# Hung-Tsai Huang I-Shou University, Kaohsiung

I-Shou University, Kaohsiung

## Computing in Mathematics, Natural Science, Engineering and Medicine

PhD

## Publications

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**ABSTRACT:**For Laplace's equation in circular domains with circular holes, the null field method (NFM) is proposed by Chen with his groups. In NFM, the fundamental solutions (FS) with the exterior field nodes to the solution domain are used in the Green formulas, where the FS are replaced by the infinite expansion series. The explicit algebraic equations are derived and reported in Li et al. (2012) [20]. The explicit algebraic equations are essential not only to practical computation, but also to the algorithm analysis, such as algorithm singularity, error and stability analysis. So far, the study of the NFM is confined to the Dirichlet problems (i.e., the Dirichlet boundary value problems) by the first kind NFM. This paper is devoted mainly to the Neumann problems (i.e., the Neumann boundary value problems) of Laplace's equation by the second kind NFM. When the field nodes are pulled to the domain boundary, this special (i.e., the optimal) NFM is equivalent to the interior field method (IFM) (Huang et al., 2013) [16]. In fact, the IFM results from the Trefftz method, where the interior field solutions are chosen to satisfy the Neumann boundary conditions. For simplicity, we call the IFM and the specific NFM as the method of field equations (MFEs). For the Neumann problems, there do not exist the degenerate scale problems, but the pseudo-singularity may be encountered if the numbers of the unknown coefficients and the collocation equations are exactly the same. To bypass this pseudo-singularity, the overdetermined system and the truncated singular value decomposition (TSVD) are solicited, to restore good stability. Interestingly, the first kind MFE can also be used for the Neumann problems. Numerical experiments with comparisons are carried out by two kinds of MFEs. The stability is made for two kinds of MFEs, and a theoretical argument is provided to verify the effectiveness of the overdetermined system. In summary, two kinds of MFEs are effective for solving the Neumann problems, and their numerical performances are excellent. -
##### Article: Stability Analysis for Singularly Perturbed Differential Equations by the Upwind Difference Scheme

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**ABSTRACT:**For solving singularly perturbed differential equations (SPDE), the upwind difference scheme (UDS) and the fitted difference method are chosen. The local refinements of grids are adopted in singular layers with the minimal meshspacing , where h is the maximal meshspacing, and a very small parameter. The traditional condition number in 2-norm is given by . For the infinitesimal in application, the huge Cond issues a dilemma regarding whether the numerical solutions by the UDS can be trusted. Although the UDS has been used for several decades, such a dilemma has not been clarified yet. The goal of this article is to clarify this dilemma. To this end, we solicit the effective condition number Cond_eff in Li et al. Numer Linear Algebra Appl 15 (2008), 575–690 Effective condition for Numerical Partial Different Equation, 2013, and develop a new actual condition number from the maximum principle. Both of them may offer much smaller bounds of the solution errors caused by perturbation, e.g., rounding, truncation, or discritization errors. We study the Dirichlet problems of SPDE by the UDS in a rectangle. When the Dirichlet boundary condition on the downwind side is homogeneous, we derive . When the entire Dirichlet boundary conditions are homogeneous, the extraordinary bound, , is achieved. Moreover, we derive the actual condition numbers as and for the homogeneous and the nonhomogeneous SPDE, respectively. Note that these bounds do not depend on ε; this is distinct from the traditional Cond. Based on the analysis of this article, the existing dilemma caused by Cond has been removed, to grant a good stability of the UDS for SPDE. © 2013 Wiley Periodicals, Inc. - [Show abstract] [Hide abstract]

