Efficient Galerkin Method Research Articles

Hydrodynamic characteristics of a thin rolling plate in finite depth of water

The two-dimensional problem of the small-amplitude forced rolling motion of a thin vertical plate in finite water depth is considered using linear water wave theory. A range of geometric cases is considered, allowing the plate to be surface-piercing, attached to the bottom, unmounted and in each case either isolated or in the presence of a vertical wall. The method of solution uses two complementary formulations in conjunction with an accurate and efficient Galerkin method which are shown to give results that are in good agreement for all cases.

Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin's method with Hermite-Gauss basis functions

An efficient vector formulation and a corresponding quasi-vector formulation for the analysis of optical waveguides are presented. The proposed method is applicable to a large class of optical waveguides with general refractive index profile in a finite region of arbitrary shape and surrounded by a homogeneous cladding. The vector formulation is based on Galerkin's procedure using Hermite-Gauss basis functions. It is shown that use of Hermite-Gauss basis functions leads to a significant increase in computational efficiency over trigonometric basis functions. The quasi-vector solution is obtained from the standard scalar formulation by including a polarization correction. The accuracy of the scalar, vector, and quasi-vector solutions is demonstrated by comparison with the exact solution for the fundamental mode in a circular fiber. Comparison of the modal solutions obtained with the various methods for optical waveguides with square, rectangular, circular, and elliptical core demonstrate the accuracy and advantage of the quasi-vector solution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>