Error Analysis of the Finite-Strip Method for Parabolic Equations
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 + M-r). M represents the Fourier truncation mode number
and h represents the finite-element grid mesh. the exponent r >= 2 increases with
the exact 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-r + N-q). The N and q represent the truncation mode number
and degree of exact solution smoothness in the additional dimension.
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