扩散反应方程的一种协调间断有限元法

A conforming discontinuous finite element method for diffusion-reaction equations

对于定常扩散反应方程的原始混合变分形式,本文基于间断有限元法和最小二乘法思想给出了一种新的协调间断有限元格式,并将其推广应用于非定常扩散反应方程,进而建立了全离散协调间断有限元格式.该格式不仅能避免LBB条件,而且适用于多边形网格.在能量范数下,本文得到了原始变量和通量的最优误差估计.最后,针对定常和非定常的扩散反应方程,本文分别以一般情形和对流占优情形下的数值算例验证了方法的有效性.

For the primal-mixed form of the steady diffusion-reaction equation,a new conforming discontinuous finite element scheme is proposed by using the idea of discontinuous finite element method and the least-square method.Then the method is extended to the case of unsteady diffusion-reaction equation and a fully discrete conforming discontinuous finite element scheme is given.This scheme avoids the LBB condition and can be applied to polygon mesh.Under the energy norm,the optimal convergence order error estimates for the primary and flux variables are established.Finally,for the steady and unsteady equations,some numerical examples in general and convection dominated cases are given to verify the effectiveness of the method.