Reference documentation for deal.II version 9.4.0
Tutorial programs

New to deal.II? You might want to start with tutorial step-1 and work your way up to step-5. At that point you can explore what features you are interested in and look at the large collection of programs listed below.

The deal.II tutorial contains a collection of programs, each more or less built atop of previous ones, which demonstrate various aspects of the library. Each such example has the following structure:

1. Introduction: What the program does, including the mathematical model, and what programming techniques are new.
2. The commented program: An extensively documented listing of the source code.
3. Results: The output of the program, with comments and interpretation.
4. The plain program: The source code stripped of all comments.

You can browse the available tutorial programs

1. as a graph that shows how the major concepts of each tutorial programs builds on previous ones (though each program may also use minor pieces from other programs not specifically connected in the graph).
2. as a list that provides a short synopsis of each program.
3. or grouped by topic.

The programs are in the examples/ directory of your local deal.II installation. After compiling the library itself, if you go into one of the tutorial directories, you can configure the program by typing cmake ., build it via make and run it using make run. The latter command also compiles the program if that has not already been done. The CMakeLists.txt files in the different directories are based on the autopilot style CMakeLists.txt example.

Note
Some of the tutorial programs also jointly form the geodynamics demonstration suite. More, often more complex but less well documented, deal.II-based programs than the ones that form the tutorial can also be found in the The deal.II code gallery .

Connections between tutorial programs

The following graph shows the connections between tutorial programs and how their major components build on each other. Click on any of the boxes to go to one of the programs. If you hover your mouse pointer over a box, a brief description of the program should appear.

Legend:

