An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space
- 51 Downloads
This paper presents an improved method for determining the invertibility of second-order linear elliptic operators with a bound on the norm of their inverses by computers in a mathematically rigorous sense. This approach is an improvement on a previous method (Nakao et al. in Jpn J Ind Appl Math 32:19–32, 2015) which used a projection and constructive a priori error estimates. Several examples confirming the effectiveness of the proposed procedure are reported.
KeywordsSolvability of linear problem Differential operator Numerical verification Computer-assisted proof
Mathematics Subject Classification65G20 47F05 35P15
The authors heartily thank the anonymous referee for her/his thorough reading and valuable comments. This work was supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 15K05012 and 15H03637) and CREST, JST. The computations were mainly carried out using the computer facilities at the Research Institute for Information Technology, Kyushu University, Japan.
- 9.Rump, S.M.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999). http://www.ti3.tu-harburg.de/rump/