Nonlinear Theory and Its Applications, IEICE
Online ISSN : 2185-4106
Special section on Recent Progress in Verified Numerical Computations
Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains
Akitoshi TakayasuXuefeng LiuShin'ichi Oishi
Author information

Volume 4 (2013) Issue 1 Pages 34-61

Download PDF (881K) Contact us

In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the Newton-Kantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.

Information related to the author
© 2013 The Institute of Electronics, Information and Communication Engineers
Previous article Next article

Recently visited articles