论文标题
用于泊松方程的插值Galerkin有限元法
An interpolated Galerkin finite element method for the Poisson equation
论文作者
论文摘要
在通过有限元方法求解泊松方程时，我们使用给定拉普拉斯的一个自由度来插值 - 部分微分方程中的右手侧函数。有限元解决方案是较小的矢量空间中的Galerkin投影。该想法类似于在标准有限元方法中插值边界条件的想法。由于点插值，我们的方法产生了一个较小的方程系统和更好的条件编号。对于$ p_k $（$ k \ ge 3 $）有限元，每个元素上的未知数大幅减少到$ 3K $。我们构建2D $ P_2 $符合和不合格的构造，以及$ P_K $（$ K \ ge3 $）在三角形网格上符合插值的有限元元素。事实证明，这种插值的Galerkin有限元法以最佳顺序收敛。提出了数值测试和与标准有限元元素的比较，从而验证了理论并显示了插值的Galerkin有限元方法的优势。
When solving the Poisson equation by the finite element method, we use one degree of freedom for interpolation by the given Laplacian - the right hand side function in the partial differential equation. The finite element solution is the Galerkin projection in a smaller vector space. The idea is similar to that of interpolating the boundary condition in the standard finite element method. Due to the pointwise interpolation, our method yields a smaller system of equations and a better condition number. The number of unknowns on each element is reduced significantly from $(k^2+3k+2)/2$ to $3k$ for the $P_k$ ($k\ge 3$) finite element. We construct 2D $P_2$ conforming and nonconforming, and $P_k$ ($k\ge3$) conforming interpolated Galerkin finite elements on triangular grids. This interpolated Galerkin finite element method is proved to converge at the optimal order. Numerical tests and comparisons with the standard finite elements are presented, verifying the theory and showing advantages of the interpolated Galerkin finite element method.
