Abstract
Mathematical modeling of real-life problems in engineering, physics or life sciences often gives rise to partial differential problems that cannot be solved analytically but need a numerical scheme to obtain a suitable approximation. Dealing with numerical modeling requires first of all an understanding of the underlying differential problem. The type of differential problem, as well as issues of well-posedness and regularity of the solution may indeed drive the selection of the appropriate simulation tool. A second requirement is the analysis of numerical schemes, in particular their stability and convergence characteristics. Last, but not least, numerical schemes must be implemented in a computer language, and often aspects which look easy “on paper” arise complex implementation issues, particularly when computational efficiency is at stake.
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- 1.
An isometry is a map between metric spaces that preserves distances.
- 2.
The support of a function is the closure of the set of points where the function is not equal to zero.
- 3.
The expansion has a well-known scalar analogue: if |q| < 1, the identity ∑ n j = 0 q j = (1 − q n + 1)/(1 − q) implies ∑ ∞ j = 0 q j = (1 − q)− 1.
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© 2012 Springer-Verlag Italia
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Formaggia, L., Saleri, F., Veneziani, A. (2012). Some fundamental tools. In: Solving Numerical PDEs: Problems, Applications, Exercises. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2412-0_1
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DOI: https://doi.org/10.1007/978-88-470-2412-0_1
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2411-3
Online ISBN: 978-88-470-2412-0
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