平面弹性问题的高次有限元离散系统的局部多重网格法

自适应算法的每一次加密过程中,只需要在旧网格中增加少数加密节点,从而使得基于相邻网格的有限元函数空间,仅有少数高次有限元基函数需要发生改变.利用这一特性,本文针对平面弹性问题的自适应高次有限元离散系统,设计了一种基于局部松弛的多重网格法,即在每一次迭代过程中,先对高次有限元分层基函数中最高次齐次部分进行一次对称Gauss-Seidal 磨光,然后将残量方程投影到线性有限元空间,得到线性有限元离散系统,最后对该线性有限元离散系统进行一次局部磨光. 数值实验表明该方法对求解自适应网格下的高次有限元方程具有鲁棒性.

Local multigrid method for higher-order finite element discretizations of elasticity problems in two dimensions

Due to few limited vertexes increase during every refinement of adaptive finite element method (AFEM), only some limited basis functions change between two finite element spaces based on two adjacent refinement meshes. By use of this special property, a type of multigrid method based on the local relaxation is applied to the high-order AFEM discrete systems of elasticity problems in two dimensions, that is, during each iteration, the part of the homogeneous high-order systems based on hierarchical basis functions is solved by a symmetric Gauss-Seidal method once, and then these residual systems are projected onto linear finite element space, and discrete systems based on linear finite element spaces are generated. Finally, these linear finite element discretizations are solved by a symmetric local Gauss-Seidal method once. The numerical experiments show that the local multigrid method is robust.