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Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S . The source function q ln | P Q * ¯ | is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by q ln | P Q * ¯ | as the source node Q * is located near the boundary Γ ( = ∂ S ) . So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q * located near Γ . In this paper, an additional FS as, d 0 ln | P Q 0 ¯ | , is added to the original source nodes in the traditional MFS, and the point charge d 0 ( = q ) and the source node Q 0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q * is located inside S but near Γ , both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function q ln | P Q * ¯ | as the source node Q * is located near Γ .

Consider Laplace’s equation in a bounded simply-connected domain S, and use the method of fundamental solutions (MFS). The error and stability analysis is made for circular/elliptic pseudo-boundaries in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020), and polynomial convergence rates and exponential growth rates of the condition number (Cond) are obtained. General pseudo-boundaries are suggested for more complicated solution domains in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020, Section 5). Since the ill-conditioning is severe, the success in computation by the MFS mainly depends on stability. This paper is devoted to stability analysis for smooth closed pseudo-boundaries of source nodes. Bounds of the Cond are derived, and exponential growth rates are also obtained. This paper is the first time to explore stability analysis of the MFS for non-circular/non-elliptic pseudo-boundaries. Circulant matrices are often employed for stability analysis of the MFS; but the stability analysis in this paper is explored based on new techniques without using circulant matrices as in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020). To pursue better pseudo-boundaries, the sensitivity index is proposed from growth/convergence rates of stability via accuracy. Better pseudo-boundaries in the MFS can be found by trial computations, to develop the study in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020) for the selection of pseudo-boundaries. For highly smooth and singular solutions, better pseudo-boundaries are different; an analysis of the sensitivity index is explored. Circular/elliptic pseudo-boundaries are optimal for highly smooth solutions, but not for singular solutions. In this paper, amoeba-like domains are chosen in computation. Several useful types of pseudo-boundaries are developed and their algorithms are simple without using nonlinear solutions. For singular solutions, numerical comparisons are made for different pseudo-boundaries via the sensitivity index.

The goals of this paper are twofold: selection of pseudo-boundaries for sources nodes in the method of fundamental solutions (MFS), and comparisons of the MFS, the method of particular solutions (MPS) and the MFS-QR of Antunes. To pursue better pseudo-boundaries, we provide new estimates of the condition number (Cond) by the MFS for arbitrary pseudo-boundaries, and propose a new sensitivity index of stability via accuracy. Numerical experiments and comparisons are carried out to verify the analysis made. For five-pedal-flower-like domains, numerical comparisons are made by the sensitivity index. Circular pseudo-boundaries are optimal for highly smooth solutions, but the pseudo-boundaries near the domain boundary may be better for singular solutions. In this paper the gap has been shortened between theoretical analysis and numerical computation of the MFS, to provide some guidance for users. This is the first goal of this paper. The second goal is to compare the MFS, the MPS and the MFS-QR. Characteristics of the MFS-QR are explored. The new basis functions of the MFS-QR are the very particular solutions (PS), and the MFS-QR may be regarded as a special MPS. The MFS-QR is not a variant of the MFS but a variant of the MPS. The MFS-QR also plays a role in bridging from the MFS to the MPS. Both the MFS and the MPS can also be recognized as twins via the MFS-QR in the Trefftz family. The comparisons in this paper are more comprehensive.

For Laplace's equation in a bounded simply-connected domain Ω in 3D, the method of fundamental solutions (MFS) is studied in this paper. Although some numerical computations can be found in Chen et al. [10] , the theoretical analysis is much behind (Li [23] only for unit sphere Ω). Our efforts are devoted to exploring a strict error analysis of the MFS. The error bounds are derived, and the optimal polynomial convergence rates can be achieved. Numerical experiments are carried out to support the analysis made, and several useful locations of source nodes are investigated numerically. The analysis in this paper may lay a theoretical basis of the MFS for 3D problems, as Bogomolny [8] and [24] for 2D problems. Besides, the method of particular solutions (MPS) in [26] is also studied by using the spherical harmonic functions (SHF). The optimal polynomial convergence rates and the exponential growth of condition number (Cond) are obtained. The source nodes are located based on the abscissas of quadrature rules on surfaces; they are “grid-like”. Since most of 3D problems, in reality, can not be simplified to 2D problems, and since the MFS has more advantages for 3D problems in algorithm simplicity and wide application, the study in this paper is essential and important to the MFS.