**ABSTRACT:**For circular domains with circular holes, the null field method (NFM) is proposed by Chen and his co-researchers when solving boundary integral equation (BIE). The explicit algebraic equations of the NFM are recently derived in Li et al. (2012) [33], and their conservative schemes are proposed in Lee et al. (2013) [28]. However, even for the Dirichlet problem of Laplace׳s equation, there may exist a singularity of the original boundary integral equation (BIE) and/or its numerical algorithms such as the NFM. Such a singularity is called the degenerate scale problem due to special domain scales, and was studied in Christiansen (1975) [22]. Since to bypass the singularity is imperative for both theory and computation, the degenerate scale problem has been extensively discussed in the literature. An algorithm singularity means the singularity of the coefficient matrix of collocation methods, but we confine ourselves to the singularity caused by the degenerate scale problem. So far, for the algorithm singularity of the NFM of degenerate scales, no advanced analysis exists, although a preliminary discussion was given in Chen and Shen (2007) and Lee et al. (2013) 15 and 28. In this paper, all kinds of field nodes of degenerate scales leading to algorithm singularity are revealed in detail. To remove singularity of discrete matrices and to restore good stability, several effective techniques are proposed. Numerical experiments are carried out to verify the theoretical analysis made. Based on the analysis and computation in this paper, not only can the algorithm singularity of the NFM be bypassed, but also the highly accurate solutions with good stability may be achieved. - [Show abstract] [Hide abstract]

**ABSTRACT:**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. - [Show abstract] [Hide abstract]

**ABSTRACT:**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) , and . In Bogomolny (1985) [4] and Li (2009) [5] the error analysis of the MFS is given for Laplace’s 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. -
##### Conference Paper: The null field and interior field methods for Neumann problems of Laplace equations

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**ABSTRACT:**Recently, the null fields method (NFM) is proposed by Chen with his groups \cite{CS2009}. In NFM, the fundamental solutions with the field nodes outside of the solution domain are used in the Green formulas. The Fourier expansions of the known and the unknown boundary conditions on the circular boundaries are chosen, so that the explicit discrete collocation equations are easily contained by means of orthogonality of Fourier functions. Recently, a new interior field method (IFM) is proposed in \cite{HLLC2013, LLHL2013a}, which is the special case of the null field method (NFM) when the field nodes are located exactly on the domain boundary.This paper is devoted to Neumman problems of Laplace's equation by NFM and IFM. The IFM is derived by the Trefftz method, based on the harmonic functions satisfying the Neumann boundary conditions, called the second kind Field Equations (FE). In fact, the explicit collocation equations for Dirichlet problems can also be used for Neumman problem, called the first kind FE. Accuracy and stability are the most important criteria. Convergence of the solution error is optimal by two kinds FE. Stability is also our main goal, which is measured by the traditional condition number and the new effective condition number of discrete matrices. %The IFM is strongly recommenced in \cite{LLHL2013a} for applications. From stability the first kind FE is superior to the second FE. Based on our best knowledge, this is the first time to deal with Neumann problems of Laplace's equation by two kinds field equations of NFM and IFM, accompanied with analysis, computation and comparisons. - [Show abstract] [Hide abstract]

**ABSTRACT:**Recently, the null-field method (NFM) has been proposed by Chen and his co-researchers for solving boundary value problems involving circular domains with circular holes. The explicit algebraic equations of the NFM are derived in our recent paper [31]. However, even for the Dirichlet problem of Laplace's equation, when the logarithmic capacity (transfinite diameter) CΓ=1CΓ=1 is given, the solutions may not exist, or not unique if existing, to cause a singularity of the discrete algebraic equations. The non-uniqueness of the solutions of Dirichlet problems by the boundary integral equations is first reported in Christiansen [20] due to some special geometry, and then in 14 and 15 called the degenerate scale problems. In this paper, the new conservative schemes of NFM are proposed. The conservative schemes can always bypass the degenerate scale problems; though numerically it causes a severe instability. A new pseudo-singularity property is discovered that only the minimal singular value σminσmin of the discrete matrices is infinitesimal to cause the instability. To restore good stability of the conservative schemes, the over-determined systems and the truncated singular value decomposition (TSVD) are proposed. The over-determined systems are more favorable than TSVD due to simpler algorithms and slightly better performances in error and stability. More importantly, such numerical techniques can also be used to deal with all the degenerate scale problems of the original NFM in 11, 12 and 13 as well as the boundary element method (BEM). - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, the boundary errors are defined for the null-field method (NFM) to explore the convergence rates, and the condition numbers are derived for simple cases to explore numerical stability. The optimal convergence (or exponential) rates are discovered numerically. This paper is also devoted to seek better choice of locations for the field nodes of the fundamental solutions (FS) expansions. It is found that the location of field nodes Q does not affect much on convergence rates, but do have influence on stability. Let δδ denote the distance of Q to ∂S∂S. The larger δδ is chosen, the worse the instability of the NFM occurs. As a result, δ=0δ=0 (i.e., Q∈∂SQ∈∂S) is the best for stability. However, when δ>0δ>0, the errors are slightly smaller. Therefore, small δδ is a favorable choice for both high accuracy and good stability. This new discovery enhances the proper application of the NFM. - [Show abstract] [Hide abstract]

