A New Cascadic Multigrid For Parabolic Problems

Cascadic multigrid is a new type of multigrid methods, which requires no coarse grid corrections at all and may be viewed as a "one way" multigrid. Recently, Shi Zhongci and Xu Xuejun have constructed a new cascadic algorithm for solving elliptic problems. This algorithm permits the use of two different finite element spaces on the coarse and on the finest mesh levels. The advantage of this approach is that we can use rather simple finite element space on coarse meshes to carry out effective cascadic iterations for some complicated finite element discretization problems.A new cascadic multigrid method proposed by Shi Zhongci and Xu Xuejun for elliptic problems has been extended in this paper to parabolic problems. We consider the second order linear parabolic problems. We propose appropriate assumptions on the intergrid transfer operator and smoothing operator. Then we establish a general theorem of the convergence order and the computational complexity. The theorem has been used to treat two nonconforming finite elements. In the first case, we use P1 nonconforming element on the finest mesh level and linear Lagrange elements on the coarse meshes. In the second case, we consider Wilson nonconforming element on the finest mesh level and bilinear Lagrange elements on the coarse meshes. In such two cases, the optimality of the corresponding cascadic multigrid methods is derived (only quasi-optimality is obtained for 1-D case). We carry out the numerical experiments in the first case by using Richardson method, SOR iteration and the conjugate gradient method (CG) as smoothers, respectively. Numerical experiments show that the method is effective.