Tutorial Part V (Miscellaneous Techniques)

This section is a collection of various examples and techniques that are worth showing even though they do not fit exactly into the previous sections.

Space H(curl) (30)

Git reference: Tutorial example 30-space-hcurl.

In example 02-space we first saw how a finite element space over a mesh is created. That was an space suitable for continuous approximations. Another widely used Sobolev space, H(curl), is typically present in Maxwell’s problems of electromagnetics. H(curl) approximations are discontinuous, elementwise polynomial vector fields that behave like gradients of functions. (Recall that in electrostatics .) In particular, H(curl) functions have continuous tangential components along all mesh edges. For the application of the H(curl) space check examples related to Maxwell’s equations in the previous sections. Below is a simple code that shows how to set up an H(curl) space and visualize its finite element basis functions:

int INIT_REF_NUM = 2;      // Initial uniform mesh refinement.
int P_INIT = 3;            // Polynomial degree of mesh elements.

int main(int argc, char* argv[])
{
  // Load the mesh.
  Mesh mesh;
  H2DReader mloader;
  mloader.load("square.mesh", &mesh);

  // Initial mesh refinement.
  for (int i = 0; i < INIT_REF_NUM; i++) mesh.refine_all_elements();

  // Create an Hcurl space with default shapeset.
  // (BC types and essential BC values not relevant.)
  HcurlSpace space(&mesh, NULL, NULL, P_INIT);

  // Visualize FE basis.
  VectorBaseView bview("BaseView", 0, 0, 700, 600);
  bview.show(&space);

  // Wait for all views to be closed.
  View::wait();
  return 0;
}

The class VectorBaseView allows the user to browse through the finite element basis functions using the left and right arrows. A few sample basis functions (higher-order bubble functions) are shown below. The color shows magnitude of the vector field, arrows show its direction.

The space H(curl) is implemented for both quadrilateral and triangular elements, and both elements types can be combined in one mesh.

Space H(div) (31)

Git reference: Tutorial example 31-space-hdiv.

The space H(div) in 2D is very similar in nature to the space H(curl), except its functions behave like (vector-valued) divergences of functions. Finite element basis functions in the space H(div) are discontinuous across element interfaces but their normal components are continuous. The following code shows how to set up an H(div) space and visualize its basis functions:

int INIT_REF_NUM = 2;      // Initial uniform mesh refinement.
int P_INIT = 3;            // Polynomial degree of mesh elements.

int main(int argc, char* argv[])
{
  // Load the mesh.
  Mesh mesh;
  H2DReader mloader;
  mloader.load("square.mesh", &mesh);

  // Initial mesh refinement.
  for (int i = 0; i < INIT_REF_NUM; i++) mesh.refine_all_elements();

  // Create an Hdiv space with default shapeset.
  // (BC types and essential BC values not relevant.)
  HdivSpace space(&mesh, NULL, NULL, P_INIT);

  // Visualise the FE basis.
  VectorBaseView bview("BaseView", 0, 0, 700, 600);
  bview.show(&space);

  // Wait for all views to be closed.
  View::wait();
  return 0;
}

Sample edge functions of polynomial degrees 1, 2, 3, and 4 corresponding to a boundary edge are shown below:

So far the space H(div) only can be used with quadrilateral elements.

Space L2 (32)

Git reference: Tutorial example 32-space-l2.

We already saw the space in the Navier-Stokes example where it was used for pressure to keep the velocity discreetely divergence-free. This example shows how to create an space, visualize finite element basis functions, and perform an orthogonal -projection of a continuous function onto the FE space:

const int INIT_REF_NUM = 1;    // Number of initial uniform mesh refinements.
const int P_INIT = 3;          // Polynomial degree of mesh elements.

// Projected function.
scalar F(double x, double y, double& dx, double& dy)
{
  return - pow(x, 4) * pow(y, 5);
  dx = 0; // not needed for L2-projection
  dy = 0; // not needed for L2-projection
}

int main(int argc, char* argv[])
{
  // Load the mesh.
  Mesh mesh;
  H2DReader mloader;
  mloader.load("square.mesh", &mesh);

  // Perform uniform mesh refinements.
  for (int i=0; i<INIT_REF_NUM; i++) mesh.refine_all_elements();

  // Create an L2 space with default shapeset.
  L2Space space(&mesh, P_INIT);

  // View basis functions.
  BaseView bview("BaseView", 0, 0, 600, 500);
  bview.show(&space);
  View::wait(H2DV_WAIT_KEYPRESS);

  // Assemble and solve the finite element problem.
  WeakForm wf_dummy;
  LinSystem ls(&wf_dummy, &space);
  Solution sln;
  int proj_norm_l2 = 0;
  ls.project_global(F, &sln, proj_norm_l2);

  // Visualize the solution.
  ScalarView view1("Projection", 610, 0, 600, 500);
  view1.show(&sln);

  // Wait for all views to be closed.
  View::wait();
  return 0;
}

Sample basis functions:

The projection (note that this is a discontinuous function):

Adapting Mesh to an Exact Function (33)

Git reference: Tutorial example 33-exact-adapt.

Description coming soon.

Remote Computing (34)

Git reference: Tutorial example 34-remote-computing.

Description coming soon.