Preconditioning For Algebraic Discretization Systems Of Two Class Of Typical Differential Equations

In this paper, we discuss the preconditioning techniques and make some theoretical analysis that aims at different discretization methods of two class of typical differential equations. The full text contains two parts as follows.In the first part, a new preconditioned conjugate method for symmetric finite volume element systems of a class of elliptic problems on triangular mesh is presented by establishing spectrum equivalence between the coefficient matrix of symmetric finite volume element equation and that of the linear finite element equation. Moreover, it has been proved that the condition number of our PCG method is uniformly bounded. Numerical experiments verify the correctness of the theoretical results.The second part is devoted to argumentation of preconditioning technique and the implementation of the new algorithm for two class of Nedelec edge element equations on tetrahedron mesh. With regard to the first class of Nedelec edge element equations, we construct the new preconditioning algorithm presented in [37] which is called as Nodal Auxiliary Space Preconditioning , and develop some algorithms. Numerical experiments verify the correctness of the theoretical results. For the second class of Nedelec edge element equations, by choosing the first class of Nedelec edge element sapce as the subspace, we establish a kind of preconditioning algorithm for the the second class of Nedelec edge element equations. Numerical experiments show that the new preconditioning algorithm is robust and efficient.