Global superconvergence of Adini's elements coupled with the Trefftz method for singular problems

Abstract

This paper reports new results of the Trefftz method (i.e. the boundary approximation method (BAM) by Li [Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, 1998] using particular solutions which is coupled with Adini's method for singular problems. First, the computational aspects and numerical experiments are provided for the global superconvergence of Adini's elements for the Poisson equation. The superconvergence O( h 3.5) is obtained over the entire domain for the uniform rectangulation, by means of an a posteriori interpolant of the obtained solutions. Such a superconvergence is a half order higher than the optimal convergence O( h 3) of Adini's elements. Second, for the Neumann problems, we add the natural boundary constraints ( u n ) ij = g ij to the admissible functions of Adini's elements, the global superconvergence O( h 4) can also be achieved. Third, for singular problems, e.g. the Motz problem, the Adini's elements are coupled with the Trefftz method also to give the global superconvergence O( h 3.5). This paper reports the numerical results which have verified perfectly the high global superconvergence of Adini's elements and their coupling with the Trefftz method. For Motz's problem, the leading coefficient, d ̃ 0=401.1624462 with the relative errors 0.189×10 −7, is obtained at N=16, e.g. h=1/16. This is the most accurate leading coefficient ever published by the coupled method of the Trefftz method with other different methods [Engng Anal BE 18 (1996) 119; Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, 1998; Engng Anal BE 10 (1992) 75]. Since the a posteriori interpolant formulas are required to yield the global superconvergence, two useful interpolant formulas are explored in this paper. Besides, the stiff matrix is provided with exact entries of Adini's elements on the unit square.