. Finally, some conclusions are given.