A fast algorithm for solving the generalized airfoil equation
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Rates of convergence for collocation with Jacobi polynomials for the airfoil equation
Journal of Computational and Applied Mathematics
Numerical Solution of the Generalized Airfoil Equation for an Airfoil with a Flap
SIAM Journal on Numerical Analysis
A Collocation Method for the Generalized Airfoil Equation for an Airfoil with a Flap
SIAM Journal on Numerical Analysis
A fast solver for a hypersingular boundary integral equation
Applied Numerical Mathematics
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In this paper, we develop an efficient Petrov-Galerkin method for the generalized airfoil equation. In general, the Petrov-Galerkin method for this equation leads to a linear system with a dense coefficient matrix. When the order of the coefficient matrix is large, the complexity for solving the corresponding linear system is huge. For this purpose, we propose a matrix truncation strategy to compress the dense coefficient matrix into a sparse matrix. Subsequently, we use a numerical integration method to generate the fully discrete truncated linear system. At last we solve the corresponding linear system by the multilevel augmentation method. An optimal order of the approximate solution is preserved. The computational complexity for generating the fully discrete truncated linear system and solving it is estimated to be linear up to a logarithm. The spectral condition number of the truncated matrix is proved to be bounded. Numerical examples complete the paper.