LEGENDRE POLYNOMIAL BASIS FOR CONSTRUCTING THE GREEN’S FUNCTION OF THE BIHARMONIC OPERATOR ON A SQUARE

This paper is devoted to the numerical construction of the Green’s function for the biharmonic operator on the unit square under clamped boundary conditions. The problem models the bending of a thin elastic Kirchhoff – Love plate subjected to an arbitrary transverse load. Within the framework of the Galerkin – Ritz projection method, a comparison is made between the trigonometric basis of sine squares and a polynomial system based on shifted Legendre polynomials with a weight “bubble” factor that exactly satisfies the homogeneous boundary conditions. The completeness of the polynomial system in the energy space H02(Ω), the quasi-optimality of the approximation by Cea’s lemma, the exactness of the method on polynomial data, and the exponential (spectral) convergence rate for analytic solutions are demonstrated. It is shown that the trigonometric system of sine squares is not complete in H02(Ω); the closure of its linear span in the energy norm coincides with the subspace of functions whose second derivative is symmetric with respect to the midpoint of the interval. Consequently, the Galerkin approximations with this basis converge only for solutions with a symmetric Laplacian and stagnates otherwise. Numerical experiments confirm the theoretical conclusions and verify the algorithm against classical reference coefficients for the deflection of a clamped square plate (uniform load and concentrated force). The Legendre polynomial basis provides high-accuracy approximation of higher derivatives, which is unattainable for the trigonometric analogue.