L-Shape (Elliptic)¶

Git reference: Benchmark lshape.

This is a standard adaptivity benchmark whose exact solution is smooth but contains singular gradient in a re-entrant corner.

Model problem¶

Equation solved: Laplace equation

(1)-\Delta u = 0.

Domain of interest:

Exact solution¶

(2)u(x, y) = r^{2/3}\sin(2a/3 + \pi/3)

where r(x,y) = \sqrt{x^2 + y^2} and a(x,y) = \mbox{atan}(x/y).

In the code:

// Exact solution.
static double fn(double x, double y)
{
  double r = sqrt(x*x + y*y);
  double a = atan2(x, y);
  return pow(r, 2.0/3.0) * sin(2.0*a/3.0 + M_PI/3);
}

static double fndd(double x, double y, double& dx, double& dy)
{
  double t1 = 2.0/3.0*atan2(x, y) + M_PI/3;
  double t2 = pow(x*x + y*y, 1.0/3.0);
  double t3 = x*x * ((y*y)/(x*x) + 1);
  dx = 2.0/3.0*x*sin(t1)/(t2*t2) + 2.0/3.0*y*t2*cos(t1)/t3;
  dy = 2.0/3.0*y*sin(t1)/(t2*t2) - 2.0/3.0*x*t2*cos(t1)/t3;
  return fn(x, y);
}

Boundary conditions¶

These are nonconstant Dirichlet, so we first define a callback:

// Essential (Dirichlet) boundary condition values.
scalar essential_bc_values(double x, double y)
{
  return fn(x, y);
}

The callback is registered in BCValues:

// Enter boundary markers.
BCTypes bc_types;
bc_types.add_bc_dirichlet(BDY_DIRICHLET);

// Enter Dirichlet boudnary values.
BCValues bc_values;
bc_values.add_function(BDY_DIRICHLET, essential_bc_values);

Weak forms¶

// Bilinear form corresponding to the Laplace equation.
template<typename Real, typename Scalar>
Scalar bilinear_form(int n, double *wt, Func<Scalar> *u_ext[], Func<Real> *u,
                     Func<Real> *v, Geom<Real> *e, ExtData<Scalar> *ext)
{
  return int_grad_u_grad_v<Real, Scalar>(n, wt, u, v);
}