# 210c 8 analysis of the finite element method with

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Unformatted text preview: olation error estimates available, we can establish convergence of the nite element method. Theorem 7.2.3. Suppose: 1 N 1. u 2 H0 and U 2 S0 H01 satisfy (7.2.8) and (7.2.7), respectively 2. A(v u) is a symmetric, continuous, and H 1 -elliptic bilinear form N 3. S0 consists of complete piecewise-polynomial functions of degree p with respect to a uniform family of meshes h and 1 4. u 2 H0 \ H p+1. Then ku ; U k1 C hpkukp+1 (7.2.11a) and A(u ; U u ; U ) C h2pkuk2+1: p (7.2.11b) Proof. From Theorem 7.2.2 A(u ; U u ; U ) = infN A(u ; V u ; V ) A(u ; W u ; W ) V 2S0 where W is an interpolant of u. Using (7.2.1) with s = 1 and v and u replaced by u ; W yields A(u ; W u ; W ) ku ; W k2: 1 Using the interpolation estimate (7.2.10a) with s = 1 yields (7.2.11b). In order to prove (7.2.11a), use (7.2.2) with s = 1 to obtain ku ; U k2 1 A(u ; U u ; U ): The use of (7.2.11b) and a division by yields (7.2.11a). Since the H 1 norm dominates the L2 norm, (7.2.11a) trivially gives us an error estimate in L2 as ku ; U k0 C hp ku...
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