Biharmonic equation

This demo illustrates how to:

  • Solve a linear partial differential equation

  • Use a discontinuous Galerkin method

  • Solve a fourth-order differential equation

Equation and problem definition

Strong formulation

The biharmonic equation is a fourth-order elliptic equation. On the domain \(\Omega \subset \mathbb{R}^{d}\), \(1 \le d \le 3\), it reads

\[ \nabla^{4} u = f \quad {\rm in} \ \Omega, \]

where \(\nabla^{4} \equiv \nabla^{2} \nabla^{2}\) is the biharmonic operator and \(f\) is a prescribed source term. To formulate a complete boundary value problem, the biharmonic equation must be complemented by suitable boundary conditions.

Weak formulation

Multiplying the biharmonic equation by a test function and integrating by parts twice leads to a problem of second-order derivatives, which would require \(H^{2}\) conforming (roughly \(C^{1}\) continuous) basis functions. To solve the biharmonic equation using Lagrange finite element basis functions, the biharmonic equation can be split into two second-order equations (see the Mixed Poisson demo for a mixed method for the Poisson equation), or a variational formulation can be constructed that imposes weak continuity of normal derivatives between finite element cells. This demo uses a discontinuous Galerkin approach to impose continuity of the normal derivative weakly.

Consider a triangulation \(\mathcal{T}\) of the domain \(\Omega\), where the set of interior facets is denoted by \(\mathcal{E}_h^{\rm int}\). Functions evaluated on opposite sides of a facet are indicated by the subscripts \(+\) and \(-\). Using the standard continuous Lagrange finite element space

\[ V = \left\{v \in H^{1}_{0}(\Omega)\,:\, v \in P_{k}(K) \ \forall \ K \in \mathcal{T} \right\} \]

and considering the boundary conditions

(1)\[\begin{align} u &= 0 \quad {\rm on} \ \partial\Omega, \\ \nabla^{2} u &= 0 \quad {\rm on} \ \partial\Omega, \end{align}\]

a weak formulation of the biharmonic problem reads: find \(u \in V\) such that

\[ a(u,v)=L(v) \quad \forall \ v \in V, \]

where the bilinear form is

\[\begin{align*} a(u, v) &= \sum_{K \in \mathcal{T}} \int_{K} \nabla^{2} u \nabla^{2} v \, {\rm d}x \\ &\qquad+\sum_{E \in \mathcal{E}_h^{\rm int}}\left(\int_{E} \frac{\alpha}{h_E} [\!\![ \nabla u ]\!\!] [\!\![ \nabla v ]\!\!] \, {\rm d}s - \int_{E} \left<\nabla^{2} u \right>[\!\![ \nabla v ]\!\!] \, {\rm d}s - \int_{E} [\!\![ \nabla u ]\!\!] \left<\nabla^{2} v \right> \, {\rm d}s\right) \end{align*}\]

and the linear form is

\[ L(v) = \int_{\Omega} fv \, {\rm d}x. \]

Furthermore, \(\left< u \right> = \frac{1}{2} (u_{+} + u_{-})\), \([\!\![ w ]\!\!] = w_{+} \cdot n_{+} + w_{-} \cdot n_{-}\), \(\alpha \ge 0\) is a penalty parameter and \(h_E\) is a measure of the cell size.

The input parameters for this demo are defined as follows:

  • \(\Omega = [0,1] \times [0,1]\) (a unit square)

  • \(\alpha = 8.0\) (penalty parameter)

  • \(f = 4.0 \pi^4\sin(\pi x)\sin(\pi y)\) (source term)


The implementation is in two files: a form file containing the definition of the variational forms expressed in UFL and a C++ file containing the actual solver.

Running this demo requires the files: demo_biharmonic/main.cpp, demo_biharmonic/ and demo_biharmonic/CMakeLists.txt.

UFL form file

The UFL file is implemented in demo_biharmonic/


TODO: explanation on how to run cmake and/or shell commands for ffcx. To compile using FFCx with an option for PETSc scalar type float64 one would execute the command

ffcx --scalar_type=float64

C++ program

The main solver is implemented in the demo_biharmonic/main.cpp file.

At the top we include the DOLFINx header file and the generated header file “biharmonic.h” containing the variational forms for the Biharmonic equation, which are defined in the UFL form file. For convenience we also include the DOLFINx namespace.

#include "biharmonic.h"
#include <basix/finite-element.h>
#include <cmath>
#include <dolfinx.h>
#include <dolfinx/common/types.h>
#include <dolfinx/fem/Constant.h>
#include <dolfinx/fem/petsc.h>
#include <numbers>
#include <utility>
#include <vector>
using namespace dolfinx;
using T = PetscScalar;
using U = typename dolfinx::scalar_value_type_t<T>;

Inside the main function, we begin by defining a mesh of the domain. As the unit square is a very standard domain, we can use a built-in mesh provided by the UnitSquareMesh factory. In order to create a mesh consisting of 32 x 32 squares with each square divided into two triangles, and the finite element space (specified in the form file) defined relative to this mesh, we do as follows

int main(int argc, char* argv[])
  dolfinx::init_logging(argc, argv);
  PetscInitialize(&argc, &argv, nullptr, nullptr);
    //  Create mesh
    auto part = mesh::create_cell_partitioner(mesh::GhostMode::shared_facet);
    auto mesh = std::make_shared<mesh::Mesh<U>>(
        mesh::create_rectangle<U>(MPI_COMM_WORLD, {{{0.0, 0.0}, {1.0, 1.0}}},
                                  {32, 32}, mesh::CellType::triangle, part));

