# What does fully nonlinear second order pde

DC FieldValueLanguagedc.contributor.advisorKühn, Christian-dc.contributor.authorJawecki, Tobias-dc.date.accessioned2020-06-28T00: 49: 16Z-dc.date.issued2017-dc.date.submitted2017-03-dc.identifier.urihttps://resolver.obvsg.at/urn:nbn:at:at-ubtuw:1-95655-dc.identifier.urihttp://hdl.handle.net/20.500.12708/2214-dc.descriptionDifferent title according to the author's translation-dc.description.abstractIn this thesis we deal with numerical bifurcation analysis of equilibrium solutions. We consider parameter dependent dynamic systems generated by nonlinear elliptical PDGs. In the first chapter we give theoretical results on nonlinear elliptical PDGs and Fredholm operators. With the help of the Fredholm operators we can show analytical results of the bifurcation theory, on which the numerical methods are later based. For the numerical bifurcation analysis we mainly need path tracking algorithms and stability analyzes of the equilibrium solutions. Both can be solved numerically with FEM. Our theory of numerical methods also includes a convergence analysis of FEM applied to nonlinear elliptic operators and discrete eigenvalue problems. We use the numerical methods to analyze different PDGs defined on rectangular areas in $ R 2 $ and dependent on one-dimensional parameters. All of our numerical examples are well documented, but the source code is only shown for simple problems in the appendix. The software itself was written in Python and is based on different packages, for example Fenics to apply the FE methods or PETSc to solve linear algebra problems. In simple examples we analyze the Bratu and Allen-Cahn equation with simple boundary conditions. Further examples show the difficulties that can arise with symmetrical domains or equations with masses. Fourth order equations can be discretized numerically with a system of second order equations. In order to apply bifurcation analysis to mixed formulations, the numerical problem needs to be rewritten. An example of a fourth order equation is also calculated, the fourth order of the equation being given by a biharmonic function. To analyze this problem, we use a mixed formulation and implement our theoretical ideas.dedc.description.abstractThe focus of this thesis lies on numerical bifurcation analysis of steady states. We use dynamical systems generated by nonlinear elliptic PDEs which depend on one-dimensional bifurcation parameters. In the first chapter we work on the theory of nonlinear elliptic differential equations and Fredholm operators. Based on Fredholm operators we can give some results of the analytical bifurcation theory which are later used for numerical methods. Numerical bifurcation analysis is mainly based on path following and stability analysis of steady states. Both problems can be solved numerically with FEM. Our theory on numerical methods includes convergence results of FEM applied to nonlinear elliptic operators and discrete eigenvalue problems. In all of our practical examples we use PDEs on rectangular domains in $ \ R 2 $ with an one-dimensional bifurcation parameter. We use different equations for our numerical examples. Any of those examples is fully documented, but we only give the source code for the basic examples. The software is written in Python and is mainly based on the FEM packages FEniCS and the linear algebra package PETSc. We also use SLEPc to solve eigenvalue problems and some other packages for lower-level tasks. The basic examples consist of numerical bifurcation analyzes of the Bratu and Allen-Cahn problem with simple boundary conditions. We give additional examples to show difficulties with symmetric domains and mass constraints. For fourth-order problems we use a mixed formulation and simple FE spaces. To apply the numerical bifurcation theory with a mixed formulation we need to adapt the dynamical system for the stability analysis. At the end, we give a numerical bifurcation analysis of a fourth-order problem of Biharmonic type. To solve this problem a mixed formulation is used and the system is adapted based on our analytical results.endc.format175 pages-dc.languageEnglish-dc.language.isoen-dc.subjectbifurcation analysisendc.subjectnumerical continuationendc.subjectFENICSendc.subjectFEMendc.subjectfourth-orderendc.titleBifurcation analysis via numerical continuation for nonlinear fourth-order partial differential equationsendc.title.alternativeBifurcation analysis and numerical path tracing for nonlinear partial differential equations of the fourth orderdedc.typeThesisendc.typeUniversity thesisdedc.publisher.placeVienna-thesis informationtechnical University of Vienna-tuw.publication.orgunitE101 - Institute for Analysis and Scientific Computing-dc.type.qualificationlevelDiploma-dc.identifier.libraryidAC13642458-dc.description.numberOfPages175-dc.identifier.urnurn: nbn: at: at-ubtuw: 1-95655-dc.thesistypeThesisdedc.thesistypeDiploma thesisenitem.grantfulltextopen-item.languageiso639-1en-item.openairecristypehttp://purl.org/coar/resource_type/c_18cf-item.openairecristypehttp://purl.org/coar/resource_type/c_18cf-item.fulltextwith fulltext-item.cerifentitytypePublications-item.cerifentitytypePublications-item.openairetypeThesis-item.openairetypeUniversity thesis-Appears in Collections:Thesis

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