**ABSTRACT:**In [27], the effective condition number Cond_eff is developed for the linear least squares problem. In this paper, we extend the effective condition number for weighted linear least squares problem with both full rank and rank-deficient cases. We apply the effective condition number to the collocation Trefftz method (CTM) [29] for Laplace's equation with a crack singularity, to prove that Cond_eff =O(L) and Cond =O(L1/2(2)L), where L is the number of singular particular solutions used. The Cond grows exponentially as L increases, but Cond_eff is only O(L). The small effective condition number explains well the high accuracy of the TM solution, but the huge Cond cannot. - [Show abstract] [Hide abstract]

**ABSTRACT:**The new interior field method (IFM) is proposed, which is the special case of the null field method (NFM) when the field nodes are just located on the domain boundary. The IFM is simpler than the NFM, because only one formula of interior solutions is needed, compared with multiple formulas used in the NFM. The IFM is more advantageous in simplicity and application. Numerical experiments are provided to support the analysis. - [Show abstract] [Hide abstract]

**ABSTRACT:**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 Poisson's 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. - [Show abstract] [Hide abstract]

**ABSTRACT:**In our previous study [Huang et al., 2008, 2009, 2010 [21], [24] and [20]; Huang and Lu, 2004 [22] and [23]; Lu and Huang, 2000 [38]], we have proposed advanced (i.e., mechanical) quadrature methods (AQMs) for solving the boundary integral equations (BIEs) of the first kind. These methods have high accuracy O(h3), where h=max1⩽m⩽dhm and hm (m=1,…,d) are the mesh widths of the curved edge Γm. The algorithms are simple and easy to carry out, because the entries of discrete matrix are explicit without any singular integrals. Although the algorithms and error analysis of AQMs are discussed in Huang et al. (2008, 2009, 2010) [21], [24] and [20], Huang and Lu (2004) [22] and [23], Lu and Huang (2000) [38], there is a lack of systematic stability analysis. The first aim of this paper is to explore a new and systematic stability analysis of AQMs based on the condition number (Cond) and the effective condition number (Cond_eff) for the discrete matrix Kh. The challenging and difficult lower bound of the minimal eigenvalue is derived in detail for the discrete matrix of AQMs for a typical BIE of the first kind. We obtain Cond=O(hmin−1) and Cond_eff=O(hmin−1), where hmin=min1⩽m⩽dhm, to display excellent stability. Note that Cond_eff = O(Cond) is greatly distinct to the case of numerical partial differential equations (PDEs) in Li et al. (2007, 2008, 2009, 2010) [26], [31], [32], [33], [34], [35], [36] and [37], Li and Huang (2008) [27], [28], [29] and [30], Huang and Li (2006) [19] where Cond_eff is much smaller than Cond. The second aim of this paper is to explore intrinsic characteristics of Cond_eff, and to make a comparison with numerical PDEs. Numerical experiments are carried out for three models with smooth and singularity solutions, to support the analysis made. - [Show abstract] [Hide abstract]

**ABSTRACT:**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 © 2011 John Wiley & Sons, Ltd. - [Show abstract] [Hide abstract]

**ABSTRACT:**Consider the over-determined system Fx = b where F ∈ R m×n , m ≥ n and rank (F) = r ≤ n, the effective condition number is defined by Cond_eff = b σ r x , where the singular values of F are given as σ max = σ 1 ≥ σ 2 ≥ · · · ≥ σ r > 0 and σ r +1 = · · · = σ n = 0. For the general perturbed system (A+A)(x +x) = b+b Communicated by N. Yan. Partial results of this paper were represented at the Mini symposium on Collocation and Trefftz Method for the 7th 123 88 Z.-C. Li et al. involving both A and b, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz's problem by the collocation Trefftz methods (CTM). Motz's problem is the benchmark of singu-larity problems. We choose the general particular solutions v L = L k=0 d k (r R p) k+ 1 2 cos(k + 1 2)θ with a radius parameter R p . The CTM is used to seek the coefficients d i by satisfying the boundary conditions only. Based on the new effective condition num-ber, the optimal parameter R p = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of R p is misleading. Under the optimal choice R p = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error anal-ysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method. - [Show abstract] [Hide abstract]

