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Differential Equations Solutions 179

# Differential Equations Solutions 179 - the Fner the mesh...

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189 10 2 10 3 10 4 10 5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 (approx) 1/h 2 error in eigenvalue Errors in eigenvalues as a function of 1/h 2 λ 1 λ 6 λ 11 λ 16 λ 21 Figure 31.2. The errors in the eigenvalue approximations. The error ratios are as follows: lambda j Mesh 1 vs. 2 Mesh 2 vs. 3 Mesh 3 vs. 4 j = 1 3.95e+00 3.98e+00 3.99e+00 j = 6 3.98e+00 3.98e+00 3.99e+00 j = 11 4.02e+00 4.00e+00 4.00e+00 j = 16 4.04e+00 4.01e+00 4.00e+00 j = 21 4.05e+00 4.01e+00 4.00e+00 Therefore, the error is reduced by a factor of 4 as the side of each triangle is reduced by a factor of 2, so the error is O ( h 2 ), as expected, but the larger the eigenvalue,
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Unformatted text preview: the Fner the mesh necessary to achieve a given accuracy. CHALLENGE 31.3. (a) Suppose (for convenience of notation) that Ω ⊂ R 2 . (Other dimensions are just as easy.) ±irst we apply integration by parts (with zero boundary conditions) to see that if w ± = 0 ( w, A w ) = − Z Z Ω w ∇ · ( a ∇ w )d x d y = Z Z Ω ∇ w · ( a ∇ w )d x d y = Z Z Ω a 1 ( x, y ) µ ∂w ∂x ¶ 2 + a 2 ( x, y ) µ ∂w ∂y ¶ 2 d x d y ≥ ,...
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