Q Qadratic Finite Element Multi-grid Of CG Iterative Method

To high-dimensional problems, multi-grid CG iterative together with su-percovergence of quadratic finite element is a feasible solution, instead of linear element with 2-order accuracy which would be insufficient for the vast scale of computing when grids are refined. This paper studies how multi-grid CG iter-ative of one-dimensional quadratic element solves high-dimensional problems.Section one:given the quadratic finite element solution in coarse grids of the two-layer grids, a more accurate solution would be obtained if the five values, i.e. two-node value in each of the modules, midpoint value, and value of two quartiles from quadratic interpolation, are taken as the initial value of quadratic element in refined grid, followed by CG iterative. A peculiar nature of quadratic interpolation is discovered that errors in the two quartiles with only three-order accuracy forming high-frequency oscillations are eliminated by 2-3 times of CG iteration with excellent effect.Section two:given the finite element solution uh,uh/2 in the two-layer grids Zh, Zh/2, high accuracy can be achieved by several times of CG iterative, since accurate node value of quadratic finite element can be obtained in the third layer Zh/4 when a new extrapolation formula is constructed, and midpoint value can be got with quadratic interpolation.