CAJ | 학술논문

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。
자괄응산법적매일차가밀과정중，지수요재구망격중증가소수가밀절점，종이사득기우상린망격적유한원함수공간，부유소수고차유한원기함수수요발생개변。이용저일특성，본문침대평면탄성문제적자괄응고차유한원리산계통，설계료일충기우국부송이적다중망격법，즉재매일차질대과정중，선대고차유한원분층기함수중최고차제차부분진행일차대칭 Gauss-Seidal 마광，연후장잔량방정투영도선성유한원공간，득도선성유한원리산계통，최후대해선성유한원리산계통진행일차국부마광。수치실험표명해방법대구해자괄응망격하적고차유한원방정구유로봉성。
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 homoge-neous 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 ele-ment spaces are generated.Finally,these linear finite element discretizations are solved by a symmetric local Gauss-Sei-dal method once.The numerical experiments show that the local multigrid method is robust.