For the exterior Dirichlet problems (EDP) of the Helmholtz equation in 2D, when the Sommerfeld radiation condition is satisfied, there always exists the unique solution. Denote the unbounded domain S ∞ outside of a bounded simply-connected domain, where the interior boundary Γ i n is non-circular. The standard MFS is first studied by using the complex Hankel function H 0 ( 1 ) ( k r ) as the fundamental solutions (FS). However, some solutions can not be obtained correctly (e.g., the spurious eigenvalues called). This paper is devoted to explore the spurious eigenvalues and their removals. For the non-circular interior boundary Γ i n , the error analysis is briefly made for the standard MFS, and some guidance is provided to bypass the spurious eigenvalues. Denote two nodes: P = ( ρ , θ ) and Q = ( R , ϕ ) , and r = | P Q ‾ | = R 2 + ρ 2 − 2 R ρ cos ⁡ ( θ − ϕ ) . To completely eliminate all spurious eigenvalues, we solicit the new combined FS: ( ∂ ∂ R ± i k ) H 0 ( 1 ) ( k r ) with i = − 1 , and propose the new modified MFS. The new algorithms are simple, and the strict analysis may be made. The bounds of errors are derived, and the polynomial convergence rates can be achieved. The bounds of condition numbers are derived for circular Γ ∘ i n only, to display the exponential growth via the number of the new FS used. Numerical experiments are carried to support the analysis made. This paper is the first time for the MFS to explore the spurious eigenvalues and their removals, accompanied with strict analysis of errors and stability.

In this paper, we study the Helmholtz equation by the method of fundamental solutions (MFS) using Bessel and Neumann functions. The bounds of errors are derived for bounded simply-connected domains, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Note that by using Bessel functions, the radius R of the source nodes is not necessarily to be larger than the maximal radius r max of the solution domain. This is against the well-known rule: r max R for the MFS. Numerical experiments are carried out, to support the analysis and conclusions made. This is the first novelty in this paper. The error analysis for the Helmholtz equation is more complicated than that for the modified Helmholtz equation in [35] , since the Bessel functions J n ( x ) have infinite zeros. We consider the curial and degenerate cases when J n ( k R ) ≈ 0 and J n ( k ρ ) ≈ 0 . There exist few reports for the analysis for such a degeneracy (e.g., Li [21] ). The error bounds are also explored for bounded simply-connected domains. The second novelty of this paper is for the analysis of the MFS in degeneracy. For the MFS using Neumann functions, the rule of the MFS, r max R , must obey. This paper is the first time to discover that the MFS using Bessel and Neumann functions suffer from the spurious eigenvalues. The spurious eigenvalues are not the true eigenvalues of the corresponding eigenvalue problems, but the correct solutions can not be obtained due to either algorithm singularity or divergence of numerical solutions. For the method of particular solutions (MPS) in [26] , however, the source nodes disappear. In this paper, we will briefly provide the analysis of the MFS using Neumann functions, and the polynomial convergence can be achieved for bounded simple-connected domains. The analysis of the MFS using Neumann functions and numerical comparisons for different methods are the third contribution in this paper.

The method of fundamental solutions (MFS) was first used by Kupradze in 1963 [21] . Since then, there have appeared numerous reports of the MFS. Most of the existing analysis for the MFS are confined to Dirichlet problems on disk domains. It seems to exist no analysis for Neumann problems. This paper is devoted to Neumann problems in non-disk domains, and the new stability analysis and the error analysis are made. The bounds for both condition numbers and errors are derived in detail. The optimal convergence rates in L 2 and H 1 norms in S are achieved, and the condition number grows exponentially as the number of fundamental functions increases. To reduce the huge condition numbers, the truncated singular value decomposition (TSVD) may be solicited. Numerical experiments are provided to support the analysis made. The analysis for Neumann problems in this paper is intriguing due to its distinct features.

