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Reference documentation for deal.II version 9.6.0
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Reference documentation for deal.II version 9.6.0
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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:
You can browse the available tutorial programs
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.
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:
| step-1 | Creating a grid. A simple way to write it to a file. |
| 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. |
| step-3 | Actually solve Laplace's problem. Object-orientation. Assembling matrices and vectors. Boundary values. |
| 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. |
| 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. |
| step-6 | Adaptive local refinement. Handling of hanging nodes. Higher order elements. Catching exceptions in the |
| 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. |
| 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. |
| 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. |
| step-10 | Higher order mappings. Do not solve equations, but rather compute the value of pi to high accuracy. |
| step-11 | Solving a Laplace problem with higher order mappings. Using mean value constraints and intermediate representations of sparsity patterns. |
| step-12 | Discontinuous Galerkin methods for linear advection problems. |
| 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. |
| step-15 | A nonlinear elliptic problem: The minimal surface equation. Newton's method. Transferring a solution across mesh refinement. |
| step-16 | 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. |
| 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. |
| step-19 | Coupling particles to the solution of partial differential equations. |
| step-20 | Mixed finite elements. Using block matrices and block vectors to define more complicated solvers and preconditioners working on the Schur complement. |
| 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. |
| step-22 | Solving the Stokes equations of slow fluid flow on adaptive meshes. More on Schur complement solvers. Advanced use of the AffineConstraints class. |
| step-23 | Finally a "real" time dependent problem, the wave equation. Fractional time stepping (explicit, fully implicit and Crank-Nicolson method). |
| step-24 | A variant of step-23 with absorbing boundary conditions, and extracting practically useful data. Implicit time stepping. |
| 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. |
| step-26 | The heat equation, solved on a mesh that is adapted every few time steps. Fractional time stepping. |
| step-27 | The hp-finite element method. |
| 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. |
| step-45 | Periodic boundary conditions. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| step-71 | Constitutive modelling: a demonstration of how automatic and symbolic differentiation can be used to rapidly implement a complex coupled constitutive law. |
| 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. |
| step-74 | The Symmetric interior penalty Galerkin (SIPG) method for Poisson's equation. |
| step-75 | Solving the Laplace equation on hp-adaptive meshes with MatrixFree methods on thousands of processors. |
| step-76 | Like step-67, but for demonstrating MPI-3.0 shared-memory features. |
| 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-81 | Solving the complex-valued time-harmonic Maxwell equations. |
| step-82 | Solving the fourth-order biharmonic equation using a lifting operator approach. |
| step-83 | Serialization, checkpoint/restart. |
| step-85 | Solving the Poisson equation using the cut finite element method. |
| step-86 | The heat equation, solved with an external time stepping library |
| step-87 | Evaluation of finite element solutions at arbitrary points within a distributed mesh with application to two-phase flow. |
| step-89 | Matrix-free operator evaluation for non-matching and Chimera methods with application to acoustic conservation equations. |
| step-90 | Solving the Laplace-Beltrami equation on a surface using the trace finite element method. Keywords: MeshWorker::mesh_loop(), NonMatching::FEImmersedSurfaceValues |
| 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 | step-9, step-28, step-32, step-44, step-48, step-51, step-69 |
| Block solvers and preconditioners | step-20, step-21, step-22, step-31, step-32, step-43, step-44, step-55, step-56, step-57, step-60, step-70 |
| Using Trilinos | step-31, step-32, step-33, step-40, step-41, step-42, step-43, step-50, step-55, step-71, step-72, step-75 |
| 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 | step-32, step-37, step-40, step-42, step-50, step-55, step-59, step-66, step-67, step-69, step-70, step-75, step-86 |
| Input parameter handling | step-28, step-29, step-32, step-33, step-34, step-35, step-36, step-42, step-44, step-60, step-62, step-69, step-70, step-71, step-72, step-86 |
| Higher order mappings | step-10, step-11, step-32, step-60, step-65, step-66, step-67 |
| Error indicators and estimators | |
| Transferring solutions across mesh refinement | step-15, step-28, step-31, step-32, step-33, step-42, step-43, step-57, step-70 |
| Discontinuous Galerkin methods | step-12, step-21, step-39, step-46, step-47, step-51, step-59, step-61, step-67, step-74, step-82 |
| 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 | step-37, step-48, step-50, step-59, step-64, step-66, step-67, step-75, step-76, step-89 |
| Advanced meshes and geometries | |
| Non-matching algorithms | |
| HDF5 and Python | |
| Particles |
| Conjugate Gradient solver | |
| Preconditioned CG solver | |
| BiCGStab | |
| Direct solvers | step-29, step-44, step-47, step-46, step-58, step-62, step-81, step-82 |
| Multilevel preconditioners | step-16, step-31, step-32, step-37, step-39, step-41, step-42, step-43, step-50, step-56, step-59, step-63, step-66, step-75 |
| Parallel solvers | step-17, step-18, step-32, step-37, step-40, step-42, step-50, step-55, step-59, step-66, step-75, step-86 |
| Block and Schur complement solvers | step-20, step-21, step-22, step-31, step-32, step-43, step-55, step-56, step-57, step-60, step-70 |
| Decoupled projection solvers | |
| Linear Newton systems from nonlinear equations | |
| Eigenvalue solvers | |
| Linear operators |
| 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 | step-22, step-31, step-32, step-35, step-46, step-55, step-56, step-57, step-70 |
| 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 | |
| The time-harmonic Maxwell equations | |
| Coupling different equations in different parts of the domain |
| Elasticity and elasto-plasticity equations | |
| Mixed Laplace | |
| Mixed Laplace plus an advection equation | |
| Incompressible Stokes and Navier-Stokes flow | step-22, step-31, step-32, step-35, step-55, step-56, step-57, step-70 |
| A complex-valued Helmholtz problem | |
| The Euler equations of compressible gas dynamics | |
| Coupling different equations in different parts of the domain |
| 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 |
1.11.0