Tutorial programs listed by number

 step-1 Creating a grid. A simple way to write it to a file. Keywords: Triangulation, GridGenerator::hyper_cube(), GridGenerator::hyper_shell(), GridOut, Triangulation::execute_coarsening_and_refinement() step-2 Associate degrees of freedom to each vertex and compute the resulting sparsity pattern of matrices. Show that renumbering reduces the bandwidth of matrices significantly, i.e. clusters nonzero entries around the diagonal. Keywords: FE_Q, DynamicSparsityPattern, DoFTools::make_sparsity_pattern(), DoFHandler::distribute_dofs(), DoFRenumbering, SparsityPattern step-3 Actually solve Laplace's problem. Object-orientation. Assembling matrices and vectors. Boundary values. Keywords: FEValues, VectorTools::interpolate_boundary_values(), MatrixTools::apply_boundary_values(), SolverCG, Vector, SparseMatrix, DataOut step-4 This example is programmed in a way that it is independent of the dimension for which we want to solve Laplace's equation; we will solve the equation in 2D and 3D, although the program is exactly the same. Non-constant right hand side function. Non-homogeneous boundary values. Keywords: VectorTools::point_value(), VectorTools::compute_mean_value() step-5 Computations on successively refined grids. Reading a grid from disk. Some optimizations. Using assertions. Non-constant coefficient in the elliptic operator (yielding the extended Poisson equation). Preconditioning the CG solver for the linear system of equations. Keywords: PreconditionSSOR, GridIn, SphericalManifold step-6 Adaptive local refinement. Handling of hanging nodes. Higher order elements. Catching exceptions in the main function. Keywords: DoFTools::make_hanging_node_constraints(), AffineConstraints::distribute_local_to_global(), KellyErrorEstimator, GridRefinement::refine_and_coarsen_fixed_number() step-7 Helmholtz equation. Non-homogeneous Neumann boundary conditions and boundary integrals. Verification of correctness of computed solutions. Computing the error between exact and numerical solution and output of the data in tables. Using counted pointers. Keywords: FEFaceValues, VectorTools::integrate_difference(), VectorTools::compute_global_error(), TableHandler step-8 The elasticity equations will be solved instead of Laplace's equation. The solution is vector-valued and the equations form a system with as many equations as the dimension of the space in which it is posed. Keywords: FESystem step-9 Linear advection equation, assembling the system of equations in parallel using multi-threading, implementing a refinement criterion based on a finite difference approximation of the gradient. Keywords: TensorFunction, WorkStream::run(), SolverGMRES step-10 Higher order mappings. Do not solve equations, but rather compute the value of pi to high accuracy. Keywords: MappingQ, FE_Nothing, ConvergenceTable, GridOut, FEFaceValues step-11 Solving a Laplace problem with higher order mappings. Using mean value constraints and intermediate representations of sparsity patterns. Keywords: AffineConstraints, DoFTools::extract_boundary_dofs(), TableHandler step-12 Discontinuous Galerkin methods for linear advection problems. Keywords: FEInterfaceValues, MeshWorker::mesh_loop(), DoFTools::make_flux_sparsity_pattern() step-13 Software design questions and how to write a modular, extensible finite element program. step-14 Duality based error estimators, more strategies to write a modular, extensible finite element program. Keywords: KellyErrorEstimator step-15 A nonlinear elliptic problem: The minimal surface equation. Newton's method. Transferring a solution across mesh refinement. Keywords: SolutionTransfer step-16 Multigrid preconditioning of the Laplace equation on adaptive meshes. Keywords: Multigrid, PreconditionMG, mg::Matrix, MGTransferPrebuilt, MeshWorker::mesh_loop(), MGLevelObject, MGConstrainedDoFs step-16b A variant of step-16 but with MeshWorker for assembly: Multigrid preconditioning of the Laplace equation on adaptive meshes. step-17 Using PETSc for linear algebra; running in parallel on clusters of computers linked together by MPI. Keywords: PETScWrappers::MPI::SparseMatrix, ConditionalOStream, PETScWrappers::PreconditionBlockJacobi step-18 A time dependent problem; using a much simplified version of implementing elasticity; moving meshes; handling large scale output of parallel programs. Simple implicit (backward Euler) time stepping. Keywords: parallel::shared::Triangulation, DataOutInterface::write_vtu_with_pvtu_record() step-19 Coupling particles to the solution of partial differential equations. Keywords: Particles step-20 Mixed finite elements. Using block matrices and block vectors to define more complicated solvers and preconditioners working on the Schur complement. Keywords: FEValuesExtractors, LinearOperator, TensorFunction, FE_RaviartThomas step-21 The time dependent two-phase flow in porous media. Extensions of mixed Laplace discretizations. More complicated block solvers. Simple explicit (forward Euler) time stepping. Keywords: TensorFunction, FE_RaviartThomas, VectorTools::project(), DiscreteTime step-22 Solving the Stokes equations of slow fluid flow on adaptive meshes. More on Schur complement solvers. Advanced use of the AffineConstraints class. Keywords: AffineConstraints, VectorTools::compute_no_normal_flux_constraints(), SparseILU, SparseDirectUMFPACK, BlockDynamicSparsityPattern step-23 Finally a "real" time dependent problem, the wave equation. Fractional time stepping (explicit, fully implicit and Crank-Nicholson method). Keywords: MatrixCreator, VectorTools::project() step-24 A variant of step-23 with absorbing boundary conditions, and extracting practically useful data. Implicit time stepping. Keywords: VectorTools::point_value() step-25 The sine-Gordon soliton equation, which is a nonlinear variant of the time dependent wave equation covered in step-23 and step-24. Fractional time stepping. Keywords: FunctionTime, VectorTools::integrate_difference() step-26 The heat equation, solved on a mesh that is adapted every few time steps. Fractional time stepping. Keywords: KellyErrorEstimator, SolutionTransfer, VectorTools::interpolate(), VectorTools::create_right_hand_side() step-27 The hp-finite element method. Keywords: hp::FECollection, hp::QCollection, hp::Refinement, FESeries::Fourier, Triangulation::create_triangulation() step-28 Multiple grids for solving a multigroup diffusion equation in nuclear physics simulating a nuclear reactor core. step-29 Solving a complex-valued Helmholtz equation. Sparse direct solvers. Dealing with parameter files. step-30 Anisotropic refinement for DG finite element methods. step-31 Time-dependent Stokes flow driven by temperature differences in a fluid. Adaptive meshes that change between time steps. Implicit/explicit