Hung-Tsai Huang

New expansions of numerical eigenvalues for -Δ= by nonconforming elements

The paper explores new expansions of the eigenvalues for -Δu = Λpu in S with Dirichlet boundary conditions by the bilinear element (denoted Q 1 ) and three nonconforming elements, the rotated bilinear element (denoted Q 1 rot ), the extension of Q 1 rot (denoted EQ 1 rot ) and Wilsons elements. The expansions indicate that Q 1 and Q 1 rot provide upper bounds of the eigenvalues, and that EQ 1 rot and Wilsons elements provide lower bounds of the eigenvalues. By extrapolation, the O(h 4 ) convergence rate can be obtained, where h is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.

Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation

Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains. Then the new bound of Cond is derived for bounded simply connected domains with mixed types of boundary conditions. Numerical results are reported for Motz’s problem by adding singular functions. The values of Cond grow exponentially with respect to the number of fundamental solutions used. Note that there seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Moreover, the expansion coefficients obtained by the MFS are oscillatingly large, to cause the other kind of instability: subtraction cancelation errors in the final harmonic solutions.

Ill‐conditioning of the truncated singular value decomposition, Tikhonov regularization and their applications to numerical partial differential equations

This paper explores some intrinsic characteristics of accuracy and stability for the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TR), which can be applied to numerical solutions of partial differential equations (numerical PDE). The ill-conditioning is a severe issue for numerical methods, in particular when the minimal singular value sigmamin of the stiffness matrix is close to zero, and when the singular vector umin of σmin is highly oscillating. TSVD and TR can be used as numerical techniques for seeking stable solutions of linear algebraic equations. In this paper, new bounds are derived for the condition number and the effective condition number which can be used to improve ill-conditioning by TSVD and TR. A brief error analysis of TSVD and TR is also made, since both errors and condition number are essential for the numerical solution of PDE. Numerical experiments are reported for the discrete Laplace operator by the method of fundamental solutions. Copyright

Error analysis of the method of fundamental solutions for linear elastostatics

For linear elastostatics in 2D, the Trefftz methods (i.e., the boundary methods) using the particular solutions and the fundamental solutions satisfying the Cauchy-Navier equation lead to the method of particular solutions (MPS) and the method of fundamental solutions (MFS), respectively. In this paper, the mixed types of the displacement and the traction boundary conditions are dealt with, and both the direct collocation techniques and the Lagrange multiplier are used to couple the boundary conditions. The former is just the MFS and the MPS, and the latter is also called the hybrid Trefftz method (HTM) in Jirousek (1978, 1992, 1996) [1-3]. In Bogomolny (1985) [4] and Li (2009) [5] the error analysis of the MFS is given for Laplaces equation, and in Li (2012) [6] the error bounds of both MPS and HTM using particular solutions (PS) are provided for linear elastostatics. In this paper, our efforts are devoted to explore the error analysis of the MFS and the HTM using fundamental solutions (FS). The key analysis is to derive the errors between FS and PS of the linear elastostatics, where the expansions of the FS in Li et al. (2011) [7] are a basic tool in analysis. Then the optimal convergence rates can be achieved for the MFS and the HTM using FS. Recently, the MFS has been developed with numerous reports in computation; the analysis is behind. The analysis of the MFS for linear elastostatics in this paper may narrow the existing gap between computation and theory of the MFS.

Null Field and Interior Field Methods for Laplace’s Equation in Actually Punctured Disks

For solving Laplace’s equation in circular domains with circular holes, the null field method (NFM) was developed by Chen and his research group (see Chen and Shen (2009)). In Li et al. (2012) the explicit algebraic equations of the NFM were provided, where some stability analysis was made. For the NFM, the conservative schemes were proposed in Lee et al. (2013), and the algorithm singularity was fully investigated in Lee et al., submitted to Engineering Analysis with Boundary Elements, (2013). To target the same problems, a new interior field method (IFM) is also proposed. Besides the NFM and the IFM, the collocation Trefftz method (CTM) and the boundary integral equation method (BIE) are two effective boundary methods. This paper is devoted to a further study on NFM and IFM for three goals. The first goal is to explore their intrinsic relations. Since there exists no error analysis for the NFM, the second goal is to drive error bounds of the numerical solutions. The third goal is to apply those methods to Laplace’s equation in the domains with extremely small holes, which are called actually punctured disks. By NFM, IFM, BIE, and CTM, numerical experiments are carried out, and comparisons are provided. This paper provides an in-depth overview of four methods, the error analysis of the NFM, and the intriguing computation, which are essential for the boundary methods.

SUPERCONVERGENCE OF FEMS AND NUMERICAL CONTINUATION FOR PARAMETER-DEPENDENT PROBLEMS WITH FOLDS

We study finite element approximations for positive solutions of semilinear elliptic eigenvalue problems with folds, and exploit the superconvergence of finite element methods (FEM). In order to apply the superconvergence of FEM for Poissons equation in [Chen & Huang, 1995; Huang et al., 2004, 2006; Lin & Yan, 1996] to parameter-dependent problems with folds, this paper provides the framework of analysis, accompanied with the proof of the strong monotonicity of the nonlinear form. It is worthy to point out that the superconvergence of the nonlinear problem in this paper is different from that in [Chen & Huang, 1995]. A continuation algorithm is described to trace solution curves of semilinear elliptic eigenvalue problems, where the Adini elements are exploited to discretize the PDEs. Numerical results on some sample test problems with folds and bifurcations are reported.

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.

On solution uniqueness of elliptic boundary value problems

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213-224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.

Superconvergence of bi-k-Lagrange elements for eigenvalue problems ☆

We study superconvergence of bi-k-Lagrange elements for parameter-dependent problems where k 2. We show that the superconvergence rate of the bi-k-Lagrange elements is two orders higher than that of the kth-order Lagrange elements. This is a significant improvement of the previous results [C.-S. Chien, H.T. Huang, B.-W. Jeng, Z.C. Li, Superconvergence of FEMs and numerical continuation for parameterdependent problems with folds, Int. J. Bifurcation Chaos 18 (2008) 1321–1336], which is only one order (or a half order) higher than that of the latter. Next, we apply the bi-k-Lagrange elements to the computations of energy levels and wave functions of two-dimensional (2D) Bose–Einstein condensates (BEC), and BEC in a periodic potential. Sample numerical results are reported.