Efficient Numerical Methods For Mechanics With Randomness And Singularities

In many theoretical and engineering problems of mechanics,randomness and singularities are two of the most common properties.The emergence of randomness and singularities will bring both theoretical and numerical challenges to the study of mechanics.The direct application of traditional numerical methods would face problems like serious computational burdens and slow convergence of numerical solutions,etc.Therefore,it’s important to design efficient numerical methods with nice theoretical properties,based on the features of different problems.We first study the Schr ¨odinger equation with random potentials.The application of the stochastic collocation method on this equation is investigated.Since the stochastic collocation method is based on the theory of polynomial interpolation,the convergence w.r.t.the number of collocation points is related to the stochastic regularity of the solution.An analysis of the stochastic regularity for the Schr ¨odinger equation in the whole space is given,based on the well-posedness result in the deterministic case.Combining the time-splitting spectral method and the stochastic collocation method,we propose the time-splitting based stochastic collocation method for the Schr ¨odinger equation with random inputs on a bounded interval and present the corresponding error estimate.Both the theoretical analysis and numerical results show that the smoother the potential and initial data are w.r.t.the random variable,the faster the convergence of the time-splitting based stochastic collocation method is.We then study the Schr ¨odinger equation with a periodic potential and a random external potential,presenting the Bloch decomposition based stochastic Galerkin method by combining the Bloch decomposition based time splitting pseudospectral method and the stochastic Galerkin method.We analyse the stability and local temporal error of the Bloch decomposition based time splitting pseudospectral method and prove that the Bloch decomposition based stochastic Galerkin method conserves the discrete mass in the sense of expectation.Numerical results show that the Bloch decomposition based stochastic Galerkin method maintains the advantage of the Bloch decomposition based time splitting pseudospectral method,being able to obtain a high-resolution solution under a relatively large temporal step and possessing the property of spectral convergence in the spatial direction.In addition,when the random external potential is smoothly dependent on the random variable,this method obtains the spectral convergence w.r.t.the order of orthogonal polynomials.Moreover,besides the discrete mass,this method approximately conserves the discrete energy in the sense of expectation.We finally generalize the direct method of lines for solving elliptic problems with singularities.We first deduce the variational-differential form of the anisotropic Laplace equation in the curvilinear coordinate and obtain the direct method of lines for the Laplace equation in the anisotropic case;and then we discuss how the direct method of lines should cope with Neumann boundary conditions as exterior boundary conditions for problems in a general star-shaped domain;and subsequently the application of the direct method of lines on the Poisson equation is deduced;and by combining the idea of domain decomposition,the direct method of lines for elliptic problems with multiple singular points is proposed in the end.Numerical results show that the generalized direct method of lines can effectively solve these kinds of singular elliptic problems and maintain the advantages of the original method.