Documentation of scikit-fem

scikit-fem is a pure Python 3.7+ library for performing finite element assembly. Its main purpose is the transformation of bilinear forms into sparse matrices and linear forms into vectors. The library supports triangular, quadrilateral, tetrahedral and hexahedral meshes as well as one-dimensional problems.

Note

Installing the library is as simple as running

pip install scikit-fem[all]

Remove [all] to not install the optional dependencies meshio and matplotlib.

Table of contents

• Documentation of scikit-fem
• Getting started
  • Step 1: Clarify the problem
  • Step 2: Express the forms as code
  • Step 3: Create a mesh
  • Step 4: Define a basis
  • Step 5: Assemble the linear system
  • Step 6: Find boundary DOFs
  • Step 7: Eliminate boundary DOFs and solve
  • Step 8: Calculate error
• How-to guides
  • Finding degrees-of-freedom
  • Performing projections
  • Discrete functions in forms
  • Assembling jump terms
• Advanced topics
  • Anatomy of forms
  • Indexing of the degrees-of-freedom
• Gallery of examples
  • Poisson equation
    • Example 1: Poisson equation with unit load
    • Example 7: Discontinuous Galerkin method
    • Example 12: Postprocessing
    • Example 13: Laplace with mixed boundary conditions
    • Example 14: Laplace with inhomogeneous boundary conditions
    • Example 15: One-dimensional Poisson equation
    • Example 9: Three-dimensional Poisson equation
    • Example 22: Adaptive Poisson equation
    • Example 37: Mixed Poisson equation
    • Example 38: Point source
    • Example 40: Hybridizable discontinuous Galerkin method
    • Example 41: Mixed meshes
  • Solid mechanics
    • Example 2: Kirchhoff plate bending problem
    • Example 3: Linear elastic eigenvalue problem
    • Example 4: Linearized contact problem
    • Example 8: Argyris basis functions
    • Example 11: Three-dimensional linear elasticity
    • Example 21: Structural vibration
    • Example 34: Euler-Bernoulli beam
    • Example 36: Nearly incompressible hyperelasticity
    • Example 43: Hyperelasticity
  • Fluid mechanics
    • Example 18: Stokes equations
    • Example 20: Creeping flow via stream-function
    • Example 24: Stokes flow with inhomogeneous boundary conditions
    • Example 29: Linear hydrodynamic stability
    • Example 30: Krylov-Uzawa method for the Stokes equation
    • Example 32: Block diagonally preconditioned Stokes solver
    • Example 42: Periodic meshes
  • Heat transfer
    • Example 17: Insulated wire
    • Example 19: Heat equation
    • Example 25: Forced convection
    • Example 26: Restricting problem to a subdomain
    • Example 28: Conjugate heat transfer
    • Example 39: One-dimensional heat equation
  • Miscellaneous
    • Example 10: Nonlinear minimal surface problem
    • Example 16: Legendre’s equation
    • Example 31: Curved elements
    • Example 33: H(curl) conforming model problem
    • Example 35: Characteristic impedance and velocity factor
    • Example 44: Wave equation
• Detailed API description
  • Module: skfem.mesh
    • Abstract class: Mesh
      • Mesh
        • Mesh.elements_satisfying()
        • Mesh.facets_satisfying()
        • Mesh.load()
        • Mesh.nodes_satisfying()
        • Mesh.refined()
        • Mesh.save()
      • Class: MeshTri
        • MeshTri
        • MeshTri1
          • MeshTri1.__init__()
          • MeshTri1.init_circle()
          • MeshTri1.init_lshaped()
          • MeshTri1.init_refdom()
          • MeshTri1.init_sqsymmetric()
          • MeshTri1.init_symmetric()
          • MeshTri1.init_tensor()
          • MeshTri1.load()
        • MeshTri2
          • MeshTri2.init_circle()
          • MeshTri2.load()
      • Class: MeshQuad
        • MeshQuad
        • MeshQuad1
          • MeshQuad1.__init__()
          • MeshQuad1.init_refdom()
          • MeshQuad1.init_tensor()
          • MeshQuad1.load()
          • MeshQuad1.to_meshtri()
        • MeshQuad2
          • MeshQuad2.load()
      • Class: MeshTet
        • MeshTet
        • MeshTet1
          • MeshTet1.__init__()
          • MeshTet1.init_ball()
          • MeshTet1.init_refdom()
          • MeshTet1.init_tensor()
          • MeshTet1.load()
        • MeshTet2
          • MeshTet2.init_ball()
          • MeshTet2.load()
      • Class: MeshHex
        • MeshHex
        • MeshHex1
          • MeshHex1.__init__()
          • MeshHex1.init_tensor()
          • MeshHex1.load()
          • MeshHex1.to_meshtet()
      • Class: MeshLine
        • MeshLine
        • MeshLine1
          • MeshLine1.__init__()
  • Module: skfem.assembly
    • asm()
    • Abstract class: AbstractBasis
      • AbstractBasis
        • AbstractBasis.get_dofs()
      • Class: CellBasis
        • Basis
        • CellBasis
          • CellBasis.__init__()
          • CellBasis.interpolate()
          • CellBasis.project()
      • Class: FacetBasis
        • BoundaryFacetBasis
        • FacetBasis
          • FacetBasis.__init__()
      • Class: InteriorFacetBasis
        • InteriorFacetBasis
          • InteriorFacetBasis.__init__()
    • Abstract class: Form
      • Class: BilinearForm
        • BilinearForm
          • BilinearForm.assemble()
      • Class: LinearForm
        • LinearForm
      • Class: Functional
        • Functional
          • Functional.elemental()
  • Module: skfem.element
    • DiscreteField
  • Module: skfem.utils
    • Function: solve
      • solve()
    • Function: condense
      • condense()
    • Function: enforce
      • enforce()
  • Module: skfem.helpers
    • grad()
    • div()
    • curl()
    • d()
    • dd()
    • ddd()
    • dddd()
    • sym_grad()
    • dot()
    • ddot()
    • dddot()
    • mul()
    • trace()
    • transpose()
    • prod()
    • inv()