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《Numerical Mathematics A Journal of Chinese Universities》 1982-02
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GENERALIZED DIFFERENCE METHODS FOR 2ND ORDER ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Li Yunghua Zhu Piqi (Jilin University)  
In paper[1], the generalized difference methods (GDM) were described asfollowing. Let Th: a = x0 x1…xN=b be a subdivision and Th*:a=x0x1/2….xN-1/2xN=b be a corresponding dual subdivision, where Xt+1/2=(xt +xt+1)/2.Assume that the trial s'ubspace Uh is taken as usual finite elemente space and that the test subspace Vh has a finite basis which cons/sis of the following functions:where 0≤i≤N, k=0,1,2,… Let a(u,v) be a bilinear form associated with the two point boundary value problems. Then the generalized difference methods consists in finding an approximation uh∈Uh such thata(uh,vh)=(f,vh)Especially, if Uh consists of piecewise cubic Hermite spline functions, and the bases of Vh are φt(0) (x),φt(1) (1≤iN-1),, we can also obtain the error estimates ||u-uh||H1=O(h3). ( cf. [1] [12]) Consequently, the GDM has the same order of convergence rate as finite element method but it requires less computational expanses. In this paper, we will generalize the result of paper [1] to the two-dimensional triangulation and 2nd order elliplic partial differential equation, and prove an error estimate ||u-uh||H1=O(h) for piecewise linear trial Space and piecewise constant test space.
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