    //    A function space object, which is defined in the generated code,
    //    is created:

    auto element = basix::create_element<U>(
        basix::element::family::P, basix::cell::type::triangle, 2,
        basix::element::dpc_variant::unset, false);

    //  Create function space
    auto V = std::make_shared<fem::FunctionSpace<U>>(
        fem::create_functionspace(mesh, element));

    // The source function $f$ and the penalty term $\alpha$ are
    // declared:
    auto f = std::make_shared<fem::Function<T>>(V);
        [](auto x) -> std::pair<std::vector<T>, std::vector<std::size_t>>
          std::vector<T> f;
          for (std::size_t p = 0; p < x.extent(1); ++p)
            auto pi = std::numbers::pi;
            f.push_back(4.0 * std::pow(pi, 4) * std::sin(pi * x(0, p))
                        * std::sin(pi * x(1, p)));
          return {f, {f.size()}};
    auto alpha = std::make_shared<fem::Constant<T>>(8.0);

    //  Define variational forms
    auto a = std::make_shared<fem::Form<T>>(fem::create_form<T>(
        *form_biharmonic_a, {V, V}, {}, {{"alpha", alpha}}, {}));
    auto L = std::make_shared<fem::Form<T>>(
        fem::create_form<T>(*form_biharmonic_L, {V}, {{"f", f}}, {}, {}));

    //  Now, the Dirichlet boundary condition ($u = 0$) can be
    //  created using the class {cpp:class}`DirichletBC`. A
    //  {cpp:class}`DirichletBC` takes two arguments: the value of the
    //  boundary condition, and the part of the boundary on which the
    //  condition applies. In our example, the value of the boundary
    //  condition (0.0) can represented using a {cpp:class}`Function`,
    //  and the Dirichlet boundary is defined by the indices of degrees
    //  of freedom to which the boundary condition applies. The
    //  definition of the Dirichlet boundary condition then looks as
    //  follows:

    //  Define boundary condition
    auto facets = mesh::locate_entities_boundary(
        *mesh, 1,
        [](auto x)
          constexpr U eps = 1.0e-6;
          std::vector<std::int8_t> marker(x.extent(1), false);
          for (std::size_t p = 0; p < x.extent(1); ++p)
            U x0 = x(0, p);
            U x1 = x(1, p);
            if (std::abs(x0) < eps or std::abs(x0 - 1) < eps)
              marker[p] = true;
            if (std::abs(x1) < eps or std::abs(x1 - 1) < eps)
              marker[p] = true;
          return marker;
    const auto bdofs = fem::locate_dofs_topological(
        *V->mesh()->topology_mutable(), *V->dofmap(), 1, facets);
    auto bc = std::make_shared<const fem::DirichletBC<T>>(0.0, bdofs, V);

    //  Now, we have specified the variational forms and can consider
    //  the solution of the variational problem. First, we need to
    //  define a {cpp:class}`Function` `u` to store the solution. (Upon
    //  initialization, it is simply set to the zero function.) Next, we
    //  can call the `solve` function with the arguments `a == L`, `u`
    //  and `bc` as follows:

    //  Compute solution
    fem::Function<T> u(V);
    auto A = la::petsc::Matrix(fem::petsc::create_matrix(*a), false);
    la::Vector<T> b(L->function_spaces()[0]->dofmap()->index_map,

    fem::assemble_matrix(la::petsc::Matrix::set_block_fn(A.mat(), ADD_VALUES),
                         *a, {bc});
    MatAssemblyBegin(A.mat(), MAT_FLUSH_ASSEMBLY);
    MatAssemblyEnd(A.mat(), MAT_FLUSH_ASSEMBLY);
    fem::set_diagonal<T>(la::petsc::Matrix::set_fn(A.mat(), INSERT_VALUES), *V,
    MatAssemblyBegin(A.mat(), MAT_FINAL_ASSEMBLY);
    MatAssemblyEnd(A.mat(), MAT_FINAL_ASSEMBLY);

    fem::assemble_vector(b.mutable_array(), *L);
    fem::apply_lifting<T, U>(b.mutable_array(), {a}, {{bc}}, {}, T(1.0));
    fem::set_bc<T, U>(b.mutable_array(), {bc});

    la::petsc::KrylovSolver lu(MPI_COMM_WORLD);
    la::petsc::options::set("ksp_type", "preonly");
    la::petsc::options::set("pc_type", "lu");

    la::petsc::Vector _u(la::petsc::create_vector_wrap(*u.x()), false);
    la::petsc::Vector _b(la::petsc::create_vector_wrap(b), false);
    lu.solve(_u.vec(), _b.vec());

    //  Update ghost values before output

    // The function `u` will be modified during the call to solve. A
    // {cpp:class}`Function` can be saved to a file. Here, we output the
    // solution to a `VTK` file (specified using the suffix `.pvd`) for
    // visualisation in an external program such as Paraview.

    //  Save solution in VTK format
    io::VTKFile file(MPI_COMM_WORLD, "u.pvd", "w");
    file.write<T>({u}, 0.0);

  return 0;