Lewis John Gregg, Rehm Ronald G
National Bureau of Standards, Washington, D.C. 20234.
J Res Natl Bur Stand (1977). 1980 Sep-Oct;85(5):367-390. doi: 10.6028/jres.085.022.
In this report the combination of an iterative technique, the conjugate gradient algorithm, with a fast direct method, cyclic reduction, is used to solve the linear algebraic equations resulting from discretization of a nonseparable elliptic partial differential equation. An expository discussion of the conjugate gradient and preconditioned conjugate gradient algorithms and of their use in the solution of partial differential equations is presented. New results extending the use of the preconditioned conjugate gradients technique to singular linear equations which arise from discretized elliptic equations with Neumann boundary conditions are also given. The algorithms are applied to solve a specific elliptic equation which arises in the study of buoyant convection produced by a room fire. A code was developed to implement the algorithms for this application. Numerical results obtained through testing and use of the code are discussed.
在本报告中，迭代技术共轭梯度算法与快速直接方法循环约化相结合，用于求解由不可分离椭圆型偏微分方程离散化产生的线性代数方程。文中对共轭梯度算法和预处理共轭梯度算法以及它们在偏微分方程求解中的应用进行了说明性讨论。还给出了将预处理共轭梯度技术的应用扩展到具有诺伊曼边界条件的离散椭圆方程所产生的奇异线性方程的新结果。这些算法被应用于求解一个在室内火灾产生的浮力对流研究中出现的特定椭圆方程。开发了一个代码来实现此应用的算法。讨论了通过测试和使用该代码获得的数值结果。
