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Two-scale Boolean Galerkin discretizations for Fredholm integral equations of the second kind. (English) Zbl 1144.65087
The authors consider the Fredholm integral equation of the second kind $u+Ku=f$ in multidimensions, where $$u \in L^2(\square)$$, $$\square=(0,1)^d$$ is the unit cube in $$\mathbb R^d$$, $$d\geq 2$$, $$K$$ is a compact integral operator in $$\in L^2(\square)$$. They use a two-scale Boolean Galerkin approximation, that is (for $$d=3$$) $B^h_{H,H,H}u\equiv B_{h,H,H}u + B_{H,h,H}u + B_{h,H,H}u + B_{H,H,h}u - 2B_{H,H,H}u,$ where $$B_{h1,h2,h3}$$ is the standard Galerkin approximation with steps of discretizations $$(h1,h2,h3)$$ of variables. It is shown by both theory and numerics that this type of multiscale discretization algorithm with $$h=H^2$$, piece-wise constant approximating functions, and with the smooth exact solution significantly reduces the calculations, and produces approximations almost the same as the more expensive method $$B_{h,h,h}u$$.

##### MSC:
 65R20 Numerical methods for integral equations 45B05 Fredholm integral equations
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