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

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 methods.

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。

本项目致力于对分数阶偏微分方程设计出长时间收敛和稳定的高精度数值算法，并分析格式的收敛性，验证其高阶精度特点。经过一年的努力，基本完成了项目的预期成果，取得成果概述如下：首先，对时间分数阶扩散方程构造了高精度的局部间断有限元方法。在时间方向上用有限差分离散，空间方向上间断有限元方法离散，构造一种隐式全离散局部间断有限元方法，并给出格式的误差估计和稳定性结果。其次， 对时间分数阶KdV方程，设计无条件稳定的全离散间断有限元格式，通过构造特殊的全局投影，证明当使用交错数值流通量时，局部间断有限元解的收敛性，并在数值上进行验证。另外， 对时间分数阶四阶问题的全离散的局部间断有限元方法进行分析，给出误差估计和收敛性结果。