NIST-04 (Peak)¶

Git reference: Benchmark nist-04.

This problem has an exponential peak in the interior of the domain.

Model problem¶

Equation solved: Poisson equation

(1)-\Delta u = f.

Domain of interest: Unit Square (0, 1)^2.

Boundary conditions: Dirichlet, given by exact solution.

Exact solution¶

u(x,y) = e^{-\alpha ((x - x_{loc})^{2} + (y - y_{loc})^{2})}

where (x_{loc}, y_{loc}) is the location of the peak, and \alpha determines the strength of the peak.

Right-hand side: Obtained by inserting the exact solution into the latter equation. The corresponding code snippet is shown below:

{
public:
  CustomRightHandSide(double alpha, double x_loc, double y_loc)
    : DefaultNonConstRightHandSide(), alpha(alpha), x_loc(x_loc), y_loc(y_loc) {};

  virtual double value(double x, double y) const {
    double a_P = (-alpha * pow((x - x_loc), 2) - alpha * pow((y - y_loc), 2));

    return -(4 * exp(a_P) * alpha * (alpha * (x - x_loc) * (x - x_loc)
                                  + alpha * (y - y_loc) * (y - y_loc) - 1));
}