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Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates
Mario Bebendorf and Ying Chen
The numerical solution of nonlinear problems is usually connected with Newton's method. Due to its computational cost, variants (so-called quasi-Newton methods) have been developed in which the arising inverse of the Jacobian is replaced by an approximation. In this article we present a new approach which is based on Broyden updates. Instead of updating the inverse we introduce a method which constructs updates of the $LU$ decomposition. Since an approximate $LU$ decomposition of finite element stiffness matrices can be efficiently computed in the set of hierarchical matrices, the complexity of the proposed method scales almost linearly.
Numerical examples demonstrate the effectiveness of this new approach.