High-order Numerical Approximations To Stochastic Partial Differential Equations

Thanks to the great progress in the development of numerical methods and computer resources, many classical partial differential equations can now be solved very efficiently with high accuracy. However, in many cases, the coefficients, boundary conditions as well as the geometry of the considered partial differential equation may contain uncertainties. In order to provide meaningful predictions to problems involving uncertain data, there is a need to investigate efficient numerical methods for handling general stochastic partial differential equations. The spectral element method, which combines the high-order precision of spectral methods and the geometrical flexibility of finite element methods, has achieved great success and is becoming a useful tool for investigation of such equations.The purpose of this work is to numerically solve differential equations with uncertain input data by using stochastic Galerkin spectral methods. Some stochastic Galerkin methods, combining the Askey polynomial chaos expansion for the stochastic inputs and the spectral method in the physical space, are proposed to approximate some popular model problems. Precisely, the content of the paper is as follows:firstly, the influence of the stochastic input on the numerical solutions and the accuracy of the numerical method are discussed by performing a numerical investigation for the stochastic ordinary differential equations.secondly, we propose a stochastic Galerkin spectral method for the steady/unsteady stochastic diffusion problems. The detailed implementation is presented, and its efficiency is tested by some numerical examples.thirdly, as a main part of this work, we propose and analyze a spectral method to solve the Stokes equations with random coefficients. A stochastic Galerkin approach is used to reduce the original stochastic Stokes equations into a set of deterministic equations for the expansion coefficients. Then a P_N×P_N-2 spectral method, together with a block Jacobi iteration is applied to solve the resulting problem. We establish the well-posedness of the weak formulation and its discrete counterpart. Moreover, we provide a rigorous convergence analysis and numerical validation.finally, we investigate the parameter optimization problem via the variational adjoint assimilations in the frame of the spectral approximation of a heat model. Precisely, we consider the optimization of the initial condition for the heat equation by means of the discrete adjoint assimilations. Spectral methods and the classical Crank-Nicolson scheme are applied to deduce the full discrete system for the continuous model, then the variational adjoint assimilation approach is used to derive the gradient and the corresponding adjoint system. In Particular, a method for choosing optimal step sizes is presented.