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1.
We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results. This research is supported by Air Force Office of Scientific Research under the grant number FA9550-05-1-0133 and 985 Project of Jilin University.  相似文献
2.
A general framework is proposed for what we call the sensitivity derivative Monte Carlo (SDMC) solution of optimal control problems with a stochastic parameter. This method employs the residual in the first-order Taylor series expansion of the cost functional in terms of the stochastic parameter rather than the cost functional itself. A rigorous estimate is derived for the variance of the residual, and it is verified by numerical experiments involving the generalized steady-state Burgers equation with a stochastic coefficient of viscosity. Specifically, the numerical results show that for a given number of samples, the present method yields an order of magnitude higher accuracy than a conventional Monte Carlo method. In other words, the proposed variance reduction method based on sensitivity derivatives is shown to accelerate convergence of the Monte Carlo method. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the proposed method is a fraction of the total time of the Monte Carlo method.  相似文献
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4.
In this paper, we address the finite element method and discontinuous Galerkin method for the stochastic scattering problem of Helmholtz type in ℝ d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided to examine our theoretical results. The first author is supported by NSF under grand number 0609918 and AFOSR under grant numbers FA9550-06-1-0234 and FA9550-07-1-0154, the second author is supported by NSFC(10671082, 10626026, 10471054), and the third author is supported by NNSF (No. 10701039 of China) and 985 program of Jilin University.  相似文献
5.
In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.  相似文献
6.
The goal of this paper is to construct an efficient numerical algorithm for computing the coefficient matrix and the right hand side of the linear system resulting from the spectral Galerkin approximation of a stochastic elliptic partial differential equation. We establish that the proposed algorithm achieves an exponential convergence with requiring only O\((n\log _{2}^{d+1}n)\) number of arithmetic operations, where n is the highest degree of the one dimensional orthogonal polynomial used in the algorithm, d+1 is the number of terms in the finite Karhunen–Loéve (K-L) expansion. Numerical experiments confirm the theoretical estimates of the proposed algorithm and demonstrate its computational efficiency.  相似文献
7.
We consider a system of partial differential equations which models flows through elastic porous media. This system consists of an elasticity equation describing the displacement of an elastic porous matrix and a quasilinear elliptic equation describing the pressure of the saturating fluid (flowing through its pores). In this model, the permeability depends nonlinearly on the dilatation (divergence of the displacement) of the medium. We show that the solution has regularity. We describe the numerical approximation of solutions using a hybrid finite element‐least squares mixed finite element method. Error estimates are obtained through the introduction of an auxiliary linear elasticity equation. Numerical experiments verify the error estimates and validate the proposed poroelasticity model. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1174–1189, 2015  相似文献
8.
Domain decomposition methods for solving the coupled Stokes–Darcy system with the Beavers–Joseph interface condition are proposed and analyzed. Robin boundary conditions are used to decouple the Stokes and Darcy parts of the system. Then, parallel and serial domain decomposition methods are constructed based on the two decoupled sub-problems. Convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence.  相似文献
9.
We consider the continuum Darcy/pipe flow model for flows in a porous matrix containing embedded conduits; such coupled flows are present in, e.g., karst aquifers. The mathematical well‐posedness of the coupled problem as well as convergence rates of finite element approximation are established in the two‐dimensional case. Computational results are also provided. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1242–1252, 2011  相似文献
10.
Numerical solutions of the stochastic Stokes equations driven by white noise perturbed forcing terms using finite element methods are considered. The discretization of the white noise and finite element approximation algorithms are studied. The rate of convergence of the finite element approximations is proved to be almost first order (h|ln h|) in two dimensions and one half order ( h\frac12h^{\frac{1}{2}}) in three dimensions. Numerical results using the algorithms developed are also presented.  相似文献
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