General Initial Condition (03-newton-2)

Git reference: Tutorial example 03-newton-2.

Model problem

We will still stay with the nonlinear model problem from the previous sections,

-\nabla \cdot (\lambda(u)\nabla u) - f(x,y) = 0 \ \ \ \mbox{in } \Omega = (-10,10)^2

but now we will change the boundary conditions to nonhomogeneous Dirichlet,

u(x, y) = (x+10)(y+10)/100 \ \ \ \mbox{on } \partial \Omega

and with a general initial guess init_guess(x,y).

Defining nonhomogeneous Dirichlet boundary conditions

// This function is used to define Dirichlet boundary conditions.
double dir_lift(double x, double y, double& dx, double& dy) {
  dx = (y+10)/10.;
  dy = (x+10)/10.;
  return (x+10)*(y+10)/100.;
}

// Boundary condition types.
BCType bc_types(int marker)
{
  return BC_ESSENTIAL;
}

// Essential (Dirichlet) boundary condition values.
scalar essential_bc_values(int ess_bdy_marker, double x, double y)
{
  double dx, dy;
  return dir_lift(x, y, dx, dy);
}

Setting a nonconstant initial condition for the Newton’s method

The initial condition has the form:

// Initial condition. It will be projected on the FE mesh
// to obtain initial coefficient vector for the Newton's method.
scalar init_cond(double x, double y, double& dx, double& dy)
{
  // Using the Dirichlet lift elevated by two
  double val = dir_lift(x, y, dx, dy) + 2;
  return val;
}

As in the previous example, the initial condition is projected on the finite element space to obtain an initial coefficient vector \bfY_0 for the Newton’s iteration:

// Project the initial condition on the FE space to obtain initial
// coefficient vector for the Newton's method.
info("Projecting to obtain initial vector for the Newton's method.");
scalar* coeff_vec = new scalar[Space::get_num_dofs(&space)] ;
Solution* init_sln = new Solution(&mesh, init_cond);
OGProjection::project_global(&space, init_sln, coeff_vec, matrix_solver);
delete init_sln;

Sample results

The following figure shows the H^1-projection of the initial condition init_cond():

The converged solution looks as follows: