椭圆型偏微分方程的三角形单元有限元的数值解法

Triangle finite element numerical solution of elliptic partial differential equations

本文以椭圆型偏微分方程问题为背景研究三角形单元的有限元方法.随着计算机图形学的发展,三角形单元划分取得了巨大的成就,可以得到质量非常好的三角形单元,进而提高偏微分方程的有限元方法的数值解的精度.本文采用节点增量算法,对问题区域进行三角形单元划分,得到的三角形单元满足Delaunay条件,再对三角形单元的所有节点采用自适应编号,最后运用三角形单元的有限元方法得到椭圆型偏微分方程的数值解.通过数值实验,得出相比传统的三角形单元的有限元方法,本文的三角形单元的有限元方法减小了舍入误差,提高了计算精度.

In the present work,the finite element method for triangular element is studied based on the elliptic partial differential equation.With the development of computer graphics,the triangle element division has made great achievements.High-quality triangular elements are able to be generated,which improved the accuracy of numerical solutions for partial differential equations.Here,the triangular element satisfying the Delaunay conditions is calculated through unit division of the region with the node incremental algorithm.All nodes of the triangular element are adopted with adaptive numbers and the numerical solution of elliptic partial differential equations is obtained utilizing the finite element method of triangular elements.Finally,through numerical experiments,the finite element method is compared with the traditional one.Results indicate that the finite element method of triangular element reduces the rounding error and elevates the calculation precision.