What is the complexity of weakly singular integral equations?
Arvet Pedas, Gennadi Vainikko
Consider the integral equation where ν ∈ (0, 1), f ∈ C m [0, 1], a, b ∈ C 2m ([0, 1]×[0, 1]), m ∈ N, and the corresponding homogeneous equation has only the trivial solution. By a fast (C, C m ) solver of (1) we mean a solver which produces approximate solutions u n , n ∈ N, such that • given the values of a, b and f at certain not more than n points depending on the solver (with n → ∞ as n → ∞), the parameters of u n can be determined at the cost of γ m n arithmetical operations, and an
... y is achieved where u is the solution of (1); • having the parameters of u n in hand, the value of u n at any point x ∈ [0, 1] is available at the cost of γ m operations. Here the constants c m , γ m , γ m are independent of f and n. It occurs that estimate (2) is information optimal -in the worst case, under above assumptions, a higher order of error estimate cannot be achieved allowing more arithmetical work. In a fast (L p , C m ) solver, u − u n L p (0,1) ≤ c m n −m f C m [0,1] is required instead of (2). In a quasifast (C, C m ) solver, u − u n C[0,1] ≤ c m n −m log n f C m [0,1] is required instead of (2). In the literature, fast (C, C m ) solvers have been constructed only for integral equations without singularities that for (1) corresponds to the case a ≡ 0. We consider (1) in general case and construct a solver which is (C, C m ) quasifast and (L p , C m ) fast for 1 ≤ p < ∞; under slightly strengthened smoothness assumptions that the mth derivatives of a and b are Hoelder continuous, this solver is also (C, C m ) fast. Hence the complexity of (1) is the same as that for integral equations with smooth kernels or close to it. Actually some boundary singularities of a, b ∈ C 2m ([0, 1] × (0, 1)) and f ∈ C[0, 1] ∩ C m (0, 1) are allowed in the final formulations.