The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is frequently pointed out as one of the bottlenecks in the computations. The second bottleneck being the large and numerous linear systems to be solved arising from the use of Newton's method to solve nonlinear systems of equations. In this paper, we will address the first issue. We will see how under mild assumptions the assemblage procedure may be rewritten using a completely loop-free algorithm. Our approach leads to a small matrix-matrix multiplication for which we may count on highly optimized algorithms.
翻译:有限元素法是线性和非线性部分差异方程(PDEs)数字解决方案的既定方法,但非线性PDE的有限元素矩阵的反复重新评估经常被指出为计算中的瓶颈之一。第二个瓶颈是因使用牛顿法解决非线性方程系统而需要解决的庞大和众多线性系统。在本文件中,我们将讨论第一个问题。我们将看到,在轻度假设下,如何使用完全无环的算法重写组合程序。我们的方法导致一个小矩阵矩阵矩阵法的倍增,我们可以依靠高度优化的算法。