time stepping. step-32 A massively parallel solver for time-dependent Stokes flow driven by temperature differences in a fluid. Adapting methods for real-world equations. Implicit/explicit time stepping. step-33 A nonlinear hyperbolic conservation law: The Euler equations of compressible gas dynamics. Fractional time stepping. step-34 Boundary element methods (BEM) of low order: Exterior irrotational flow. The ParsedFunction class. step-35 A projection solver for the Navier–Stokes equations. step-36 Using SLEPc for linear algebra; solving an eigenspectrum problem. The Schrödinger wave equation. step-37 Solving a Poisson problem with a multilevel preconditioner without explicitly storing the matrix (a matrix-free method) in a massively parallel context. step-38 Solving the Laplace-Beltrami equation on curved manifolds embedded in higher dimensional spaces. step-39 Solving Poisson's equation once more, this time with the interior penalty method, one of the discontinuous Galerkin methods developed for this problem. Error estimator, adaptive meshes, and multigrid preconditioner, all using the MeshWorker framework. step-40 Techniques for the massively parallel solution of the Laplace equation (up to 10,000s of processors). step-41 Solving the obstacle problem, a variational inequality. step-42 A solver for an elasto-plastic contact problem, running on parallel machines. step-43 Advanced techniques for the simulation of porous media flow. Explicit time stepping. step-44 Finite strain hyperelasticity based on a three-field formulation. Implicit time stepping. Keywords: CellDataStorage, FEValuesExtractors, WorkStream::run, BlockSparseMatrix, BlockVector, ComponentSelectFunction, Physics::Elasticity, FullMatrix::extract_submatrix_from(), FullMatrix::scatter_matrix_to(), LinearOperator, SolverSelector, PreconditionSelector, ReductionControl, MappingQEulerian step-45 Periodic boundary conditions. Keywords: GridTools::collect_periodic_faces(), GridTools::PeriodicFacePair, Triangulation::add_periodicity() step-46 Coupling different kinds of equations in different parts of the domain. step-47 Solving the fourth-order biharmonic equation using the $$C^0$$ Interior Penalty (C0IP) method. Keywords: FEInterfaceValues step-48 Explicit time stepping for the Sine–Gordon equation based on a diagonal mass matrix. Efficient implementation of (nonlinear) finite element operators. step-49 Advanced mesh creation and manipulation techniques. step-50 Geometric multigrid on adaptive meshes distributed in parallel. Keywords: Multigrid, MGLevelObject, MGConstrainedDoFs, IndexSet, MGTools, PreconditionMG, MatrixFree, FEInterfaceValues, MeshWorker::mesh_loop() step-51 Solving the convection-diffusion equation with a hybridizable discontinuous Galerkin method using face elements. step-52 Solving the time dependent neutron diffusion equation using Runge-Kutta methods. Explicit and implicit time stepping. step-53 Describing the geometry of complex domains and curved boundaries. step-54 Using CAD files to describe the boundary of your domain. Keywords: Manifold, OpenCASCADE::read_IGES(), OpenCASCADE::NormalProjectionBoundary step-55 Solving the Stokes problem in parallel. step-56 Geometric Multigrid for Stokes. step-57 Incompressible, stationary Navier Stokes equations. step-58 The nonlinear Schrödinger equation. step-59 Solving a Poisson problem discretized with an interior penalty DG method and a multilevel preconditioner in a matrix-free fashion using a massively parallel implementation. step-60 Distributed Lagrange multipliers for the solution of Poisson problems in complex domains with constraints defined on non-matching grids. step-61 Solving the Poisson problem with the "weak Galerkin" finite element method. step-62 Resonance frequency and bandgap of a phononic crystal. Elastic wave equation in the frequency domain with Perfectly Matched Layer boundary conditions. Parallelization via MUMPS and MPI. step-63 Block smoothers for geometric multigrid. A scalar convection diffusion equation is solved with different additive or multiplicative multigrid smoothers. Keywords: Multigrid, MeshWorker::mesh_loop(), MGSmootherPrecondition, RelaxationBlock, DoFRenumbering::downstream() step-64 Solving a Helmholtz problem using matrix-free methods on the GPU with MPI parallelization. step-65 The TransfiniteInterpolationManifold and MappingQCache classes for advanced manifold operations. step-66 A matrix-free geometric multigrid solver for a nonlinear problem. Keywords: MatrixFree, Multigrid, MGTransferMatrixFree::interpolate_to_mg(), MatrixFreeTools::compute_diagonal(), TransfiniteInterpolationManifold, MappingQGeneric step-67 Solving the Euler equations of compressible gas dynamics with an explicit time integrator and high-order discontinuous Galerkin methods based on matrix-free implementations. step-68 Simulation of the motion of massless tracer particles in a vortical flow. Parallel simulation of the advection of particles with load balancing. Keywords: Particles step-69 Hyperbolic conservation laws: a first-order guaranteed maximum wavespeed method for the compressible Euler equations. Explicit time stepping. step-70 A fluid structure interaction problem on fully distributed non-matching grids, using penalty methods, and a coupling constructed through a ParticleHandler object. Keywords: Particles step-71 Constitutive modelling: a demonstration of how automatic and symbolic differentiation can be used to rapidly implement a complex coupled constitutive law. Keywords: Automatic differentiation, Symbolic differentiation, Constitutive modelling step-72 A nonlinear elliptic problem: The minimal surface equation. Newton's method. Using automatic differentiation to linearize the residual, or to compute the full linear system from an energy functional. Keywords: Automatic differentiation, MeshWorker::mesh_loop() step-74 The Symmetric interior penalty Galerkin (SIPG) method for Poisson's equation. Keywords: MeshWorker::mesh_loop(), FEInterfaceValues, ConvergenceTable step-75 Solving the Laplace equation on hp-adaptive meshes with MatrixFree methods on thousands of processors. Keywords: parallel::distributed::Triangulation, hp::Refinement, MatrixFree step-76 Like step-67, but for demonstrating MPI-3.0 shared-memory features. Keywords: MatrixFree, MatrixFree::cell_loop() step-77 The nonlinear minimal surface equation revisited. Interfacing with SUNDIALS' KINSOL nonlinear solver. step-78 Solving the Black-Scholes equation for options pricing in 1d. step-79 A topology optimization program for elastic media using the solid isotropic material penalization (SIMP) formulation. step-82 Solving the fourth-order biharmonic equation using a lifting operator approach. step-85 Solving the Poisson equation using the cut finite element method.

