分数阶偏微分方程的有限元高精度快速算法研究

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.

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．

首先，对时间分数阶慢扩散方程和扩散波方程构造高精度的有限元快速算法。利用积分恒等式技巧，证明了格式的无条件稳定性和超收敛性。为了降低算法的存储量和计算时间，我们采用SOE快算法逼近L1公式，构造快速的有限元数值格式，并给出误差分析。最后通过数值算例验证了算法的精度，并且数值例子说明SOE快速算法在减少计算时间和降低存储量上是有效的。对非线性时间分数阶次扩散方程构造了几个线性化的有限元数值格式，利用L1型离散的Gronwall不等式，严格地证明了格式的无条件收敛性。采用SOE快速算法逼近L1公式，构造快速的有限元数值格式，并给出误差分析。最后，研究了带有初始奇异性的多项时间分数阶扩散方程的一种全离散数值方法。基于$L1$公式在graded 网格下离散多项Caputo时间分数阶导数，构造了多项时间分数阶扩散方程的时间半离散格式，证明了时间格式通过选取合适的网格参数$r$，时间方向的误差可以达到最优的收敛阶$2-\alpha_1$，其中$\alpha_1$$(0<\alpha_1<1)$为多项时间分数阶导数阶数的最大值。然后，空间采用谱方法进行离散，得到了全离散格式，证明了全离散格式的无条件稳定性和收敛性。为了降低计算量和储存量，对多项时间分数阶扩散方程又构造了时间方向的快速算法，同时证明了该格式的收敛性。最后，数值算例验证了算法的有效性,通过算例进一步说明了快速算法的高效性。