Jupyter tutorials

To work through the i-tutorials we refer to the repository https://github.com/ngsxfem/ngsxfem-jupyter with instructions and more material. Here, we give a selection of notebooks. You can run it through binder here: https://mybinder.org/v2/gh/ngsxfem/ngsxfem-jupyter/master?filepath=tutorials.ipynb .

• Geometry description with the level set function
  • The background mesh
  • Geometry approximation through level set approximation
  • Cut Information
    • Marking elements
    • DOMAIN_TYPE
    • COMBINED_DOMAIN_TYPE
  • Updating the CutInformation
  • Marking with facets
    • Example 1:
    • Example 2:
  • Restriction to active mesh/dofs
    • Marking dofs (GetDofsOfElements)
    • Marking integrators (definedonelements)
    • Restricted Bilinearforms
• Integration on level set domains
  • Piecewise linear geometry approximation
    • Convergence study
  • Higher order geometry approximation with an isoparametric mapping
    • Convergence study with isoparametric mapping applied
• Unfitted FEM discretizations
  • Interface problem
    • Cut FE spaces
    • Let us visualize active dofs:
    • Improvement: use Compress to reduce unused dofs
  • Nitsche discretization
    • Cut ratio field
    • Normal direction
    • Averages and jumps
    • Integrals
    • Assembly
  • Higher order accuracy
  • XFEM spaces
  • Similar examples and extensions (python demo files)
• Fitted space-time finite elements
• Unfitted space-time finite elements
  • Background geometry and mesh:
  • Handling of the time variable
  • Data functions (depending on \(t\))
    • A View on the time-dependent level set function on \([0,\)tend\(]\)
  • Space-Time finite elements
  • Levelset mesh adaptation class
    • Space-Time version of the CutInfo class
    • Solution GridFunction
  • Variational formulation
  • Upwind DG in time
  • Ghost penalty stabilization
    • Implementation of space-time integrals
      • Transformation from reference interval to \((t_{n-1},t_n)\):
    • Solution of linear systems in every time step
    • At the end of every time step, we
    • The final time loop
• Integration over domains defined by multiple level sets
  • Integrating over regions of co-dimension 0
  • Integrating over regions of co-dimension > 0
  • Convenience Layer: xfem.mlset
    • Working with DomainTypeArrays
    • Geometrical features
    • Visualisation Tools
    • Removing unnecesary regions
    • Generating outward pointing unit normal vectors
  • Integrating using DomainTypeArrays
• Solving PDEs on domians defined via multiple level sets
  • Multiple level set cut-information
    • Marking the correct elements
  • Defining the weak form
    • Integrators
  • Solving the system
• Cell and basis aggregation in ngsxfem
  • Introduction: Cut cells’ best friends: their neighbors
  • Patches
    • ElementAggregation
  • Basis aggregation
    • L2 low order
    • L2 higher order
    • H1
    • Vector-valued spaces
    • How it works
      • A formulation as a harmonic extension
      • A formulation as a patch-wise harmonic extension
      • A generic semi-norm \(\vert \varphi_i + w_h \vert_{\ast,\omega}\)
  • Applications of patch aggregations
• Unfitted Mixed FEM (using HDiv FE spaces)
  • The Method
    • Discretization parameters
    • Geometry approximation
  • Solving the problem
    • Finite element spaces
    • GridFunction, trial- and test functions, and some standard coefs
    • Differential symbols for different integration domains:
    • Definition of variational formulation:
    • Assembly and solution
  • Patchwise Post-processing
  • What’s left?
  • Appendix: extensions
    • Extension 1: Patchwise approximation of \(f\)
    • Extension 2: Element-wise Post-processing