The numerical solution of the generalized airfoil equation
The authors consider the singular integral equation \[ {1\over\pi} \int_{-1}^ 1 {u(y) \over {y-x}}dy-{c\over y} \int_{-1}^ 1 \ln| y-x| u(y)dy+\int_{-1}^ 1 k(x,y)u(y)dy=f(x), \qquad x\in(-1,1),\tag{1} \] where \(f\) and \(k\) are given Hölder continuous functions, \(c\) is a complex number, and \(u\) is the unknown function. Using some mapping properties of the operator with logarithmic kernel, they establish a quadrature method for the approximate solution of equation (1), where the discretization of the first two integral operators in (1) can be done without any integration. The paper contains stability and convergence results for the quadrature method and discusses iterative two-grid and multi-grid methods for determining an approximate solution to equation (1) with a computational complexity of \(O(n^ 2)\).
iterative two-grid and multi-grid methods
scientific article; zbMATH DE number 86118
scientific article