Interpolation and Related Coarsening Techniques for the Algebraic Multigrid Method
G. Golubovici, Constantin Popa · Birkhäuser Basel eBooks · 1994
Let A be a symmetric and positive definite matrix and b ∊ R r . We consider the system $$Au = b$$ with the (unique) exact solution u ∊ R r . For q ≥ 2 let C 1, C2, …, C q be a sequence of nonvoid subsets of {1,..., r} such that $$\left\{ {1, \cdots ,r} \right\} = {C_1} \supset {C_2} \supset \cdots \supset {C_q}$$ $$\left| {{C_m}} \right| = {n_{m,}}m = 1, \cdots ,q$$ $$r = {n_1} > {n_2} > \cdots > {n_q} \ge 1$$ where by |C m | we denoted the number of elements in the set C m . Furthermore, for m = 1, 2,..., q − 1 we consider the matrices A 1 − A and A m+1 and the linear operators $$I_{m + 1}^m:{R^{{n_{m + 1}}}} \to {R^{{n_m}}},I_m^{m + 1}:{R^{{n_m}}} \to {R^{{n_{m + 1}}}}$$ with the properties: I +1 has full rank, $$I_m^{m + 1} = {(I_{m + 1}^m)^t}$$ $${A^{m + 1}} = I_m^{m + 1}{A^m}I_{m + 1}^m$$ We also define the coarse grid correction operators T m by $${T^m} = {I_m} - I_{m + 1}^m{({A^{m + 1}})^{ - 1}}I_m^{m + 1}{A^m}$$ and the smoothing process $$u_{new}^m = {G^m}u_{old}^m + \left( {{I_m} - {G^m}} \right){\left( {{A^m}} \right)^{ - 1}}{b^m}$$ where I m is the identity and $${A^m}{u^m} = {b^m}$$ are the systems corresponding to the coarse levels.