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

Abstract

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<x<1, 0<y<1}. In this paper, we study three crack models of singularity problems in [1]. In Figure 1, the section O D represents an interior crack under the clamped condition u = u n = 0. The symmetric conditions, u n = u nnn = 0 on O A∪ BC ∪C D, are required. Here n is the outward normal direction to the boundary *S. On AB, when the clamped conditions are provided, we propose the biharmonic boundary value problem with the following conditions, called Model I in this paper, EFFECTIVE CONDITION NUMBER OF TREFFTZ METHODS ‡ The terminology 'energy' is often used for the least-squares deviation from the exact solution both in S and on *S, see Chien [12]. § The weights w i are chosen to balance different errors terms of (18), and w i = 1/(N +1) i are derived from some simple domains.

The Gravitational-wave physics II: Progress

“…For a general spacetime without symmetry, we note that the generalized harmonic formalism of Einstein equations is Hamilton-Jacobi-like [739]. Accordingly, we apply the finite element method to the generalized harmonic formalism of Einstein equations based on the Parallel Hierarchical Grid (PHG) library [733,740,741]. Since the continuous Galerkin module is more well developed than the discontinuous Galerkin module in PHG library, we used continuous Galerkin instead of discontinuous Galerkin in ref.…”