The method of fundamental solutions (MFS) was first proposed by Kupradze in 1963. Since then, the MFS has been extensively applied for solving various kinds of problems in science and engineering. However, few theoretical works have been reported in the literature. In this paper, we devote our work to the error analysis and stability of the MFS for the case of the modified Helmholtz equation. For disk domains, a convergence analysis of the MFS was provided by Li (J. Comput. Appl. Math., 159:113125, 2004) for solving the modified Helmholtz equation. In this paper, the error bounds of the MFS for bounded and simply connected domains are derived for smooth solutions of the modified Helmholtz equation. The exponential convergence rates can be achieved for analytic solutions. The bounds of condition numbers of the MFS are derived for both disk domains and the bounded and simply connected domains, to give the exponential growth via the number of fundamental solutions used. Numerical experiments are carried out to support the theoretical analysis. Moreover, the analysis of this paper is applied to parabolic equations, and some reviews of proof techniques and the analytical characteristics of the MFS are addressed.

The null field method (NFM) was proposed by Chen and his co-researchers, and has been discussed in numerous papers, see Chen et al. (2007 [11] ), Chen et al. (2002 [12] ), Chen et al. (2001 [13] ), and Chen and Shen (2009 [14] ). The further developments of the NFM have been made in our recent papers (Huang et al., 2013 [21] ; Lee et al., 2013 [23] ; Lee et al., 2014 [25] ; Li et al., 2012 [29] ). In this paper, the interior field method (IFM) is proposed, which offers the best performance of the NFM when the field nodes are located exactly on the domain boundary. The algorithms of the IFM are much simpler than those of the NFM, because only one formula of the interior solutions is needed, compared with multiple formulas in the NFM. Since the IFM can be classified into the family of the Trefftz method (Li et al., 2008 [31] ), a new error analysis of the IFM and the collocation IFM (CIFM) can be explored, to achieve the optimal convergence rates. Moreover, new proof techniques for aliasing errors in this paper are straightforward, heuristic and much easier to follow, because of direct derivations from trigonometric functions, which are distinct from Canuto and Quarteroni (1982 [8] ), Canuto et al. (2006 [9] ), and Kreiss and Oliger (1979 [22] ). Based on this paper, the IFM and the CIFM may be recommended for those problems solvable by the NFM before.

The boundary knot method (BKM) is applied to the Helmholtz equation in 2D bounded simply-connected domains, to show high accuracy of the solutions obtained by not many fundamental solutions (FS) used. When the Bessel function is chosen as the FS, the optimal polynomial convergence rates are obtained for disk domains. Moreover, the bounds of condition number (Cond) are derived for disk domains, to show a super-exponential growth via the number of FS used. The super-exponential growth of Cond is new and intriguing in the numerical methods for partial differential equations (PDE). It is also imperative for the BKM that good numerical solutions may be achieved by balancing accuracy and instability. Numerical experiments are carried out to support the analysis made. Comparisons between the BKM and the MFS are also made.

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.

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] , [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.

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 hmin = O(he), where h is the maximal meshspacing, and e( 1) a very small parameter. The traditional condition number in 2-norm is given by Cond = O(h−2e−1). For the infinitesimal e(≤ 10−8) 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 Cond_eff = O( 1 √ h ). When the entire Dirichlet boundary conditions are homogeneous, the extraordinary bound, Cond_eff = O(1), is achieved. Moreover, we derive the actual condition numbers as Cond_actual = O( 1 √ h ) and Cond_actual = O( 1 h ) for the homogeneous and the nonhomogeneous SPDE, respectively. Note that these bounds do not depend on e; 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. Numer Methods Partial Differential Eq 000: 000–000, 2013

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 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.

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) C Γ = 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] , [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 σ min 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] , [13] as well as the boundary element method (BEM).

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.

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 δ denote the distance of Q to ∂ S . The larger δ is chosen, the worse the instability of the NFM occurs. As a result, δ = 0 (i.e., Q ∈ ∂ S ) is the best for stability. However, when δ > 0 , the errors are slightly smaller. Therefore, small δ is a favorable choice for both high accuracy and good stability. This new discovery enhances the proper application of the NFM.

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 ( L 1 / 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.

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 rea...

In our previous study [Huang et al., 2008, 2009, 2010 [21] , [24] , [20] ; Huang and Lu, 2004 [22] , [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 = max 1 ⩽ m ⩽ d h m and hm (m=1,…,d) are the mesh widths of the curved edge Γ m . 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] , [20] , Huang and Lu (2004) [22] , [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 h min = min 1 ⩽ m ⩽ d h m , 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] , [37] , Li and Huang (2008) [27] , [28] , [29] , [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.

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.

