奇异摄动问题各向异性自适应有限元

Singularly perturbed problems have wide applications. Because they contain a small perturbed parameter, these problems have singularities, not only may have singular corners, but also may have inner and out boundary layers，the solutions are anisotropic. Using usual finite element methods often do not obtain convergences uniformly for the perturbed parameter. In this item we study singularly perturbed problems, especially reaction-convection-diffusion equations, Darcy-Stokes equation, 4-order singularly perturbed equation, Cahn-Hilliard equations, find constructing methods and convergence mechanism of the finite elements on anisotropic meshes which are convergent uniformly for perturbed parameter, and construct effective elements; study how to construct anisotropic a posteriori error estimators, containing explicit and implicit ones for singularly perturbed problems. These estimators improve existed results, can find not only the domains which need mesh fined, but also direction which need fined, in order that the anisotropic meshes fit better the singularities of singularly perturbed problems and make computing less. These researches are founded very little in existed papers. Then we do numerical experiments to check effectiveness of the constructed elements and anisotropic a posteriori error estimators, improve constructive method of the elements and anisotropic a posteriori error estimators according to the numerical experiments.

奇异摄动问题有广泛的实际应用背景，由于小参数的存在，此类问题有奇性，不仅可能有点奇性，而且可能有内、外边界层奇性，解呈现各向异性特征，用通常的有限元方法求解往往不能对摄动参数一致收敛。本项目研究奇异摄动问题，重点是反应-对流-扩散方程，Darcy-Stokes方程，四阶奇异摄动方程，Cahn-Hilliard方程的在各向异性网格上对摄动参数一致收敛单元的构造方法和收敛机理，构造出有效单元，这方面的研究还很少。研究如何针对奇异摄动问题构造各向异性后验误差估计子，包括显式的和隐式的，这些后验误差估计子突破已有的框架，使得不仅能检验出需要加密的区域，而且能检验出局部需要加密的方向，以便使各向异性网格更能适应奇异摄动问题的奇性，这在以往的研究中很少见到的。对由此构造出的单元和各向异性后验误差估计子进行数值试验，检验理论分析的有效性，并由数值试验结果反过来改进单元和各向异性后验误差估计子的构造方式。

奇异摄动问题有广泛的实际应用背景。由于小参数的存在，此类问题有奇性，不仅可能有点奇性，而且可能有内、外边界层奇性，解呈现各向异性特征。用通常的有限元方法求解往往不能对摄动参数一致收敛。在单元构造方面，本项目针对四阶奇异摄动问题和Darcy-Stokes问题，构造出具有一致收敛性的非协调单元；针对线弹性问题，构造了一组协调的矩形单元和一组协调的立方体单元；针对板弯曲问题，构造了一个C0连续的非协调虚拟单元和非C0连续的Morley-型虚拟单元；在离散格式设计方面，本项目针对线弹性问题，提出一个无闭锁稳定化有限元方法和一个低阶混合元的稳定化方法；针对多孔介质中的non-Fickian流，利用非协调Wilson元构造了一个半离散格式和一个全离散格式；在后验误差估计方面，针对Poisson问题的线性单元，推导了有保证的各向同性后验误差估计；针对二阶椭圆问题的混合元新格式，推导了各向异性残量型后验误差估计；针对扩散问题和奇异摄动反应扩散问题的协调逼近，推导了各向异性后验误差估计；针对对流-扩散问题的各种协调或非协调有限元方法，推导了各向同性残量型后验误差估计；在自适应计算方面，基于所得的各向异性后验误差估计，推导出相应的各向异性误差指示子，从而用于引导网格的各向异性自适应加密，最终实现了各向异性自适应计算。