Examples

The examples each highlight a specific aspect, possibly without much explanation.

• Example: prolongations and computing products and norms
  • This is work in progress (WIP), still missing:
  • 1: prolongation onto a finer grid
  • 2: computing products and norms
  • 3: computing errors w.r.t. a known reference solution
• Example: Grid adaptation
  • keeping track of refined elements
• Example: structured 1d grids (YASP_1D vs OneDGrid)
  • Two ways to mesh the unit interval
  • The stationary heat equation on both grids
  • The equidistant-offset feature: a structured grid on [a, b]
• Example: the ALU cube grid (3d hexahedra)
  • Structured YASP vs. unstructured ALU cube
  • A finite-element space on the ALU cube grid
  • How this reaches the property-test suite
• Example: a posteriori error estimates based on flux reconstruction
• Example: working with gmsh grids
  • 1: creating a gmsh file
  • 2: discretization using pyMOR
  • 3: using the gmsh grid in dune
• Example: mixed-element and prism grids
  • 1: a mixed cube/simplex grid in 2d
  • 2: assembling and solving on the mixed grid
  • 3: a prism grid in 3d
• Example: Taylor-Hood P2/P1 Stokes
  • grid and spaces
  • the exact solution
  • assembling \(A\) (velocity Laplacian) and \(B\) (pressure/velocity divergence coupling)
  • solving the saddle-point system
  • comparing against the exact solution
• Example: custom Python functions as C++ (grid) functions
  • 1: GenericFunction vs. ExpressionFunction
  • 2: GenericGridFunction with post_bind – an element-dependent coefficient
  • 3: overhead of a Python callback vs. a parsed expression
• Example: residual-based a posteriori error estimates
  • 1: inspect grid properties and relations between elements and intersections
  • 2: solving an elliptic PDE
  • 3: computing the a posterior error estimates and indicators from [MNS2002]
  • 4: data oscialltion
• Tutorial: discontinuous IPDG for the stationary heat equation
  • 1.9: everything in a single function
• Example: SIPDG for the instationary heat equation
  • This is work in progress (WIP), still missing:
  • stationary data functions
  • timedependent right hand side
• Example: LA eigensolvers
  • assembling the Laplace stiffness and mass matrices
  • restricting to the interior DoFs
  • reducing to a standard eigenvalue problem
  • computing eigenvalues via the bound eigen-solver
  • comparison with the known spectrum of the unit square
  • matrix-inverter cross-check
• Example: linear transport via finite volumes
  • 1: a moving Gaussian bump in 1d
  • 2: experimental order of convergence (EOC)
  • 3: the same setup in 2d
  • 4: choosing dt automatically
  • 5: adaptive time stepping
  • 6: higher order in space: discontinuous Galerkin
  • 7: piecewise linear reconstruction of cell averages
  • 8: operator splitting in 2d