Adapting Mesh to an Exact Function (06-exact-adapt)

Git reference: Tutorial example 06-exact-adapt.

This technique can be useful, for example, when a time-dependent proces starts from a complicated initial condition that would not be represented with sufficient accuracy on a coarse initial mesh.

As usual, the adaptivity algorithm expects a pair of solutions on the coarse and globally refined meshes. So the adaptivity loop begins with refining the coarse mesh:

// Construct globally refined reference mesh and setup reference space.
Space* ref_space = construct_refined_space(&space);

Instead of calculating a solution on the fine mesh, we set the exact function:

// Assign the function f() to the fine mesh.
info("Assigning f() to the reference mesh.");
bool is_linear = true;
ref_sln.set_exact(ref_space->get_mesh(), f);

The coarse mesh solution is obtained by projecting ‘sln_fine’:

// Project the fine mesh solution onto the coarse mesh.
info("Projecting reference solution on coarse mesh.");
OGProjection::project_global(&space, &ref_sln, &sln, matrix_solver);

Error estimates are calculated as usual:

// Calculate element errors and total error estimate.
info("Calculating exact error.");
Adapt* adaptivity = new Adapt(&space, HERMES_H1_NORM);
// Note: the error estimate is now equal to the exact error.
bool solutions_for_adapt = true;
double err_exact_rel = adaptivity->calc_err_est(&sln, &ref_sln, solutions_for_adapt,
                       HERMES_TOTAL_ERROR_REL | HERMES_ELEMENT_ERROR_REL) * 100;

Mesh adaptation is standard as well:

// If err_exact_rel too large, adapt the mesh.
if (err_exact_rel < ERR_STOP) done = true;
else
{
  info("Adapting coarse mesh.");
  done = adaptivity->adapt(&selector, THRESHOLD, STRATEGY, MESH_REGULARITY);

  // Increase the counter of performed adaptivity steps.
  if (done == false)  as++;
}
if (Space::get_num_dofs(&space) >= NDOF_STOP) done = true;

Sample solution and mesh are shown below: