Technical Report CS0899

Title: Effect of Boundary Conditions in Legendre Spectral Methods
Authors: Uri Zrahia, Steve Orszag, Moshe Israeli
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Abstract: In this paper, the effect of boundary conditions on the solution of one dimensional singular boundary layer problems (with a small parameter $\epsilon$ multiplying the highest derivative) is analyzed. In particular, solutions obtained by polynomial spectral methods may be degraded throughout the entire domain unless the polynomial degree $N$ of the spectral solution is high enough to resolve thin boundary and internal layers. For some choices of boundary conditions, it is shown that the strength of boundary layers are reduced considerably. Replacing the original boundary conditions with such alternative conditions will result in a problem for which the numerical solution is much more acurate for a given $N$. This solution can be matched with a solution which is accurate within the boundary layer. In the present work we analyze the effect of different sets of boundary conditions on the numerical solution, and we show that the error induced far from the layer can be reduced by a factor which is a power of the small parameter $\epsilon$ when compared to a straightforward numerical solution of the problem.
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