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Differential Equations Solutions 131

Differential Equations Solutions 131 - 141 Infinity norm of...

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Unformatted text preview: 141 Infinity norm of the error at the grid points for various methods and numbers of interior grid points M M = 9 99 999 1st order finite difference 1.5702e-01 1.6571e-01 1.6632e-01 2nd order finite difference 1.5702e-01 1.6571e-01 1.6632e-01 Linear finite elements 1.4974e-01 1.6472e-01 1.6622e-01 Quadratic finite elements 1.4975e-01 1.6472e-01 1.6622e-01 Discussion: Clearly, the finite difference methods are easier to program and therefore are almost always used when x is a single variable. Finite elements become useful, though, when x has 2 or more components and the shape of the domain is nontrivial. The bulk of the work in these methods is in function evaluations. We need O ( M ) evaluations of a , c , and f in order to form each matrix. For finite differences, the constant is close to 1, but quad (the numerical integration routine) uses many function evaluations per call (on the order of 10), making formation of the finite element matrices about 10 times as expensive....
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