A fast Petrov-Galerkin method for solving the generalized airfoil equation

In this paper we develop a fast Petrov-Galerkin method for solving the generalized airfoil equation using the Chebyshev polynomials. The conventional method for solving this equation leads to a linear system with a dense coefficient matrix. When the order of the linear system is large, the computational complexity for solving the corresponding linear system is huge. For this we propose the matrix truncation strategy, which compresses the dense coefficient matrix into a sparse matrix. We prove that the truncated method preserves the optimal order of the approximate solution for the conventional method. Moreover, we solve the truncated equation using the multilevel augmentation method. The computational complexity for solving this truncated linear system is estimated to be linear up to a logarithmic factor.