Numerical experiments with FEMLAB® to support mathematical research
Independent thesis Basic level (professional degree), 20 points / 30 hpStudent thesis
Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(u) ≡ u2xuxx + 2uxuyuxy + u2yuyy = 0. For numerical reasons ∆q(u) = div (|▼u|q▼u) = 0, which (formally) approaches as ∆∞(u) = 0 as q → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(u) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.
Place, publisher, year, edition, pages
Matematiska institutionen , 2005. , 77 p.
FEMLAB, numerical experiments, Dirichlet problems for the Infinity-Laplace equation, AMLE, Absolutely Minimizing Lipschitz Extension, sets of uniqueness, critical segments, lines of singularity
IdentifiersURN: urn:nbn:se:liu:diva-3724ISRN: LITH-MAT-EX--05/14--SEOAI: oai:DiVA.org:liu-3724DiVA: diva2:20436