自由项在W_1~r([0,1])上的第二类Fredholm积分方程的计算复杂性

The Complexity of the Fredholm Equation of the Second Kind with Free Term in W_1~r ( [0,1 ] )

本文研究了近似求解自由项f∈W_1([0,1])的第二类Fredholm积分方程u-T_ku=f的计算复杂性.首先,证明此问题的第n信息半径具有弱渐近式r(n)=θ(n～(-r))(n→∞).然后证明了利用f与次数为k的有限元子空间的基的内积为信息的有限元方法(FEM)具有几乎最优误差的充要条件是k≥r-1.在这两个结果的基础上得出如下结论:问题的固有ε复杂性为comp(ε)=θ(ε～(-1/r))(ε→0+),而FEM的ε复杂性为FEM(ε)=θ(ε^(-1/μ))(ε→0+),其中μ=min(k+1,r).对于f∈W_p^r([0,1])(1<P≤∞),类似的问题已由Werschulz(1985)解决.

This paper deals with the approximate solution of the Fredholm equation Lu=f of the second kind with f ∈ W1r([0,1]). We show that the finite element method (FEM) of degree k using n inner products of f has minimal error if and only if k≥r-1. For f∈ Wpr([0,1])(1<p≤∞), the similar results have been obtained by [Werschulz(1985a), J. Integral Equations, 9(1985), 213-241].