Estimation of the convergence rate of difference solutions to generalized solutions of the Dirichlet problem for the Helmholtz equation of class \(\overset\circ W^ 1_ 2(\Omega)\) in a convex domain
For the Helmholtz equation with Dirichlet condition in a convex polygon \(\Omega\), a difference scheme was constructed by the authors and \textit{A. A. Samarskij} [Dokl. Akad. Nauk SSSR 273, 1040-1044 (1983; Zbl 0556.65080)] which, on any nonuniform mesh \({\hat \omega}\) in mesh norm \(L_ 2({\hat \omega})\), has accuracy O(h) if the solution u(x) of the initial problem belongs to space \(\overset\circ W^ 1_ 2(\Omega)\). In the present article we take the case when \(\Omega\) is any convex domain with boundary \(\Gamma \in C^ 2\). By meas of operators of exact difference schemes, we construct a scheme whose solution converges to the solution of the initial problem at a rate \(O(h^{1/2})\) in mesh norm \(L_ 2(\Omega)\).
scientific article; zbMATH DE number 3982443