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Unformatted text preview: e, kekA 2 = Z 1 (e )2 + e2 ]dx: 0 0 The integral is the square of the norm used on the Sobolev space H 1 thus, kek1 := Z 1 (e ) + e ]dx 0 0 2 2 1=2 : (1.3.23b) Other global error measures will be important to our analyses however, the only one 1.3. A Simple Finite Element Problem 19 that we will introduce at the moment is the L2 norm kek0 Z1 := 0 e (x)dx 2 1=2 : (1.3.23c) Results for the L2 and strain energy errors, presented in Table 1.3.2 for this example, indicate that kek0 = O(h2) and kekA = O(h). The error in the H 1 norm would be identical to that in strain energy. Later, we will prove that these a priori error estimates are correct for this and similar problems. Errors in strain energy converge slower than those in L2 because solution derivatives are involved and their nodal convergence is O(h) (Table 1.3.1). N 4 8 16 32 64 128 kek0 0.265(-2) 0.656(-3) 0.167(-3) 0.417(-4) 0.104(-4) 0.260(-5) kek0 =h2 0.425(-1) 0.426(-1) 0.427(-1) 0.427(-1) 0.427(-1) 0.427(-1) kekA 0.390(-1) 0.19...
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