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Existence Techniques I: Methods Based on the Maximum Principle

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Book cover Partial Differential Equations

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 214))

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Abstract

The basic idea of the difference methods consists in replacing the given differential equation by a difference equation with step size h and trying to show that for h ;→ ;0, the solutions of the difference equations converge to a solution of the differential equation. This is a constructive method that in particular is often applied for the numerical (approximative) computation of solutions of differential equations.

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Notes

  1. 1.

    The boundary values here are not continuous as in the maximum principle, but they can easily be approximated by continuous ones satisfying the same bounds. This easily implies that the maximum principle continues to hold in the present situation.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Jürgen Jost

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  1. Jürgen Jost
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Jost, J. (2013). Existence Techniques I: Methods Based on the Maximum Principle. In: Partial Differential Equations. Graduate Texts in Mathematics, vol 214. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4809-9_4

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