A fast trigonometric collocation method for some elliptic pseudodifferential equations

摘要

In this paper, we propose and analyze a fast trigonometric collocation method for a class of periodic elliptic pseudodifferential equations, whose pseudodifferential operators can always be represented as the sum of a principal part and a smoothing operator. We show that the whole matrix representation for the principal part in our discrete linear system can be generated by only computing O(n) entries rather than computing all entries of the matrix, where 2n or 2n+1 is the size of the matrix. The dense matrix for the smoothing operator can be compressed into a sparse matrix with only O(nlogn) nonzero entries. We also prove that our proposed method preserves the optimal convergent order the same as without compression. Some numerical experiments for solving three cases of boundary integral equations are presented to demonstrate its approximate accuracy and computational efficiency, verifying the theoretical estimates.