Tutorial programs grouped by topics

Basic techniques

 Creating a grid. A simple way to write it to a file Degrees of freedom Solving the Laplace equation Dimension independent programming, non-zero data Computing on uniformly refined meshes Adaptivity Evaluating errors Nonlinear problems, Newton's method

 Multithreading Block solvers and preconditioners Using Trilinos Parallelization via PETSc and MPI Parallelization via Trilinos and MPI Parallelization via MUMPS and MPI Parallelization via CUDA and MPI Parallelization on very large numbers of processors Input parameter handling Higher order mappings Error indicators and estimators Transferring solutions across mesh refinement Discontinuous Galerkin methods hp-finite elements Anisotropic refinement for DG finite element methods Computing Jacobians from residuals, automatic and symbolic differentiation Operator splitting Boundary element methods, curved manifolds Periodic boundary conditions Matrix-free methods with sum factorization Advanced meshes and geometries Non matching algorithms HDF5 and Python Particles

Linear solvers

 Conjugate Gradient solver Preconditioned CG solver BiCGStab Direct solvers Multilevel preconditioners Parallel solvers Block and Schur complement solvers Decoupled projection solvers Linear Newton systems from nonlinear equations Eigenvalue solvers Linear operators

Other equations

 Helmholtz equation Elasticity and elasto-plasticity equations Heat equation Minimal surface equation Quasi-static elasticity equations Transport (advection) equations The nonlinear hyperbolic Euler system of compressible gas dynamics Mixed Laplace, Darcy, Porous media Stokes and incompressible Navier-Stokes flow The wave equation, in linear and nonlinear variants A multigroup diffusion problem in neutron transport Irrotational flow An eigenspectrum problem Fourth-order biharmonic equation The obstacle problem, a variational inequality The nonlinear Schrödinger equation Coupling different equations in different parts of the domain

Vector problems

 Elasticity and elasto-plasticity equations Mixed Laplace Mixed Laplace plus an advection equation Incompressible Stokes and Navier-Stokes flow A complex-valued Helmholtz problem The Euler equations of compressible gas dynamics Coupling different equations in different parts of the domain

Time dependent problems

 The heat equation Quasi-static elasticity Porous media flow The wave equation, in linear and nonlinear variants Time dependent Stokes flow driven by buoyancy The Euler equations of compressible gas dynamics The nonlinear Schrödinger equation Time dependent neutron diffusion equation Time dependent fluid structure interaction problems step-70