论文标题
Richardson推断了与Green的内核的Urysohn积分方程的迭代galerkin解决方案
Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels
论文作者
论文摘要
我们考虑使用Green函数类型的内核来考虑Urysohn积分运算符$ \ MATHCAL {K} $。对于$ r \ geq 1 $，选择相对于统一分区的度量$ \ leq r-1 $的分段多项式空间被选择为近似空间，并且选择该投影是正交投影。迭代的Galerkin方法应用于积分方程$ x- \ Mathcal {k}（x）= f $。众所周知，迭代的Galerkin解决方案的收敛顺序为$ R+2 $，在上述分区点为$ 2R $。我们在上述Urysohn积分方程的分区点获得了迭代的Galerkin溶液的渐近膨胀。理查森外推用于改善收敛顺序。考虑了一个数值示例来说明我们的理论结果。
We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type of Green's function. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1 $ with respect to a uniform partition is chosen to be the approximating space and the projection is chosen to be the orthogonal projection. Iterated Galerkin method is applied to the integral equation $x - \mathcal{K}(x) = f$. It is known that the order of convergence of the iterated Galerkin solution is $r+2$ and, at the above partition points it is $2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.
