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In this paper, we focus on developing a high efficient algorithm for solving d -dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order O ( 2 - J J ) with O ( 2 J J ) degrees of freedom (DOF) for d = 1 and O ( 2 Jd ) DOF for d > 1 , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment., Competing Interests: Competing interestsThe authors declare no competing interests., (© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023, Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.)

In this paper, we consider the sample average approximation (SAA) approach for a class of stochastic nonlinear complementarity problems (SNCPs) and study the corresponding convergence properties. We first investigate the convergence of the SAA counterparts of two-stage SNCPs when the first-stage problem is continuously differentiable and the second-stage problem is locally Lipschitz continuous. After that, we extend the convergence results to a class of multistage SNCPs whose decision variable of each stage is influenced only by the decision variables of adjacent stages. Finally, some preliminary numerical tests are presented to illustrate the convergence results., (© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.)

In this paper, a finite element analysis to approximate the solution of an obstacle problem for a static shallow shell confined in a half space is presented. To begin with, we establish, by relying on the properties of enriching operators, an estimate for the approximate bilinear form associated with the problem under consideration. Then, we conduct an error analysis and we prove the convergence of the proposed numerical scheme., (© The Author(s) 2020.)