The Generalized Inverse of a Class of Block Matrix from Models for Flow in a Dual-Permeability System
The generalized inverse method has grown to become an effective and important method for solving linear mathematical problem as well as applications of linear mathematics, such as computational fluid dynamic, constrained least-squares problems, constrained quadratic programming, and so on. The commonly used generalized inverse methods mainly include minus inverse, plus inverse, least square generalized inverse, reflexive generalized inverse and minimum norm generalized inverse. In this paper, generalized inverse method is studied for a special class of block four-by-four matrix problems discretizing the Dual-Permeability flow in porous media model by mixed multi-scale finite element methods. Firstly, some partial differential equation models of flow in a Dual-Permeability system are given in this paper. Then a class of block matrix from the discrete systems of mathematic models by the multi-scale finite element methods is analyzed. And for both singular and nonsingular cases, the minus inverse of the block two-by-two block matrices are discussed, respectively. Finally, the generalized inverse of the block four-by-four matrix is obtained by transforming the block four-by-four matrix problems into the two-by-two matrix problems, and the algorithm for computing the minus inverse is presented.
KeywordsGeneralized inverse Block matrix Partial difference equation Flow model
The authors are grateful to the referees for their valuable comments and suggestions which helped to improve the presentation of the paper.
This research was funded by the National Natural Science Foundation of China under Grant No. 41766001, 41962019.
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