This paper aims to propose a novel direct method for indefinite algebraic linear systems. It is well adapted for sparse linear systems, such as those of two-dimensional (2-D) finite elements problems, especially for coupled systems.
The proposed method is developed on an example of an indefinite symmetric matrix. The algorithm of the method is given next, and a comparison between the numbers of operations required by the method and the Cholesky method is also given. Finally, an application on a magnetostatic problem for classical methods (Gauss and Cholesky) shows the relative efficiency of the proposed method.
The proposed method can be used advantageously for 2-D finite elements in stepping methods without using a block decomposition of matrices.
This method is advantageous for direct linear solving for 2-D problems, but it is not recommended at this time for three-dimensional problems.
The proposed method is the first direct solver for algebraic linear systems proposed since more than a half century. It is not limited for symmetric positive systems such as many of direct and iterative methods.
