A practical solution for KKT systems

Abstract

We consider KKT systems of linear equations with a 2 × 2 block indefinite matrix whose (2, 2) block is zero. Such systems arise in many applications. Treating such matrices would encounter some intricacies, especially when its (1, 1) block, i.e., the stiffness matrix in term of computational mechanics, is rank-deficient. It is the rank-deficiency of the stiffness matrix that leads to the so-called rigid-displacement issue. This is believed to be one of the main reasons that many programmers would unwillingly give up the Lagrange multiplier method but select the penalty method. Based on the Sherman–Morrison formula and the conventional LDLT decomposition for symmetric positive definite matrices, a robust direct solution is proposed, which is amenable to the conventional finite element codes, competent for both nonsingular and singular stiffness matrices, and particularly suitable to parallel computation. As a paradigm, the application to the element-free Galerkin method (EFGM) with the moving least squares interpolation is illustrated.