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arXiv · 2001.03854

A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations

Kazuaki Tanaka
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Abstract

This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution $ u $ and a numerically computed approximate solution $ \hat{u} $, we evaluate the number of sign-changes of $ u $ (the number of nodal domains) and determine the location of zero level-sets of $ u $ (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen-Cahn equation. The nodal line of solutions of this equation represents the interface between two coexisting phases.

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Kazuaki Tanaka. 2020-01-12. A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations. https://arxiv.org/abs/2001.03854

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