A Finite Element Splitting Extrapolation for Second Order Hyperbolic Equations
SIAM Journal on Scientific Computing
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Let $A(y) \sim A+\sum_{k=1}^\infty Q_k (\log y) y^{\sigma _k}$ as $y\rightarrow 0+$, where $y$ is a discrete or continuous variable and $Q_k( \xi) $ are polynomials in $\xi $. It is assumed that $\sigma _k$ and the degree of $Q_k(\xi) $ or an upper bound for it are known for each $k$, and that $A(y)$ is known for all possible $y\in ( 0,b] $. The aim is to find $A$, whether it is the limit or antilimit of $A(y)$ for $y\rightarrow 0+$. A very effective way of doing this is by the generalized Richardson extrapolation. In this paper this procedure is described and a very efficient recursive algorithm for its implementation is given when the set of extrapolation points is $\{y_l=y_0\omega^l,\, l=0,1,\ldots \}$ for some $\omega \in (0,1)$. In addition, a complete theory of convergence and stability for the columns and the diagonals of the corresponding extrapolation table is provided. Finally, two applications are considered in detail, one of which is to generalized Romberg integration of functions with algebraic and logarithmic endpoint singularities.