新型时空高精度稳定化有限元格式构造与分析

The space-time high order stabilized finite element methods in Galerkin or Petro-Galerkin form, based on the techniques of combining the variational form in time and stabilized term in space, are studied. The discussed schemes include the forms which are continuous in both space and time, or discontinuous in time but continuous in space. This kind of approximate formulas not only has high order accuracy in both space and time directions, but also can control nonphysical oscillations. The new numerical schemes constructed here has some good properties, such as stability, efficiency, and high resolution. In this project, some new type of space-time stabilized finite element methods, such as space-time least square stabilized method, space-time local projection stabilized methods and so on, are proposed. The theoretical analysis, including the existence and uniqueness, prior error estimates, super convergence, posterior error estimates and so on, will be proved. And based on the theoretical results, the problems of how to choose the optimized stabilized parameter are studied. The numerical simulations of some classical model equations are given to illustrate the reliability and efficiency of the new schemes constructed in this projection. In the theoretical analyses and numerical simulations, interpolations at special points, Radau, Gauss or Lobatto points etc.,are introduced in time direction. The decoupling of the space and time variables is obtained by the technique of combining the interpolation and finite element method. The decoupling techniques overcomes the disadvantages of the space-time schemes, that is the dimension of the approximate solutions increases when the schemes are constructed by discretizing in both space and time variables simultaneously. The decoupling method not only simplify the theoretical analysis and algorithm, but also save computational cost, improve operability. The projection itself has important theoretical value and practical application prospect.

将时间变分离散和空间稳定化技巧相结合，构造时间和空间都连续，或时间允许间断而空间连续的Galerkin型和Petro-Galerkin型时空高精度稳定化有限元格式，此类格式不但同时具有时间和空间方向的高精度，而且能够控制非物理振荡，是一种新型稳定、有效、高分辨率数值方法。项目拟构造时空最小二乘、局部投影等新型时空高精度稳定化有限元方法，证明近似解的存在唯一性、先验误差、超收敛性和后验误差等，在此基础上研究时空区域上稳定化最优参数的选择，结合典型模型方程的数值模拟，验证所构造新型格式的可靠性和有效性。在理论分析和数值模拟中，时间方向引入Radau点、Gauss点或Lobatto点等特殊节点处的插值，和有限元方法相结合，实现时空变量的解耦，克服时、空方向都用有限元离散而增加近似解维数的不足，不但理论分析简单可行，而且简化算法，节省计算量，可操作性强。项目本身具有重要的理论价值和实际应用前景。

本项目主要构造并分析新型时空高精度稳定化有限元方法，同时研究其他相关稳定数值方法。在数值格式构造方面，将时间变分离散和空间稳定化技巧相结合，构造时间允许间断而空间连续的Galerkin型和Petro-Galerkin型时空有限元格式、局部投影稳定化时空有限元方法、流线扩散Petro-Galerkin稳定化时空元方法等；同时，针对不同方程类型构造了时间和空间都连续的时空Galerkin有限元法、H1-Galerkin时空混合有限元分裂格式、混合时空有限元方法、时间两网格方法、时空谱方法、有限体积元方法等多种数值格式；研究的方程包括电报方程、Sobolev方程、Sine-Gordon方程、正则长波Burgers方程、具有变系数的伪双曲方程、Cable方程、对流扩散方程、分数阶和分布阶方程等。证明了解的存在唯一性、稳定性、不同范数意义下的先验误差估计等。利用时间方向引入Radau点、Gauss点或Lobatto点等特殊节点处的插值的技巧，实现时空变量的解耦，克服时、空方向都用有限元离散而增加近似解维数的不足，使理论分析简单可行，并节省计算量。项目针对所构造的数值格式，给出典型模型方程的数值模拟，验证所构造新型格式的可靠性和有效性。. 项目执行期间，发表标注该基金资助的论文49篇，出版学术专著1部，培养博士毕业生4人，硕士毕业生9人。圆满完成了项目的研究目标。

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