三维Poisson方程的三种有限元解及特征值下界

THREE FINITE ELEMENT SOLUTIONS OF THREE-DIMENSIONAL POISSON EQUATION AND THE LOWER BOUNDS OF EIGENVALUES

本文给出了三种有限元的特征值方法求解三维Laplace算子特征值和边值问题的数值计算结果;探索了已有的非协调元和协调元的一些理论性质;猜测新的七个自由度三维NF_1元的数值规律.数值实验表明:七个自由度三维NF_1元和三维EQ_1^(rot)元特征值都下逼近准确特征值;七个自由度三维NF_1元和三维EQ_1^(rot)元二网格离散方案特征值都下逼近准确特征值;七个自由度三维NF_1元外推特征值下逼近准确特征值;七个自由度三维NF_1元比三维EQ_1^(rot)元有较好的数值表现;八节点等参数元特征值上逼近准确特征值.

In this paper, the numerical results of three-dimensional Laplace operator eigenvalue and boundary problems are given by methods of eigenvalue methods of three finite ele- ments. We explore the theoretical properties of the known non-conforming element and conforming element and conjecture the numerical law for the new three-dimensional NFi element of 7-degree of freedom. It is shown by the numerical experiments results that the three-dimensional NF1 non-conforming element of 7-degree of freedom and the three- dimensional EQ1rot non-conforming element eigenvalues approximate exact eigenvalues from below, the three-dimensional NF1 non-conforming element of 7-degree of freedom and the three-dimensional EQ1rot non-conforming element two-grid discretization schemes eigenval- ues approximate exact eigenvalues from below, the three-dimensional NF1 non-conforming element of 7-degree of freedom extrapolation eigenvalues approximate exact eigenvalues from below, the numerical accuracy of the three-dimensional NF1 non-conforming element of 7- degree of freedom is much better than that of the three-dimensional EQ1rot non-conforming element, 3-rectangle of degree 2 conforming element eigenvalues approximate exact eigenval- ues from above.