New type of a posteriori error estimates for elliptic boundary value and spectral problems (Q5942466)

A new scale of Hilbert spaces \(A_{1,m}\) is introduced for which the Sobolev space \(H\) is a subspace. It allows on the basis of an appropriate penalty method with the parameter \(\varepsilon \to +0\) for \(m\geq 1/2\) to consider related variational problems in \(A_{1,m}\) with minima \(m_{\varepsilon}\) of the energetic functionals such that \(0 \leq m - m_{\varepsilon} \leq K{\varepsilon}^2.\) The case of spaces with \(m=1\) is especially remarkable. Particularly, it leads to new effective algorithms for obtaining lower estimates for eigenvalues of symmetric positive grid elliptic operators.