高阶分数阶偏微分方程的全离散局部间断有限元方法研究

项目名称： 高阶分数阶偏微分方程的全离散局部间断有限元方法研究

项目编号： No.11426090

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 韦雷雷

作者单位： 河南工业大学

项目金额： 3万元

中文摘要： 近年来，随着分数阶偏微分方程在越来越多的领域中得到应用，已经引起了国内外越来越多的学者及工程技术人员的兴趣和重视。对这类方程的数值解法进行研究有着重要的理论和实践意义。目前对于含有高阶空间导数的分数阶偏微分方程数值方法方面的研究非常有限。本项目致力于研究几类高阶分数阶偏微分方程的局部间断有限元方法的超收敛性和误差估计。超收敛性能够有效地保证数值解与真解的误差在很长一段时间内不会增长，尤其当网格很密时，该性质体现的更为明显，为数值解的长时间形态提供了坚实的理论依据。该项目的研究结果能够显示间断有限元方法用于求解此类方程的有效性和优越性，同时进一步丰富间断有限元方法的超收敛性理论。

中文关键词： 分数阶偏微分方程；稳定性；收敛性；；

英文摘要： Fractional partial differential equations have attracted much interest and attention of more and more domestic and international scholars and engineers with this kind of equations have been applied in more and more fields in recent years. The numerical method for such problems is very important in the theory and practice. At present the research on the numerical methods for fractional partial differential equations with higher derivatives is very limited. The project studies superconvergence property and error estimates of fully discrete discontinuous Galerkin methods for solving a class of fractional partial differential equations with higher derivatives. Superconvergence is a effective way to improve the convergence rate and solve the high-dimensional problems. An important motivation for investigating such superconvergence is to lay a solid theoretical foundation for the fact that the error between the discontinuous Galerkin solution and the exact solution does not grow over a long time period. This property is especially prominent for fine meshes,and provides a solid theoretical basis for making numerical simulation for a long time. The results will show that the methods has a unique advantage to solve this kind of equations, which will further strengthen the convergence theory of discontinuous Galerkin met

英文关键词： Fractional partial differential equations；Stability；Convergence；；