Fast solvers of integral and pseudodifferential equations on closed curves

Abstract

On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order ‖ u N − u ‖ λ ≤ c λ , μ N λ − μ ‖ u ‖ μ \|u_N-u\|_\lambda \leq c_{\lambda ,\mu }N^{\lambda -\mu }\|u\|_\mu , λ ≤ μ \lambda \leq \mu (Sobolev norms of periodic functions) in O ( N log ⁡ N ) \mathrm {O}(N\log N) arithmetical operations.