近年来,发展空间分数阶和时间分数阶偏微分方程的高效数值算法是计算数学领域的一个热点问题.已有的数值算法存在一些的局限性,比如对求解几何区域的适应性很差,数值精度偏低,存储量大,计算耗时等.如何进一步提高计算精度,加快计算速度仍需进行深入研究.本项目拟采用加权平均的技巧来获得L1逼近公式的超收敛点,以及高次插值逼近等方法,提高时间分数阶导数的离散精度.利用积分恒等式技巧,加权范数,平均值方法以及导数转移等技巧提高空间方向的逼近精度,从而建立线性/非线性时空分数阶偏微分方程的高精度有限元数值格式,并探讨算法长时间运行的稳定性, 收敛性和超收敛分析.同时,通过FFT、预处理等技巧,对所得到的时空高精度有限元算法进行加速计算,从而得到求解分数阶偏微分方程的高精度快速算法．

In recent years, developing effective numerical algorithms for solving the space/time fractional partial differential equations is one of the hot issues in the field of computational mathematics.However, the existing numerical algorithms have some limitations; such as poor adaptability on complex geometric domain, low-order accuracy, large memory cost, time-consuming computation and so on.How to improve calculation accuracy and expedite calculation speed still needs further study.Therefore, this project mainly aims at developing the fast finite element algorithms with high-order accuracy for the fractional partial differential equations. With the help of the techniques of using the weighted average for the L1 approximation formula of superconvergence points and high order interpolation approximation, we improve the time fractional order derivative of discrete accuracy.Using the integral identities techniques, weighted norm, average method and derivative transfer techniques improve the accuracy of spatial direction, we establish finite element numerical algorithms with high-order accuracy for solving the space/time linear/nonlinear fractional partial differential equations.The stability of the algorithm for a long time running, convergence and superconvergence will also be studied. Meanwhile, some fast algorithms will be investigated by using the FFT, pre-processing for the obtained time and space high accuracy finite element algorithms in order to achieve high-order accuracy methods.