Consider the over-determined system Fx = b where F ∈ Rm × n, m ≥ n and rank (F) = r ≤ n, the effective condition number is defined by Cond_eff = ||b||/σ1||x||, where the singular values of F are given as σmax = σ1 ≥ σ1 ≥...≥ σr > 0 and σr+1 = ... = σn = 0. For the general perturbed system (A + ΔA)(x+ Δx) = b+ Δb involving both ΔA and Δb, 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 singularity problems. We choose the general particular solutions vL = Σk=0L dk(r/Rp)k+1/2 cos(k + 1/2)θ with a radius parameter Rp. The CTM is used to seek the coefficients di by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter Rp = 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 Rp is misleading. Under the optimal choice Rp = 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 analysis in [14, 15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.

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.

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.

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. © 2009 Elsevier B.V. All rights reserved.

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.

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 Cond=O(h"m"i"n^-^2) in Strang and Fix [G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973], where h"m"i"n is the minimal length of elements. Since h"m"i"n 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 h(

We study Adini’s elements for nonlinear Schrodinger equations (NLS) defined in a square box with periodic boundary conditions. First we transform the time-dependent NLS to a time-independent stationary state equation, which is a nonlinear eigenvalue problem (NEP). A predictor–corrector continuation method is exploited to trace solution curves of the NEP. We are concerned with energy levels and superfluid densities of the NLS. We analyze superconvergence of the Adini elements for the linear Schrodinger equation defined in the unit square. The optimal convergence rate Oðh 6 Þ is obtained for quasiuniform ele

In this paper, the stability study is further made for three kinds of collocation methods. (I) The general collocation Trefftz methods (GCTMs) using piecewise particular solutions, while in Li et al. [Stability analysis for the penalty plus hybrid and the direct Trefftz methods for singularity problems. Eng Anal Boundary Elem 2004;31:163–75] the collocation Trefftz methods (CTMs) using the uniform particular solutions in the entire domain. (II) The method of fundamental solutions (MFSs), i.e., the CTM using the fundamental solution (FS). (III) The collocation method using the radial basis functions (CM-RBFs). In this paper, the new bounds of the traditional condition number Cond and the effective condition number Cond_eff are derived, and comparisons are made to confirm that the effective condition number is a better criterion for stability analysis. The main results are as follows. (1) For Motz's problem by GCTM, Cond _ eff = O ( N ) and Cond = O ( a - N ) ( 0 a 1 ) are obtained by a suitable scaling, where N is the number of admissible functions used. (2) For MFS and CM-RBF, both Cond and Cond_eff are exponential with respect to N , but Cond_eff is much smaller than Cond. (3) For MFS and CM-RBF, the expansion coefficients of FS and radial basis functions may be oscillatingly large, to cause a severe subtraction cancellation in the final solutions. Different behaviours of Cond_eff compared to Cond are explored for different collocation methods in this paper, to provide a comprehensive and intrinsic nature of Cond_eff.

The simplified hybrid Trefftz method was first proposed in Trefftz [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7] in 1926 for solving Laplace's equation, where the harmonic functions are chosen as admissible functions, and their linear combination is sought to satisfy the boundary conditions. The error analysis of the hybrid TM is provided in [Li ZC, Chen YL, Georgiou GG, Xenohontos C. Special boundary approximation methods for Laplace equation problems with boundary singularities—applications to the Motz problem. Int Comput Math Appl 2006;51:115–42; Li ZC, Lu TT, Hu HY, Cheng AH-D. Trefftz and collocation methods. Southampton: WIT Publishers; 2007], but no stability analysis exists so far. Also the simplified hybrid techniques have been applied for the TM to couple with the finite element method (FEM) in our previous study and only the error analysis has been made. Hence, the stability analysis is important for the simplified hybrid TM. In this paper, we will apply the effective condition number Cond _ eff . For the original hybrid TM [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7], uniform particular solutions satisfying the governed equation (e.g., the harmonic functions satisfying Laplace's equation) were chosen. In general, piecewise particular solutions can be used for wide application of the hybrid TM, and the interior continuity conditions may be dealt with by hybrid techniques. Their algorithms and error analysis are provided in [Huang HT, Li ZC, Herrera I. Coupling techniques of Trefftz methods. Technical Report, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan; 2006] without stability analysis. In this paper, two cases of the simplified hybrid TM are considered: Case I: uniform particular solutions used, and Case II: piecewise particular solutions used. It is proved that both Cond _ eff and Cond grow exponentially, with respect to the number of particular solutions used. In Case I, Cond is huge and Cond _ eff is significantly smaller than Cond; but in Case II, Cond is moderately large, and Cond _ eff is significantly smaller than Cond. Hence the ill-conditioning of the simplified hybrid TM for Case II is not severe. Such theoretical results have been validated by the numerical experiments. The study of Cond _ eff in this paper provides a complete and comprehensive knowledge of the simplified hybrid TM.