**ABSTRACT:**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. KeywordsStability analysis-Condition number-Effective condition number-Method of fundamental solutions-Mixed boundary problem-Motz’s problem-Laplace’s equation Mathematics Subject Classification (2000)65N10-65N30 - [Show abstract] [Hide abstract]

**ABSTRACT:**In the Trefftz method (TM), the admissible functions satisfying the governing equation are chosen, then only the boundary conditions are dealt with. Both fundamental solutions (FS) and particular solutions (PS) satisfy the equation. The TM using FS leads to the method of fundamental solutions (MFS), and the TM using PS to the method of particular solutions (MPS). Since the MFS is one of TM, we may follow our recent book [20,21] to provide the algorithms and analysis. Since the MFS and the MPS are meshless, they have attracted a great attention of researchers. In this paper numerical experiments are provided to support the error analysis of MFS in Li [15] for Laplace's equation in annular shaped domains. More importantly, comparisons are made in analysis and computation for MFS and MPS. From accuracy and stability, the MPS is superior to the MFS, the same conclusion as given in Schaback [24]. The uniform FS is simpler and the algorithms of MFS are easier to carry out, so that the computational efforts using MFS are much saved. Since today, the manpower saving is the most important criterion for choosing numerical methods, the MFS is also beneficial to engineering applications. Hence, both MFS and MPS may serve as modern numerical methods for PDE. - [Show abstract] [Hide abstract]

**ABSTRACT:**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. - [Show abstract] [Hide abstract]

**ABSTRACT:**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 parameter-dependent 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. - [Show abstract] [Hide abstract]

**ABSTRACT:**For Laplace's equation and other homogeneous elliptic equations, when the particular and fundamental solutions can be found, we may choose their linear combination as the admissible functions, and obtain the expansion coeffi-cients by satisfying the boundary conditions only. This is known as the Trefftz method (TM) (or boundary approximation methods). Since the TM is a meshless method, it has drawn great attention of researchers in recent years, and Inter. Work-shops of TM and MFS (i.e., the method of fundamental solutions). A number of efficient algorithms, such the collocation algorithms, Lagrange multiplier methods, etc., have been developed in computation. However, there still exists a gap of con-vergence and errors between computation and theory. In this paper, convergence analysis and error estimates are explored for Laplace's equations with the solution u ∈ H k (k > 1 2), to achieve polynomial convergence rates. Such a basic theory is im-portant for TM and MFS and their further developments. Numerical experiments are provided to support the analysis and to display the significance of its applica-tions. - [Show abstract] [Hide abstract]

**ABSTRACT:**This is a continued study but at advanced levels of effective condition number in [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, J. Comput. Appl. Math. 198 (2007) 208–235; Z.C. Li, H.T. Huang, Effective condition number for numerical partial differential equations, Numer. Linear Algebra Appl. 15 (2008) 575–594] for stability analysis. To approximate Poisson's equation with singularity by the finite element method (FEM), the adaptive mesh refinements are an important and popular technique, by which, the FEM solutions with optimal convergence rates can be obtained. The local mesh refinements are essential to FEM for solving complicated problems with singularities, and they have been used for three decades. However, the traditional condition number is given by in Strang and Fix [G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973], where hmin is the minimal length of elements. Since hmin is infinitesimal near the singular points, Cond is huge. Such a dilemma can be bypassed by small effective condition number. In this paper, the bounds of the simplified effective condition number Cond_EE are derived as O(1), O(h−1.5) or O(h−0.5), where is the maximal length of elements. Evidently, Cond_EE is much smaller than Cond. The numerical experiments are carried out, to verify the stability analysis. Small effective condition numbers explain well the satisfactory FEM solutions obtained. This paper provides a stability justification for the adaptive mesh refinements used in FEM. Compared with [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, J. Comput. Appl. Math. 198 (2007) 208–235; Z.C. Li, H.T. Huang, Effective condition number for numerical partial differential equations, Numer. Linear Algebra Appl. 15 (2008) 575–594], the analysis in this paper is more difficult and challenging, its proof techniques are new and intriguing, and the results are more important and useful.

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