Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity

We study formally self-adjoint boundary value problems for elliptic systems of differential equations in domains with periodic (in particular, cylindrical) exits to infinity. Statements of problems in a truncated (finite) domain which provide approximate solutions of the original problem are presented. The integro-differential conditions on the artificially formed end face are interpreted as a finite-dimensional approximation to the Steklov–Poincaré operator, which is widely used when dealing with the Helmholtz equation in cylindrical waveguides. Asymptotically sharp approximation error estimates are obtained for the solutions of the problem with the compactly supported right-hand side in an infinite domain as well as for the eigenvalues in the discrete spectrum (if any). The construction of a finite-dimensional integro-differential operator is based on natural orthogonality and normalization conditions for oscillating and exponential Floquet waves in a periodic quasicylindrical end.