Cascadic multilevel method



Principle ideas of our methodology

Cascadic conjugate gradient method.
The CCG method is an iterative method on a set of nested grids and associated linear systems. On the first level, which is assumed to have only a small number of degrees of freedom, the linear system is solved directly. On the next level, linear systems are solved iteratively by the conjugate gradient method. Starting point for CG-iterations is the interpolated solution from the previous level. Only a few CG-iterations (2-3) are performed on higher levels to annihilate high-frequency errors introduced by the interpolation procedure.


Material jump elliptic problems.
Let us consider the model elliptic problem:
o
The problem is discretized with the mortar finite element method. The decomposition of the computational domain into 3 subdomains is shown on the left picture below. The adaptive non-conformal grid on the 10-th multigrid level is shown on the right.

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The convergence of the cascadic iterations is demonstrated with the following figure. We plot the number of CG iterations versus the multigrid level for three values, 1, 103 and 106, of a1. The numbers on the right give some details of the cascadic iterations for the case a1=106.

o 
Level  itr      N  accuracy

    1    5     87     0.195
    2    7    173     0.126
    3   10    333     0.100
    4   10    537     0.077
    5    7    771     0.060

    6    9    939     0.053
    7    5   1559     0.040
    8    4   2435     0.030
    9    2   3455     0.025
   10    2   5683     0.019


References.