For biharmonic equations, the Hermite finite element methods (FEM) are chosen, to seek their approximate solutions. The linear algebraic equations Ax=b are obtained from the Hermite FEM, where the matrix A is symmetric and positive definite, and x and b are the unknown and known vectors, respectively. It is well known that Cond=@l"m"a"x@l"m"i"n, and @l"m"a"x and @l"m"i"n are the maximal and minimal eigenvalues of the stiffness matrix A, respectively. The bounds of Cond are derived to be O(h^-^4). Note that when h is small, the values of Cond (=O(h^-^4)) are huge, to indicate a severe instability, compared with Cond =O(h^-^2) for Poisson's equation by the FEM. In fact, for specific application problems, the instability is not so severe, a new effective condition number is defined by Cond_eff =@?b@?@?x@?@l"m"i"n in [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, Comput. Appl. Math. 198 (2007) 208-235], to provide a better upper bound of perturbation errors. It is proven that Cond_eff =O(h^-^3^.^5) for general cases, which is smaller than the traditional Cond. However, for special cases, the Cond_eff could be much smaller. For instant, for the homogeneous boundary conditions of biharmonic equations, Cond_eff =O(1), can be reached as h diminishes. This is astonishing, against our intuition from the knowledge of the Cond. From the analysis in this paper, the traditional Cond may mislead the stability analysis for practical computation of engineering problems.

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 Wilson's elements. The expansions indicate that Q 1 and Q 1 rot provide upper bounds of the eigenvalues, and that EQ 1 rot and Wilson's 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.

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 Poisson's 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.

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of two-coupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction–diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.

The finite difference method (FDM) is used for Dirichlet problems of Poisson’s equation, and the Dirichlet boundary condition is dealt with by boundary penalty techniques. Two penalty techniques, penalty-integrals and penalty-collocations (i.e., fixing), are proposed in this paper. The error bounds in the discrete H 1 norm and the infinite norms are derived. The stability analysis is based on the new effective condition number (Cond_eff) but not on the traditional condition number (Cond). The bounds of Cond_eff are explored to display that both the penalty-integral and the penalty-collocation techniques have good stability; the huge Cond is misleading. Since the penalty-collocation technique (i.e., the fixing technique) is simpler, it has been applied in engineering problem for a long time. It is worthy to point out that this paper is the first time to provide a theoretical justification for such a popular penalty-collocation (fixing) technique. Hence the penalty-collocation is recommended for dealing with the complicated constraint conditions such as the clamped and the simply support boundary conditions of biharmonic equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 24: 972–990, 2008

In this paper, we apply the effective condition number in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83] to the penalty Trefftz methods (TMs) with high order elements such as Adini's elements for Poisson's equations with singularities. The best superconvergence rates O ( h 3.5 ) in H 1 norm can be reached, where h is the maximal boundary length of elements. For the penalty and the penalty plus hybrid Trefftz methods, the bounds of effective condition numbers retain the same. However, for the penalty TM combination, not only are the algorithms simple, but also the numerical solutions are more accurate. Hence we prefer the penalty TM combination to the penalty plus hybrid TM combination. Such a recommendation is based on the effective condition number, but not on the traditional condition number, based on which the penalty TM combination was declined in Li and Yan [Global superconvergence for blending surfaces by boundary penalty plus hybrid FEMs for biharmonic equations. Appl Numer Math 2001;39:61–85]. The new stability analysis in this paper by means of effective condition number may re-evaluate the existing numerical methods; this paper reports the results of Motz's problem. Note that in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83], no stability analysis was provided.

