By Charles J.(Charles J. Stone) Stone

This author's smooth technique is meant basically for honors undergraduates or undergraduates with a very good math history taking a mathematical facts or statistical inference direction. the writer takes a finite-dimensional practical modeling standpoint (in distinction to the normal parametric process) to bolster the relationship among statistical idea and statistical technique.

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**Example text**

J . The ∂Ω space Vh (Γ ) of discrete harmonic functions is defined by Vh (Γ ) = {v ∈ Vh : Ω ∇v · ∇w dx = 0 ∀ w ∈ Vh , w Γ = 0}. We will choose v∗ from Vh (Γ ). Note that a discrete harmonic function is uniquely determined by its restriction on Γ . Let E be an edge of length H shared by two nonoverlapping subdomains ˆ2 . Let g be a function defined on E such that (i) g is piecewise ˆ1 and Ω Ω linear with respect to the uniform subdivision of E of mesh size H/8, (ii) g is identically zero within a distance of H/4 from either one of the endpoints of E, (iii) g is L2 (E)-orthogonal to all polynomials on E of degree ≤ 1.

We will describe the results in terms of the following model problem. Find uh ∈ Vh such that Ω ∇uh · ∇v dx = f v dx Ω ∀ v ∈ Vh , (1) where Ω = [0, 1]2 , f ∈ L2 (Ω), and Vh is the P1 Lagrange finite element space associated with a uniform triangulation Th of Ω. We assume that the length of the horizontal (or vertical) edges of Th is a dyadic number h = 2−k . We recall the basic facts concerning additive Schwarz preconditioners in Section 2 and present the lower bound results for one-level and two-level additive Schwarz preconditioners, Bramble-Pasciak-Schatz preconditioner and the FETI-DP preconditioner in Sections 3–6.

V. Nepomnyaschikh, A weakly overlapping domain decomposition preconditioner for the finite element solution of elliptic partial diﬀerential equations, SIAM J. Sci. , 23 (2002), pp. 1817–1841. 6. R. E. Bank and S. Lu, A domain decomposition solver for a parallel adaptive meshing paradigm, SIAM J. Sci. , 26 (2004), pp. 105–127. 7. R. E. Bank and R. F. Marcia, Interior Methods for a Class of Elliptic Variational Inequalities, vol. 30 of Lecture Notes in Computational Science and Engineering, Springer, 2003, pp.