FENICS Examples

• annulus_flow, a FENICS script which simulates flow in an annulus, goverened by the time-dependent Navier Stokes equations.
• burgers_steady_viscous, a FENICS script which solves the steady viscous Burgers equation in 1D.
• burgers_time_viscous, a FENICS script which solves the time-dependent viscous Burgers equation in 1D.
• bvp, FENICS scripts which solve two-point boundary value problems (BVP) in 1D.
• convection_diffusion, a FENICS script which simulates a 1D convection diffusion problem.
• convection_diffusion_stabilized, a FENICS script which simulates a 1D convection diffusion problem, using a stabilization scheme.
• dg_advection_diffusion, a FENICS script which uses the Discontinuous Galerking (DG) method to set up and solve an advection diffusion problem.
• domain, a FENICS script which creates a region starting with a circle, and then "subtracting" a rectangle and smaller circle.
• dpg_bvp, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a boundary value problem over an interval, by Jay Gopalakrishnan.
• dpg_laplace, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem over the unit square, by Jay Gopalakrishnan.
• dpg_laplace_adapt, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem over the unit square, with adaptivity, by Jay Gopalakrishnan.
• ell, a FENICS script which solves the Poisson equation on the L-shaped region.
• expression_test, a FENICS script which demonstrates some properties of the objects created by the FEniCS Expression() function.
• fenics_to_fem, a FENICS script which illustrate how a mesh or scalar function computed by a FENICS program can be written to FEM files, which can then be used to create images, or as input to meshing programs or other analysis tools.
• fenics_to_txyv, a FENICS script which illustrates how a mesh or scalar function computed by a FENICS program can be written to a triangulation file ("t.txt"), a coordinate file ("xy.txt", and a value file ("v.txt").
• fenics_to_txyz, a FENICS script which illustrates how a mesh or scalar function computed by a FENICS program can be written to a triangulation file ("t.txt") and a coordinate and value file ("xyz.txt").
• glacier, a FENICS script which sets up the 2D steady Stokes equations to model the movement of a glacier, by William Mitchell. Sadly, this USED to work with a previous version of FENICS...
• heat_explicit, a FENICS script which uses the finite element method to solve a version of the time dependent heat equation over a rectangular region with a circular hole. Time steps are handled using an explicit method.
• heat_implicit, a FENICS script which uses the finite element method to solve a version of the time dependent heat equation over a rectangular region with a circular hole. Time steps are handled using an implicit solver.
• heat_steady, FENICS scripts which set up the 2D steady heat equation in a rectangle.
• interpolant, a FENICS script which shows how to define a function in FENICS by supplying a mesh and the function values at the nodes of that mesh, so that FENICS works with the finite element interpolant of that data.
• membrane, a FENICS script which models the deflection of a circular elastic membrane subject to a perpendicular pressure modeled as a Gaussian function.
• mesh_test, a FENICS script which demonstrates some properties of the objects created by the various FEniCS meshing functions.
• meshes, a FENICS script which generates and plots the simple built-in mesh types that FENICS provides.
• mitchell, FENICS scripts which illustrate the implementation of the Mitchell elliptic test problems using FENICS. I only managed to complete a few of these.
• neumann, a FENICS script which solves a boundary value problem in the unit square, for which homogeneous Neumann conditions are imposed, adapted from a program by Doug Arnold.
• p_laplacian, a FENICS script which sets up and solves the nonlinear p-Laplacian PDE in the unit square.
• poisson, a FENICS script which solves the Poisson equation on the unit square, adapted from the FENICS tutorial by Langtangen and Logg.
• poisson_mixed, a FENICS script which solves the Poisson equation using a mixed formulation.
• poisson_nonlinear, a FENICS script which uses the finite element method to solve a version of the nonlinear Poisson equation over the unit square.
• poisson_source_symbolic, a FENICS script which shows how the symbolic package can be used to determine the right hand side f(x,y) of a Poisson equation, given the desired exact solution u(x,y).
• quadrilateral, a FENICS script which solves the Poisson equation on the unit square, using a quadrilateral mesh.
• step1, a FENICS script which solves a Poisson equation with constant diffusivity kappa(x,y).
• step2, a FENICS script which solves a Poisson equation with piecewise constant diffusivity kappa(x,y).
• step3, a FENICS script which solves a Poisson equation with piecewise constant diffusivity kappa(x,y), and computes and plots |grad(u)|.
• step4, a FENICS script which shows how to generate the right hand side of a Poisson problem, using the symbolic mathematics package and an exact solution formula.
• step5, a FENICS script which solves a Poisson equation whose diffusivity kappa(x,y) is defined as |grad(w)| for a given scalar field w(x,y).
• step6, a FENICS script which solves the nonlinear p-Laplacian problem directly.
• step7, a FENICS script which compares error norm and DPG error indicators for a Poisson problem.
• step8, a FENICS script which uses DPG error indicators for adaptive mesh refinement.
• step9, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem, and repeatedly refines the mesh, guided by DPG error indicators. On each refinement step, the top fifty percent of the cells are refined, as ordered by size of local error estimates. It then plots the decay of the estimated error as a function of the number of mesh elements.
• step10, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem, and repeatedly refines the mesh, guided by DPG error indicators. On each refinement step, cells are ordered by size of the local error estimates, and then enough "top" cells are refined to represent the proportion THETA of the total error estimate. It then plots the decay of the estimated error as a function of the number of mesh elements.
• step11, a FENICS script which uses the Discontinuous Petrov Galerkin (DPG) method to solve a Poisson problem, and repeatedly refines the mesh, guided by DPG error indicators. Two methods of adaptive refinement are compared, one of which refines cells which represent a fixed percentage of the error (case 0), and one of which refines a fixed percentage of the cells (case1). The problem uses a discontinuous diffusivity function kappa(x,y).

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