Analogs of Hermite cubature formulas for the Dirichlet integral of harmonic functions (Q1920149)

Consider a bounded domain \(\Omega\) in \(\mathbb{R}^n\) and the class \(X (\Omega )\) of functions harmonic in \(\Omega\) and belonging to the Sobolev space \(W^1_2 (\Omega)\). An algorithm for approximation of the Dirichlet integrals \[ D_\Omega (u) = \int_\Omega |\nabla u |^2 dx \] for \(u \in X (\Omega)\) is proposed. The formulas have exactly one node in \(\Omega\), and \(D_\Omega (u)\) is approximated by quadratic forms of the values of \(u\) and its derivatives at the node. Namely, a canonical basis at the fixed node is constructed, and \(u\) is replaced by its expansion in the functions of this basis. An explicit upper bound for the integration error is presented, and its convergence to zero established. Finally, under additional assumptions on \(u \in X (\Omega)\), the exponential decay of the error as the order of accuracy grows is proved. It is shown that this rate depends on the maximal region of harmonic continuation of \(u\). The paper is very well written and organized.