Cutting method for some multidimensional stationary problems (Q5947817)

Algorithmic problems of the numerical solution of operational equations on the Hilbert space \(A_{1, 1}\) plays an important part in getting a posteriori estimates for standard multidimensional elliptic boundary value and spectral problems. Moreover such almost decomposed problems can be a good first approximation for the initial elliptic problems. It is described how the Hilbert space \(A_{1, 1}\) (i.e. the weakened Sobolev space \(W_2^1 (\Omega)\)) is connected (in the case of \(d\) spatial variables, \(d\geq 2\)) with a decomposition \(\overline\Omega = \overline\Omega\cup, \dots, \cup \overline\Omega_k,\) for which some surfaces of \((d-1)\)-dimensional cuttings may intersect, can be applied to the problems of getting of a posteriori estimates of the errors of the grid approximations.