FEM Library

deal.II

C++ finite element library supporting adaptive meshes, high-order elements, multigrid solvers, and massively parallel computations. One of the most comprehensive open-source FEM libraries with extensive hp-adaptivity support.

C++

hp-Adaptive FEM

PETSc / Trilinos

Open Source (LGPL)

hp-Adaptive Error Estimation

deal.II uses a posteriori error estimators to drive adaptive refinement. The Kelly error estimator computes:

Where η_K is the error indicator on cell K, h_K is the cell diameter, and ⟦·⟧ denotes the jump of the normal gradient across cell faces.

Library Architecture

Triangulation

Mesh management

DoFHandler

Degree-of-freedom distribution

FiniteElement

Shape functions

LinearAlgebra

PETSc / Trilinos

Numerics

Assembly & postprocess

Key Features

Adaptive Mesh Refinement

Full support for h-refinement (mesh subdivision), p-refinement (polynomial degree elevation), and hp-adaptivity. Error estimators drive automatic refinement for optimal convergence rates.

Rich Element Library

Continuous and discontinuous Lagrange, Raviart-Thomas, Nédélec, BDM elements of arbitrary order. Supports triangles, quadrilaterals, tetrahedra, hexahedra, prisms, and pyramids.

Massively Parallel

Scales to 100,000+ MPI processes. Distributed meshes via p4est. Parallel linear algebra through PETSc/Trilinos. Demonstrated scaling on leadership-class supercomputers.

Multigrid Solvers

Geometric and algebraic multigrid (AMG) preconditioners. Matrix-free operator evaluation for memory-efficient high-order discretisations.

Example: Laplace Problem

step-3.cc

// Laplace equation with deal.II (simplified)
#include <deal.II/grid/tria.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/lac/sparse_matrix.h>
#include <deal.II/lac/vector.h>
#include <deal.II/numerics/matrix_tools.h>
#include <deal.II/numerics/vector_tools.h>

using namespace dealii;

template <int dim>
class LaplaceProblem {
  Triangulation<dim> triangulation;
  FE_Q<dim>          fe;
  DoFHandler<dim>    dof_handler;
  SparsityPattern    sparsity_pattern;
  SparseMatrix<double> system_matrix;
  Vector<double>     solution;
  Vector<double>     system_rhs;

public:
  LaplaceProblem() : fe(2), dof_handler(triangulation) {}

  void run() {
    GridGenerator::hyper_cube(triangulation, -1, 1);
    triangulation.refine_global(5);
    dof_handler.distribute_dofs(fe);
    // ... assemble system and solve
  }
};

Tutorial Programme

deal.II features 80+ well-documented tutorial steps, progressing from basics to advanced topics:

Step 1–6

Basics: Laplace, grids, refinement

Step 7–12

Elasticity, mixed FEM, multigrid

Step 20–30

Navier-Stokes, nonlinear, time-dep.

Step 40+

Parallel, matrix-free, hp-methods

Resources: Visit dealii.org for documentation, tutorials, and the video lecture series. VELSTROM-specific deal.II workflows are under development.