Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates (Q2471831)

The authors consider second-order nonlinear elliptic Dirichlet problems. Discretizing these problems results in nonlinear systems \(F(u) = 0\) which are in general solved by quadratic convergent Newton methods or variants. In this paper a new version of the only superlinear convergent but cheaper Broyden algorithm, a special quasi-Newton method, is developed for the solution of the problem. Especially, the storage of the whole update vectors, a disadvantage of Broyden's method, is avoided because the updates are explicitely added to the matrix. Employing hierarchical matrices (\(H\)-matrices) an accelerated variant is obtained caused by the efficient multiplication of \(H\)- and semiseparable matrices. Furthermore, a method, called \(H\)-Broyden-\(LU\) algorithm, is introduced which explicitely updates the factors of an initially computed \(LU\) decomposition of the Jacobian matrix during the Broyden updates preserving the superlinear convergence of the Broyden method.