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We consider a boundary value problem, an eigenvalue problem and an initialboundary value problem involving a generalized 2D Hamiltonian operator (i.e., the second order self-adjoint elliptic operator) in a rectangular domain. We apply a semi-discrete Galerkin method exploiting space approximations of the form c1(x1)χ1(x1,x2)+ ···+cN (x1)χN (x1,x2), where χ1,... ,χN are some known complex-valued basis functions and c1,... ,cN are unknown coefficients. The corresponding approximate problems are stated and their properties such as existence and uniqueness of solutions and bounds for them, positive definiteness of the related sesquilinear forms, etc. are analyzed. For a specific physically reasonable choice of the basis functions, error bounds of arbitrarily high orders are proved for all the listed problems.

On one version of a semidiscrete Galerkin method for PDE problems involving a generalized 2D Hamiltonian operator