Hsin-Yun Hu

The collocation Trefftz method for biharmonic equations with crack singularities

Abstract The purpose of this paper is to extend the boundary approximation method proposed by Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], i.e. the collocation Trefftz method called in this paper, for biharmonic equations with singularities. First, this paper derives the Green formulas for biharmonic equations on bounded domains with a non-smooth boundary, and corner terms are developed. The Green formulas are important to provide all the exterior and interior boundary conditions which will be used in the collocation Trefftz method. Second, this paper proposes three crack models (called Models I, II and III), and the collocation Trefftz method provides their most accurate solutions. In fact, Models I and II resemble Motzs problem in Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], and Model III with all the clamped boundary conditions originated from Schiff et al. [The mathematics of finite elements and applications III, 1979]. Moreover, effects on d 1 of different boundary conditions are investigated, and a brief analysis of error bounds for the collocation Trefftz method is made. Since accuracy of the solutions obtained in this paper is very high, they can be used as the typical models in testing numerical methods. The computed results show that as the singularity models, Models I and II are superior to Model III, because more accurate solutions can be obtained by the collocation Trefftz method.

Superconvergence of Solution Derivatives for the Shortley–Weller Difference Approximation of Poisson's Equation. II. Singularity Problems

Abstract This is a continued analysis on superconvergence of solution derivatives for the Shortley–Weller approximation in Li (Li, Z. C., Yamamoto, T., Fang, Q. ([2003]): Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poissons equation, Part I. Smoothness problems. J. Comp. and Appl. Math. 152(2):307–333), which is to explore superconvergence for unbounded derivatives near the boundary. By using the stretching function proposed in Yamamoto (Yamamoto, T. ([2002]): Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math. 140:849–866), the second order superconvergence for the solution derivatives can be established. Moreover, numerical experiments are provided to support the error analysis made. The analytical approaches in this article are non-trivial, intriguing, and different from Li, Z. C., Yamamoto, T., Fang, Q. ([2003]). This article also provides the superconvergence analysis for the bilinear finite element method and the finite difference method with nine nodes.

Numerical simulation of gas emission in a sanitary landfill equipped with a passive venting system

Abstract The gas emission in a sanitary landfill equipped with a passive venting system was investigated numerically. The Darcys law was employed to simulate the gas flow in the landfill. We used the first order biodegradation rate in modeling the waste biodegradability. The results show that more than 43.3% gases produced from the waste will emit from the landfill surface for R = 30 m, where R is the half of the well spacing. These indicate that the influence radii (between 30 to 50 m) were overestimated in the designing of the gas collection systems in the Sanjuku and the Taichung City sanitary landfills in Taiwan. This might pose a potential of air pollution and raise chances of fires under such designing.

Superconvergence of Solution Derivatives of the Shortley–Weller Difference Approximation to Elliptic Equations with Singularities Involving the Mixed Type of Boundary Conditions

This paper presents a superconvergence analysis for the Shortley–Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. In this analysis, we first formulate the method as a special finite element/volume method. We then analyze the convergence of the method in a finite element framework. An O(h 1.5)-order superconvergence of the solution derivatives in a discrete H 1 norm is obtained. Finally, numerical experiments are provided to support the theoretical convergence rate obtained.

Verification of reduced convergence rates

In this short article, we recalculate the numerical example in Křížek and Neittaanmäki (1987) for the Poisson solution u=xσ(1−x)sinπy in the unit square S as . By the finite difference method, an error analysis for such a problem is given from our previous study by where h is the meshspacing of the uniform square grids used, and C1 and C2 are two positive constants. Let ε=u−uh, where uh is the finite difference solution, and is the discrete H1 norm. Several techniques are employed to confirm the reduced rate of convergence, and to give the constants, C1=0.09034 and C2=0.002275 for a stripe domain. The better performance for arises from the fact that the constant C1 is much large than C2, and the h in computation is not small enough.