For solving the linear algebraic equations Ax = b , the new stability analysis is made based on the effective condition number Cond_eff. The Cond_eff may provide a better upper bound of relative errors of x resulting from the rounding errors of b , than the traditional condition number Cond, which with too large value is, in many times, misleading. In this paper, we apply the effective condition number to the Trefftz methods (TMs) for Poisson's equations with singularities. Two TMs, such as the penalty plus hybrid TM and the Lagrange multiple (i.e., direct) TM, are studied. We focus on the stability analysis of the solutions when the optimal superconvergence is achieved. When solving Motz's problem, the benchmark of singularity problems, by the penalty plus hybrid TM, we have derived that Cond _ eff = O ( N ( 2 ) N ) and Cond = O ( N 2 2 N ) , where N is the number of the singular particular functions used. For solving Motz's problem by the direct TM, we have derived that Cond / Cond _ eff = O ( N 2 N ) . Numerical experiments are provided to verify the stability analysis made. In summary, the two TMs are efficient, but the penalty plus hybrid TM is more recommended, due to simplicity of algorithms without extra-variables and less limitations in applications.

For solving the linear algebraic equations Ax=b with the symmetric and positive definite matrix A, from elliptic equations, the traditional condition number in the 2-norm is defined by Cond.=λ1/λn, where λ1 and λn are the maximal and minimal eigenvalues of the matrix A, respectively. The condition number is used to provide the bounds of the relative errors from the perturbation of both A and b. Such a Cond. can only be reached by the worst situation of all rounding errors and all b. For the given b, the true relative errors may be smaller, or even much smaller than the Cond., which is called the effective condition number in Chan and Foulser [Effectively well-conditioned linear systems, SIAM J. Sci. Statist. Comput. 9 (1988) 963–969] and Christiansen and Hansen [The effective condition number applied to error analysis of certain boundary collocation methods, J. Comput. Appl. Math. 54(1) (1994) 15–36]. In this paper, we propose the new computational formulas for effective condition number Cond_eff, and define the new simplified effective condition number Cond_E. For the latter, we only need the eigenvector corresponding to the minimal eigenvalue of A, which can be easily obtained by the inverse power method. In this paper, we also apply the effective condition number for the finite difference method for Poisson's equation. The difference grids are not supposed to be quasiuniform. Under a non-orthogonality assumption, the effective condition number is proven to be O(1) for the homogeneous boundary conditions. Such a result is extraordinary, compared with the traditional Cond.=O(hmin-2), where hmin is the minimal meshspacing of the difference grids used. For the non-homogeneous Neumann and Dirichlet boundary conditions, the effective condition number is proven to be O(h-1/2) and O(h-1/2hmin-1), respectively, where h is the maximal meshspacing of the difference grids. Numerical experiments are carried out to verify the analysis made.

In this paper, we report some new ultraconvergence results of biquadratic Lagrange elements for the Dirichlet problem of Poisson's equation, -@Du=f. The point-line-area interpolant in [V. Girault, P.A. Raviart, A Finite Element Methods for Navier-Stokes Equation, Theory and Algorithms, Springer, 1986] is chosen in this paper, instead of the traditional pure point interpolant in [P.G. Ciarlet, Basic error estimates for elliptic problems, in: P.G. Ciarlet, J.L. Lions (Eds.), Finite Element Methods, Part 1, North-Holland, Amsterdam, 1991, pp. 17-351]. Suppose that the solution is smooth enough, by means of an a posteriori interpolant, the ultraconvergence O(h^4) in H^1 norm is proved for uniform rectangles @?"i"j, and the higher ultraconvergence O(h^6^-^@?) in H^@?(@?=0,1) norm under the special case of uniform squares @?"i"j and f"x"x"y"y=0. Even when f"x"x"y"y 0, we propose two techniques: (1) the Richardson extrapolation method and (2) the correction method, to retain the same higher ultraconvergence results. Moreover, the ultraconvergence O(h^6^-^@?|lnh|) is also proved for @?(@?=0,1) order infinite norms. In this paper, the numerical experiments are provided to validate all the ultraconvergence results made. Note that the new ultraconvergence results under the special case are three order higher than the optimal convergence rate in [P.G. Ciarlet, Basic error estimates for elliptic problems, in: P.G. Ciarlet, J.L. Lions (Eds.), Finite Element Methods, Part 1, North-Holland, Amsterdam, 1991, pp. 17-351], and one order than that in [Q. Lin, N. Yan, A. Zhou, A rectangle test for interpolated finite elements, in: Proc. Sys. Sci. and Sys. Engrg., Great Wall Culture Publishers, Hong Kong, 1991, pp. 217-229].

