Westerheide, Sebastian: Unfitted discontinuous Galerkin schemes for applications with PDEs on complex-shaped surfaces. 2018
Inhalt
- List of Figures
- List of Tables
- Introduction
- Bulk PDEs and surface PDEs
- Continuity equations on static geometries
- Continuity equations on evolving geometries
- Non-conservative equations
- A class of bulk–surface models
- Numerical methods for bulk PDEs and surface PDEs
- Studying spatial features in basic cell polarization models using a classical mesh-based finite element scheme
- Challenges in applications with PDEs on complex-shaped surfaces
- Contributions and outline of this thesis
- Essential concepts from elementary differential geometry
- Surface differential operators
- A closer look at surface divergence
- Surface divergence of the tangential/normal component of a surface vector field and the notion of curvature
- Splitted representation of surface divergence
- Additional remarks
- Surface divergence in the level set framework
- Integral calculus on hypersurfaces
- Integration of those concepts into the time-dependent case
- Additional calculus on evolving hypersurfaces
- Further mathematical background
- Conservation laws and continuity equations
- Fitted DG methods for elliptic and parabolic bulk PDEs
- Obtaining DG methods by choosing numerical fluxes
- The classical SIPG formulation and related approaches
- The SWIPG formulation
- Spatial discretization of parabolic equations
- Semidiscrete conservation properties
- Implicit geometry description using the level set framework
- Unfitted DG schemes for coupled bulk–surface PDEs on complex static geometries
- Classes of static geometry model problems
- The approaches and corresponding schemes
- An extension process for surface equations
- Unfitted discontinuous Galerkin
- Recovering discrete analogues to original conservation properties
- Stabilization strategies with respect to the surface part of the solution
- Fully discrete schemes
- Numerical results
- Linear elliptic model problems
- Linear parabolic model problems
- Application: Nonlinear parabolic models for cell polarization
- Discussion
- Toward unfitted DG schemes for coupled bulk–surface PDEs on evolving geometries
- A class of evolving geometry model problems
- Simplifying the problem using operator splitting
- Operator splitting for PDEs on evolving geometries
- Specific operator splitting methods for PDEs on evolving geometries
- Related splitting approaches
- Treating the resulting subproblems
- An unfitted DG scheme for an essential type of continuity equations on evolving hypersurfaces
- Approximate reformulation of surface equations
- Unfitted discontinuous Galerkin
- Remarks on choosing extended data functions
- Global conservation properties
- Understanding the scheme in one dimension
- Numerical results
- Discussion
- Conclusion
- Software
- The condition number of a matrix
- Basic definitions and facts
- Theory from linear algebra
- The spectral condition number of a matrix
- Numerical computation of eigenvalues
- Power iteration
- Inverse iteration with shift
- Rayleigh quotient iteration
- The TLIME algorithm
- Application to computing the spectral condition number
- Implementation in the dune-istl module
- Basic terminology and facts from elementary differential geometry
- List of Symbols
- List of Acronyms
- Bibliography