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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 \(B_{h,h,h}u\).

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