The new results of this paper are twofold. (a) For solving the linear algebraic equations AxZb resulting from elliptic equations, we propose the new simplified effective condition number defined by Cond_EZ jjbjj= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjjbjj 2 Kb 2=Cond: 2 ÞCb 2 p , where bnZu Tb, un is the eigenvector of the minimal eigenvalue of matrix A, and Cond. is the traditional condition number. The Cond_E may provide a better upper bound of relative errors of x resulting from the perturbation of A and b. (b) To solve the Poisson equation with singularities, the penalty plus hybrid Trefftz method coupled with high order finite element methods (FEM) is used. New superconvergence being the best or almost the best is proven, and verified by numerical experiments for Motz’s problem. Compared with the hybrid techniques in Li and Huang [Li ZC, Huang HT. Global superconvergence of simplified hybrid combinations of the Ritz-Galerkin and FEMs of elliptic equations with singularities. II. Lagrange elements and Adini’s elements. Appl Numer Math 43;2002:253‐73.], the accuracy of the solution is higher, and the numerical stability is fairly worse, based on the numerical values of the computed Cond_E. Hence, the new comparisons are made for solving Motz’s problem, to indicate that the penalty plus hybrid Trefftz combination is superior to the hybrid Trefftz combination in Li and Huang [Li ZC, Huang HT. Global superconvergence of simplified hybrid combinations of the Ritz-Galerkin and FEMs of elliptic equations with singularities. II. Lagrange elements and Adini’s elements. Appl Numer Math 43;2002:253‐73.]. This paper integrates both the error and the stability analyses, to show a significance of the new simplified effective condition number. q 2005 Elsevier Ltd. All rights reserved.

A landing curve of an airplane is a smooth curve described by three functions x(s), y(s), and z(s) which are governed by a system of linear ordinary differential equations (ODEs) with certain boundary conditions. The separation techniques are proposed in this paper first for the landing curves of the airplane to the airport. The separation techniques are particularly significant for nonlinear boundary conditions. When a space ship is approaching a shuttle station, the landing curve is ending at the surface of the station. Since the station surface is curved, the ending boundary conditions are nonlinear. This nontrivial nonlinear landing problem can be easily solved by the separation techniques not only for great reduction of computation complexity but also for facile nonlinear error analysis. The error analysis in this paper is made to derive three significant results.(1)The exact blending curves by the separation techniques with the fundamental solutions. (2)The errors O(h^1^/^2) and O(h^5^/^2) in H^2 norms of the blending curves by the separation techniques with the cubic Hermite solutions and their a posteriori interpolant, respectively. (3)The errors O(h^2^.^5) in H^2 norms by those with the fifth-order Hermite method for general cases. Numerical experiments for nonlinear boundary conditions are provided in this paper to display the efficiency of the separation techniques, and to verify the error analysis made.

In this paper, we report some new discoveries of Adini's elements for Poisson's equation in error estimates, stability analysis and global superconvergence. It is well known that the optimal convergence rate ||u - uh||I = O(h3|u|4) can be obtained, where uh and u are the Adini's solution and the true solution, respectively. In this paper, for all kinds of boundary conditions of Poisson's equations, the supercloseness ||uIA - uh||I = O(h3.5||u||5) can be obtained for uniform rectangles □ij, where uIA is the Adini's interpolation of the true solution u. Moreover, for the Neumann problems of Poisson's equation, new treatments adding the explicit natural constraints (un)ij = gij on the boundary are proposed to yield the Adini's solution uh* having supercloseness ||uIA - uh*||I = O(h4||u||5). Hence, the global superconvergence ||u-Π5 uh*||I = O(h4||u||5) can be achieved, where Π5uh* is an a posteriori interpolant of polynomials with order five based on the obtained solution uh*. New basic estimates of errors are derived for Adini's elements. Numerical experiments in this paper are also provided to verify the supercloseness and superconvergences, O(h3.5) and O(h4), and the standard condition number O(h-2). It is worthy pointing out that for the Neumann problems on rectangular domains, the traditional finite element method is not as good as the newly proposed method interpolating the Neumann condition in this paper. Not only is the new method more accurate, but also economical in computation, as the discrete system has less unknowns.