Trefftz, collocation, and other boundary methods—A comparison

Li, Zi‐Cai; Lu, Tzon‐Tzer; Huang, Hung‐Tsai; Cheng, Alexander H.‐D.

doi:10.1002/num.20159

Cited by 110 publications

(117 citation statements)

References 151 publications

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“…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. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 94%

“…Eqs. (45) -(47) are equivalent to (43) with the central rule, see [14,15]. When other rules such as the Gaussian rule are chosen, the collocation equations (45) − (47) are modified as …”

Section: Collocation Trefftz Methods For Motz's Problemmentioning

confidence: 99%

“…The equation (51) is the over-determined system, and the equation (41) is its normal equation. Solving (51) directly is more advantageous for better stability, see [14,15].…”

Section: Collocation Trefftz Methods For Motz's Problemmentioning

confidence: 99%

“…The collocation Trefftz method (CTM) has been proved to most effective method among several boundary method in [14,15]. However, only the error analysis is made, but no stability exists so far.…”

Section: Introductionmentioning

confidence: 99%

“…Since the convergence radius r = 2 of (8) is proved in [21], Eq. (8) has been used for Motz's problem by many researchers, see [9,10,13,14,15,16,21].…”

Section: Introductionmentioning

confidence: 99%

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Effective condition number and its applications

Huang

Chen

et al. 2010

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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 σmax = σ1 ≥ σ2 ≥ ... ≥ σr > 0 and σr+1 = ... = σn = 0. For the general perturbed system (A + ∆A)(x + ∆x) = b + ∆b 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 singularity problems. We 1The CTM is used to seek the coefficients Di and di by satisfying the boundary conditions only. Based on the new effective condition number, 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 analysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.

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Section: Introductionmentioning

confidence: 94%

“…Eqs. (45) -(47) are equivalent to (43) with the central rule, see [14,15]. When other rules such as the Gaussian rule are chosen, the collocation equations (45) − (47) are modified as …”

Section: Collocation Trefftz Methods For Motz's Problemmentioning

confidence: 99%

“…The equation (51) is the over-determined system, and the equation (41) is its normal equation. Solving (51) directly is more advantageous for better stability, see [14,15].…”

Section: Collocation Trefftz Methods For Motz's Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Since the convergence radius r = 2 of (8) is proved in [21], Eq. (8) has been used for Motz's problem by many researchers, see [9,10,13,14,15,16,21].…”

Section: Introductionmentioning

confidence: 99%

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Effective condition number and its applications

Huang

Chen

et al. 2010

Computing

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Effective condition number for numerical partial differential equations

Huang

2008

Numerical Linear Algebra App

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In this paper, the new computational formulas are derived for the effective condition number Cond eff, and the new error bounds involved in both Cond and Cond eff are developed. A theoretical analysis is provided to support some conclusions in Banoczi et al. (SIAM J. Sci. Comput. 1998; 20:203-227). For the linear algebraic equations solved by the Gaussian elimination or the QR factorization (QR), the direction of the right-hand vector is insignificant for the solution errors, but such a conclusion is invalid for the finite difference method for Poisson's equation. The effective condition number is important to the numerical partial differential equations, because the discretization errors are dominant. EFFECTIVE CONDITION NUMBER FOR NUMERICAL PDE 577 where = A / n <1. In this paper the following new computational formula is derived: Dx x ‡ The exact solution of (5) may not exist. Then, the solutions are regarded as the least-squares solutions as min x∈R n Fx−b . This displays an important role of Cond eff for the stability in the numerical PDE due to the dominant discretization errors. Note that N 32 and then h 1 32 1 145 , Equation (87) coincides with Proposition 4.2 and the analysis made above. † † The conclusions in [2] and this paper are also valid if under a relaxed assumption: S f w k O(1), where w k is the eigenfunction of the leading eigenvalue k of (107).

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

Li¹,

Lu²,

Wei³

2008

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The paper presents the new stability analysis for the collocation Trefftz method (CTM) for biharmonic equations, based on the new effective condition number Cond eff. The Trefftz method is a special spectral method with the particular solutions as admissible functions, and it has been widely used in engineering. Three crack models in Li et al. (Eng. Anal. Boundary Elements 2004; 28:79-96; Trefftz and Collocation Methods. WIT Publishers: Southampton, Boston, 2008) are considered, and the bounds of Cond eff and the traditional condition number Cond are derived, to give the polynomial and the exponential growth rates, respectively. The stability analysis explains well the numerical experiments. Hence, the new Cond eff is more advantageous than Cond. Besides since the bounds of Cond eff and Cond involve the estimation of the minimal singular value min of the discrete matrix F, and since the estimation of min is challenging and difficult, the proof for lower bounds of min in this paper is important and intriguing.where u represents the transverse displacement in the thin plate theory, u = * 2 u/*x 2 +* 2 u/*y 2 , and the solution domain is the rectangle: S = {(x, y)|−1

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Trefftz, collocation, and other boundary methods—A comparison

Cited by 110 publications

References 151 publications

Effective condition number and its applications

Effective condition number and its applications

Effective condition number for numerical partial differential equations

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

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