Multigrid methods of the second kind.(English)Zbl 0577.65118

Multigrid methods for integral and differential equations, Workshop Bristol/Engl. 1983, Inst. Math. Appl. Conf. Ser., New Ser. 3, 11-83 (1985).
[For the entire collection see Zbl 0563.00017.]
This paper treats a wide variety of aspects related to the application of multigrid methods to the numerical solution of Fredholm second kind integral equations. The linear form of these equations is given by $$u=Ku+f$$, where K is an appropriate integral operator. These aspects include various linear and nonlinear problems (e.g., integral equations that arise from elliptic differential equations and those that do not, eigenproblems, control problems, time dependent equations, and integro- differential operators), several variants of the multigrid scheme theoretical analyses, complexity results, various practical applications and numerical performance reports. In particular, this work establishes, in a very general setting, that multigrid ”solves” the discrete systems at cost of a few discrete operator evaluations. For discretizations that yield explicit $$n\times n$$ full matrices, this translates to a computational complexity of $$O(n^ 2)$$.
Reviewer: S.F.McCormick

MSC:

 65R20 Numerical methods for integral equations 45B05 Fredholm integral equations

Zbl 0563.00017