Error analysis of the finite‐strip method for parabolic equations

Abstract

The finite-strip method (FSM) is a hybrid technique which combines spectral and finite-element methods. Finite-element approximations are made for each mode of a finite Fourier series expansion. The Galerkin formulated method is set apart from other weighted-residual techniques by the selection of two types of basis functions, a piecewise linear interpolating function and a trigonometric function. The efficiency of the FSM is due in part to the orthogonality of the complex exponential basis: The linear system which results from the weak formulation is decoupled into several smaller systems, each of which may be solved independently. An error analysis for the FSM applied to time-dependent, parabolic partial differential equations indicates the numerical solution error is O(h2 + W r ) .M represents the Fourier truncation mode number and h represents the finiteelement grid mesh. The exponent r 2 2 increases with the cxact solution smoothness in the respective dimension. This error estimate is verified computationally. Extending the result to the finite-layer method, where a two-dimensional trigonometric basis is used, the numerical solution error is O(h2 + M-' + N -4 ) . The N and q represent the truncation mode number and degree of exact solution smoothness in the additional dimension. GI 1993 John Wilev ti Sons. Inc.

Convergence and exact solutions of spline finite strip method using unitary transformation approach

“…For problems with general boundary conditions, the spline FSM can be much more versatile, providing a complementary method to the classical finite strips. Although the accuracy and the rate of convergence of the classical FSM were studied by Smith and Allen [9] and Li [10] , the efficacy of spline finite strips was only demonstrated in the literature by comparing numerical results with analytical solutions or other numerical solutions. Explicit mathematical derivation of the rate of convergence and the exact solution form of spline finite strips are not available in the literature, to the best of the authors' knowledge.…”