Finite Element Method and Natural Boundary Reduction
Finite Element Method and Natural Boundary Reduction
复制标题
DOI：
--
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Feng KASOc;Feng Eang
中科院分区：
文献类型：
--
作者：
Feng KASOc;Feng Eang
近二十年来，偏微分方程数值方法的主要进展之一是有限元法。该方法是基于椭圆型方程的变分公式和三角近似。第一个组成部分，变分原理，是一个古老的，导致经典的BayleighBitz方法，虽然在过去是成功的，但遭受数值不稳定性和几何不稳定性，起源于所采用的解析近似，但在计算机时代之前由于有限的大小和复杂性的问题，然后攻击被忽视。第二个组成部分，三角形的局部近似，使用，但没有充分利用有限差分方法，是更基本和更古老。追溯到古代，它在很长一段时间内被后来的解析近似成就所掩盖，但最终由于其固有的稳定性和灵活性而复兴，这在计算机时代变得重要。一个明智的组合的两个旧的组件，传统上在并列，引起了有限元法，创新的普遍适用性，特别适合于问题的巨大复杂性，以及计算机的使用。在FEM中，椭圆算子的所有基本性质，例如，在离散化之后，对称性、连续性和局部性得到了很好的保持。这导致，一方面，到一个有效的计算方案，另一方面，到一个健全的理论基础，在此基础上，椭圆方程的Sobolev空间理论被调用在一个自然的方式，确保该方法在实践中的可靠性。而且有限元法的逻辑简单、直观，易于在计算机上实现，
One of the major advances in numerical methods for partial differential equations made in the recent twenty years is the finite element method (FEM). The method is based on the variational formulation of elliptic equations and on the triangulated approximations. The first component, the variational principle, is an old one ajid leads to the classical BayleighBitz method, which, though successful in the past, suffers from numerical instability and geometric inflexibility, originating from the analytic approximations adopted, but unnoticed in the pre-computer times due to the limited size and complexity of the problems then attacked. The second component, the triangulated local approximations, used but not exploited in full in the finite difference methods, is more elementary and much older. Dating back to ancient times, it was for a long time overshadowed by the later achievements in analytic approximations, but revived eventually due to its innate stability and flexibility, which becomes important in the computer era. A judicious combination of the two old components, conventionally in juxtaposition, gives rise to the FEM, an innovation of general applicability, especially suited for problems of great complexity as well as for computer usage. In FEM, all the essential properties of elliptic operators, e.g., symmetry, coerciveness and locality are well preserved after discretization. This leads, on the one hand, to an efficient computational scheme and, on the other hand, to a sound theoretical foundation, on which the Sobolev space theory of elliptic equations is invoked in a natural way, ensuring the reliability of the method in practice. Moreover, the logic of FEM is simple, intuitive and easy to be implemented on the computer,