Tzon-Tzer Lu

Semi-convergence analysis of Uzawa methods for singular saddle point problems

Recently, Zheng, Bai and Yang studied the parameterized Uzawa method for solving singular saddle point problems (B. Zheng, Z.-Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl. 431 (2009) 808-817). In this paper, we discuss the inexact Uzawa method, which covers the Uzawa method, the preconditioned Uzawa method, and the parameterized Uzawa method to solve the singular saddle point problems. We prove the semi-convergence result under restrictions by verifying two necessary and sufficient conditions, that is, all elementary divisors associated with the eigenvalue 1 of its iterative matrix are linear, and the pseudo-spectral radius of the iterative matrix is less than 1. Sufficient conditions for the semi-convergence of several Uzawa-type methods are also provided. In addition, numerical examples are given to demonstrate the semi-convergence of Uzawa-type methods.

Inverses of 2 × 2 block matrices

Abstract In this paper, the authors give explicit inverse formulae for 2 × 2 block matrices with three different partitions. Then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as Hamiltonian, per-Hermitian, and centro-Hermitian matrices.

The collocation Trefftz method for biharmonic equations with crack singularities

Abstract The purpose of this paper is to extend the boundary approximation method proposed by Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], i.e. the collocation Trefftz method called in this paper, for biharmonic equations with singularities. First, this paper derives the Green formulas for biharmonic equations on bounded domains with a non-smooth boundary, and corner terms are developed. The Green formulas are important to provide all the exterior and interior boundary conditions which will be used in the collocation Trefftz method. Second, this paper proposes three crack models (called Models I, II and III), and the collocation Trefftz method provides their most accurate solutions. In fact, Models I and II resemble Motzs problem in Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], and Model III with all the clamped boundary conditions originated from Schiff et al. [The mathematics of finite elements and applications III, 1979]. Moreover, effects on d 1 of different boundary conditions are investigated, and a brief analysis of error bounds for the collocation Trefftz method is made. Since accuracy of the solutions obtained in this paper is very high, they can be used as the typical models in testing numerical methods. The computed results show that as the singularity models, Models I and II are superior to Model III, because more accurate solutions can be obtained by the collocation Trefftz method.

Adomian Decomposition Method Using Integrating Factor

This paper proposes a new Adomian decomposition method by using integrating factor. Nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, the complete error analysis for this method is presented.

Effective condition number of Trefftz methods for biharmonic equations with crack singularities

The paper presents the new stability analysis for the collocation Trefftz method (CTM) for biharmonic equations, based on the new effective condition number Cond_eff. The Trefftz method is a special spectral method with the particular solutions as admissible functions, and it has been widely used in engineering. Three crack models in Li et al. (Eng. Anal. Boundary Elements 2004; 28:79–96; Trefftz and Collocation Methods. WIT Publishers: Southampton, Boston, 2008) are considered, and the bounds of Cond_eff and the traditional condition number Cond are derived, to give the polynomial and the exponential growth rates, respectively. The stability analysis explains well the numerical experiments. Hence, the new Cond_eff is more advantageous than Cond. Besides since the bounds of Cond_eff and Cond involve the estimation of the minimal singular value σmin of the discrete matrix F, and since the estimation of σmin is challenging and difficult, the proof for lower bounds of σmin in this paper is important and intriguing. Copyright

Effective Condition Number of Finite Difference Method for Poisson's Equation Involving Boundary Singularities

For solving the linear algebraic equations Ax = b with the symmetric and positive definite matrix A, the effective condition number Cond_eff is defined in [6, 10] by following Chan and Foulser [2] and Rice [14]. The Cond_eff is smaller, or much smaller, than the traditional condition number Cond. Besides, the simplest condition number Cond_EE is also defined in [6, 10]. This article studies a popular model of Poissons equation involving the boundary singularities by the finite difference method using the local refinements of grids. The bounds of Cond_EE are derived to display theoretically that the effective condition number is significantly smaller than the Cond. In this article, by exploring local refinement properties, we derive the bounds of effective condition numbers up to O(1) and at least o(h −1/2) for the maximal step size h. They are significant improvements compared with the bound O(h −3/2), which is established in [6, 10]. Therefore, the study of effective condition number in this article reaches a new comprehensive and advanced level.

Global superconvergence of finite element methods for biharmonic equations and blending surfaces

Abstract In this paper, new numerical algorithms of finite element methods (FEM) are reported for both biharmonic equations and 3D blending surfaces, to achieve the global superconvergence O(h3)-O(h4) in H2 norms. This is significant, compared with the optimal convergence O(h2). The algorithms are simple because only an a posteriori interpolant solution is needed. Such a global convergence method was originated by Lin and his colleagues in [1–3] for only the clamped boundary conditions. Recently, we extended the global superconvergence to other boundary conditions, such as the simple support condition, the periodic boundary, and the natural boundary condition. Moreover, we apply in [4–6] this global superconvergence to the FEM using the penalty techniques for biharmonic equations and blending problems, also to reach O(h3) and O(h4) for quasiuniform and uniform □ij, respectively. Currently, we develop in [7,8] and in this paper the FEM using the penalty plus hybrid techniques to reduce the condition number down to O(h−4)-O(h−5) of the associated matrix, while retaining superconvergence O(h3)-O(h4). Since instability is severe for biharmonic equations, any reduction of the condition number is crucial. By the new algorithms in this paper, not only can a great deal of CPU time be saved, but also the complicated biharmonic equations and blending surfaces may be solved in double precision. Numerical experiments are carried out to support the theoretical conclusions.