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Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up
KTH, School of Electrical Engineering and Computer Science (EECS), Computational Science and Technology (CST).
Univ Bern, Math Inst, Sidlerstr 5, CH-3012 Bern, Switzerland..
2018 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 138, no 3, p. 767-799Article in journal (Refereed) Published
Abstract [en]

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

Place, publisher, year, edition, pages
SPRINGER HEIDELBERG , 2018. Vol. 138, no 3, p. 767-799
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-225293DOI: 10.1007/s00211-017-0918-2ISI: 000426063200008Scopus ID: 2-s2.0-85030543287OAI: oai:DiVA.org:kth-225293DiVA, id: diva2:1195865
Note

QC 20180406

Available from: 2018-04-06 Created: 2018-04-06 Last updated: 2018-04-06Bibliographically approved

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