By Tatsien Li, Pingwen Zhang

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Academic Press, 1984, 61-77. [9] J. Knap and M. Ortiz, An analysis of the quasicontinuum method, J. Mech. Phys. , 49 (2001), 1899-1923. [10] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model, Math. , 72 (2003), 657-675. 32 Weinan E, Pingbing Ming [11] R. Miller and E. B. Tadmor, The quasicontinuum method: overview, applications and current directions, J. Computer-aided Material Design, 9 (2002), 203-239. [12] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximation, Math.

Math. Comput, 48:449-470, 1987. [29] L. Halpern and M. Schatzman. Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal, 20:308-353, 1989. [30] H. D. Han. A new class of variational formulations for the coupling of finite and boundary element methods. Journal of Computational Mathematics, 8:223-232, 1990. [31] H. D. Han. Boundary integro-differential equations of elliptic boundary value problems and their numerical solutions. Scientia Sinica, 31:1153-1165, 1998. [32] H.

1) has a solution UH satisfying \\UH - J W C B | | I , O C < e(QC) 1 / 2 + H1-^, \\UCB ~ UH\\i,oc < C(e(QC) 1 /2 + j j i - a / p ) , where PHUCB is defined as A(UCB;PHUCB,V) = A{UCB;UCB,V) for all VeXH. 1) is locally unique. Proof. 13), we infer that there exists a constant H0 > 0 such that for 0 < H < H0, A{U ™;X,W) sup >CWVh for all 7 6 l f f . 5) and \\UCB < CH1-^. 8) by i ( t / C B ; T(V), W ) = i ( t / C B ; C^CB, W ) - R(UCB, + V, W) A(V,W)-AH(V,W) for any W € XH- Obviously T is continuous.